logic circute by ali alhejaili



‫א‬ ‫א‬١٦٧

‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



   W،،‫א‬‫א‬،‫א‬

  

  ‫א‬ ‫א‬ ‫א‬ ‫א‬‫א‬  ‫א‬ ‫א‬  ‫א‬ ‫א‬ 



 

‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K ‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬



  ‫א‬        ،‫א‬ ‫א‬‫א‬  

‫א‬  ‫א‬ ‫א‬ ‫א‬ ‫א‬ ‫א‬   ‫א‬    ،  

‫א‬،‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬،‫א‬

 K‫א‬‫א‬،‫א‬

???‫א‬?‫א‬‫א‬



‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬



 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

،‫א‬،‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬

   W    ‫א‬ ‫א‬  ‫א‬    ‫א‬



 K‫א‬

‫א‬‫א‬‫א‬‫א‬

 



 ‫ א‬١٦٧

‫א‬

 ‫א‬



 

 

 ،KKK‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬،‫א‬‫א‬،‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

،‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬

،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬،‫א‬‫א‬‫א‬

‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬  K‫א‬‫א‬،‫א‬

،‫א‬‫א‬‫א‬‫א‬،

‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬  K‫א‬

 KKKKKK،‫א‬  



‫א‬‫א‬‫א‬

 ‫א‬‫א‬

‫א‬‫א‬



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



 ‫א‬‫א‬‫א‬  

 W‫א‬‫א‬‫א‬ K‫א‬‫א‬‫א‬ •

K‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫• א‬ K‫א‬‫א‬‫א‬‫א‬‫• א‬

K‫א‬‫א‬‫א‬‫א‬‫א‬ •

-١-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



 Introduction١ J١

K  ‫א‬  ‫א‬

‫א‬ (Binary Number System)‫א‬‫א‬‫א‬‫א‬‫א‬ K(Digital Electronic Circuits)   ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ Decimal )‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬K‫א‬(Number System Octal )‫א‬‫א‬‫א‬‫א‬‫א‬

K(Hexadecimal Numbering System)  ‫א‬‫א‬‫א‬(Number System ‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬

‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬‫א‬  

 K‫א‬K١

 K‫א‬‫א‬‫א‬K٢

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K٣  K‫א‬‫א‬‫א‬‫א‬K٤

 K‫א‬‫א‬‫א‬‫א‬‫א‬K٥  

‫א‬(Digit)‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬(Symbol)،(Number)

‫א‬‫א‬ ‫א‬ ‫א‬(0,1,2,3,4, ... , 8,9)،‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬ ،‫א‬(123)‫א‬‫א‬‫א‬(14)‫א‬‫א‬‫א‬،‫א‬

‫א‬‫א‬‫א‬‫א‬(4‫و‬1)(14)‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬(6)(3‫و‬2‫و‬1)(123)  K‫א‬

-٢-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



  Decimal Numbering System‫א‬‫א‬‫א‬٢ J١ ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ (10)‫א‬‫א‬‫א‬‫א‬K  ‫א‬‫א‬‫א‬

   (10)(10)‫א‬   K9  8  7  6  5  4  3  2  1  0

‫א‬(128)‫א‬‫א‬(Positional Weight)‫א‬‫א‬

،‫א‬ E‫א‬F‫א‬‫א‬(8)‫א‬ ‫א‬(2)‫א‬‫א‬،(8 × 1 = 8) 1‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬E  ‫א‬‫א‬F‫א‬ E  ‫א‬F‫א‬‫א‬E1F‫א‬‫א‬،(2 × 10 = 20)10

‫א‬K(1 × 100 = 100) 100‫א‬‫א‬‫א‬  W،‫א‬‫א‬‫א‬‫א‬‫א‬  

(1 × 100) + (2 × 10) + (8 × 1) = 100 + 20 + 8 = 128

‫א‬‫א‬‫א‬E10F‫א‬‫א‬‫א‬

 W100 = 110 ‫א‬‫א‬‫א‬

........ 105 104 103

102 101

 

100

 W128‫א‬ 1

  

 (128)10

 ‫א‬‫א‬‫א‬‫א‬ 102 1 × 102 = 100

101 100 + 2 × 101 + 8 × 100 + 20 + 8



‫א‬‫א‬10‫א‬‫א‬(128)‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬(Subscript) -٣-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 W10-1

102

101 100 • 10-1 10-2

 

10-3 ........

 

‫א‬‫א‬ (Decimal Point)

 

 Binary Numbering System‫א‬‫א‬‫א‬٣ J١  ‫א‬‫א‬‫א‬‫א‬‫א‬ (2)(2)  (2)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ .(0‫ و‬1)‫א‬ ..... 2

..... 16

2 8

2 4

2 2

 W

2 1

 W‫א‬‫א‬‫א‬‫א‬

 W(11001)‫א‬‫א‬ 24 23 22 21 20 1 1 0 0 1 4 3 = (1 × 2 ) + (1 × 2 ) + (0 × 22) + (0 × 21) + (1 × 20) = 16 + 8 + 0 + 0 + 1 = (25)10

 

‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬

  ‫א‬‫א‬(2)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K(11001)2‫א‬‫א‬

 W‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬ (Binary Digit)‫א‬(Bit)‫א‬‫ א‬W(Bit)‫א‬‫א‬■

‫א‬E  ‫א‬F‫א‬‫א‬‫א‬K  ‫א‬‫א‬‫א‬

‫א‬(4-bits)(1001)2‫א‬،‫א‬‫א‬

 K‫א‬(7-bits)(1101101)2

-٤-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



‫א‬‫א‬ W(Number of Binary Combinations)‫א‬‫א‬■ K(bits)‫א‬‫א‬‫א‬‫א‬ N = 2n

 W‫א‬‫א‬‫א‬

 ‫א‬‫א‬‫א‬= NW  (bits)‫א‬= n

 W‫א‬‫א‬(2)‫א‬‫א‬

N = 22 = 4 N = 23 = 8 4

N = 2 = 16

 W‫א‬‫א‬(3)‫א‬‫א‬  W‫א‬‫א‬(4)‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬W(Bit)‫א‬‫א‬■ ‫א‬(1)(1)20‫א‬‫א‬‫א‬‫א‬

22‫א‬(2)21‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬(4) ‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬،‫א‬‫א‬

‫א‬‫א‬‫א‬(LSB)‫א‬‫א‬(Least Significant Bit)  K(MSB)‫א‬‫א‬ (Most Significant Bit) ‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬(Bit)‫א‬‫א‬WEByte)   ‫א‬ ■ (0)‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬EF‫א‬‫א‬(1)‫א‬‫א‬ ‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬ ‫א‬  K

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‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



(Byte)‫א‬

K ‫א‬‫א‬‫א‬‫א‬

 W‫א‬‫א‬‫א‬

1 byte = 8 bits

Decimal-to-Binary Conversion ‫א‬‫א‬‫א‬‫א‬‫א‬٤ J١ ‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬

(2)‫א‬‫א‬‫א‬‫א‬(Sum of Weights Method) ‫א‬‫א‬‫א‬(Repeated Division–by–2 Method)  K‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬١ J٤ J١

‫א‬،214‫א‬،‫א‬(14)10‫א‬‫א‬

 .(0)‫א‬2‫א‬ ‫א‬‫א‬‫א‬K  ‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬

‫א‬‫א‬،(MSB)‫א‬‫א‬‫א‬‫א‬(LSB)

 W

‫ א‬

14 ÷ 2 = 7 7 ÷2=3 3 ÷2=1 1 ÷2=0

0 1 1 1



  



1 1 1 0 (MSB)

(LSB)

 W

(14)10 = (1110)2

 

 K‫א‬(25)10‫א‬‫א‬WE١ J١F

 W‫א‬

-٦-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



 

‫א‬ 25 ÷ 2 = 12 12 ÷ 2 = 6 6 ÷2=3 3 ÷2=1 1 ÷2=0

1 0 0 1 1

(LSB)

(MSB)

 W‫א‬

(25)10 = (11001)2

 K‫א‬(87)10‫א‬‫א‬WE٢ J١F 

 W‫א‬

‫א‬ 87 ÷ 2 = 43 43 ÷ 2 = 21 21 ÷ 2 = 10 10 ÷ 2 = 5 5 ÷2=2 2 ÷2=1 1 ÷2=0

1 1 1 0 1 0 1

(LSB)

(MSB)

 W‫א‬

(87)10 = (1010111)2

 ‫א‬‫א‬‫א‬‫א‬‫א‬٢ J٤ J١

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬ (Decimal Fractions)‫א‬‫א‬‫א‬K(2)‫א‬

 K(2)‫א‬‫א‬

،(2)(0.3125)‫א‬‫א‬‫א‬‫א‬(0.3125)‫א‬‫א‬

‫א‬ ‫א‬‫א‬‫א‬(2)‫א‬‫א‬‫א‬

‫א‬(Carried Digits)‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬(0) ‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬K(LSB)‫א‬‫א‬‫א‬(MSB)‫א‬‫א‬

-٧-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



‫א‬

0.3125 × 2 = 0.625

0.625 × 2 = 1.25

0.25 × 2 = 0.5

0.5



 

× 2 = 1.00

1 (LSB) 1 0 1 0 (MSB)

‫א‬ 

 

‫א‬‫א‬‫א‬

 

‫א‬‫א‬‫א‬

 

 KE‫א‬ F

 K‫א‬(39.25)10‫א‬‫א‬WE٣ J١F

 W(2)‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬

 

‫א‬ 39 ÷ 2 = 19 19 ÷ 2 = 9 9 ÷2=4 4 ÷2=2 2 ÷2=1 1 ÷2=0

 

1 1 1 0 0 1

(LSB)

(MSB)

 W‫א‬

     W(2)‫א‬‫א‬‫א‬‫א‬

(39)10 = (100111)

0.25 × 2 = 0.5 0.5 × 2 = 1.00



 

‫א‬ 0

 W

(0.25)10 = (0.01)2

 W‫א‬‫א‬‫א‬ -٨-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



(39.25)10 = (100111.01)2

  Binary-to-Decimal Conversion ‫א‬‫א‬‫א‬‫א‬‫א‬٥ J١   (2)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬K‫א‬ 1‫ و‬2‫ و‬4‫ و‬8‫و‬16‫א‬‫א‬‫א‬‫א‬ (1)‫א‬‫א‬(Bit)‫א‬‫א‬

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬

 K‫א‬1101001‫א‬‫א‬WE٤ J١F 

‫א‬‫א‬‫א‬‫א‬(1)W‫א‬ 26

23 22

21 20 : ‫א‬

 W

1 : ‫א‬‫א‬

= 1 × 26 + 1 × 2 5 + 0 × 2 4 + 1 × 2 3 + 0 × 2 2 + 0 × 2 1 + 1 × 2 0 = 64 + 32 + 8 + 1 = (105)10

(Bits ) ‫ﺧﺎﻧ ﺎت‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬ (Binary Point)‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬(Decimal Point)‫א‬‫א‬ ……24

20 • 2-1

2-2

2-3

2-4…….

 W

 ‫א‬‫א‬

 

 K‫א‬(0.1011)2‫א‬‫א‬‫א‬WE٥ J١F  • 2-1 0• 1

2-2 0

2-3 1

 W‫א‬

2-4 1

∴(0.1011)2 = 1 × 2-1 + 1 × 2-3 + 1 × 2-4 = 0.5 + 0.125 + 0.0625 = (0.6875)10

-٩-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



 Binary Arithmetic ‫א‬‫א‬‫א‬‫א‬٦ J١ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬

 Binary Addition ‫א‬‫א‬١ J٦ J١

‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬ 0 0 1 1

+ + + +

0 1 0 1

= = = =

0 1 1 0 carry E‫א‬F 1 ⇒ = 10

 W(Binary Digits)

1 + 1 = ‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬(1)‫א‬‫א‬،(2)10  W‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬  K011, 110‫א‬‫א‬‫א‬WE٦ J١F 

 W‫א‬‫א‬‫א‬‫א‬W‫א‬ 1 6 +3 EF 9

4 +3 EF 7

+ 1

1 1 0 0

 

1 1 0

0 1 1

 K011, 100‫א‬‫א‬‫א‬WE٧ J١F  1 0 1

0 1 1

W‫א‬

 Binary Subtraction ‫א‬‫א‬٢ J٦ J١  W‫א‬‫א‬

 K‫א‬‫א‬‫א‬ J١

K‫א‬‫א‬ J٢

-١٠-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



‫א‬K ‫א‬‫א‬،‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E ‫א‬F‫א‬‫א‬ 0–0=0 1–0=1 1–1=0 0–1=1

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

(1) ‫א‬ (1)‫א‬

 W‫א‬‫א‬‫א‬

 K‫א‬‫א‬ •

W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫• א‬ K(0)‫א‬(1)(1)(0)(0)

K(1)‫א‬(1)(0)

‫א‬F‫א‬‫א‬(0)(1)‫א‬(0)(1) K(0)(1)(1)E

K‫א‬‫א‬‫א‬‫א‬‫א‬

 K(011)‫א‬‫א‬(101)‫א‬‫א‬‫א‬WE٨ J١F 

(0)‫א‬ (1)‫א‬ ‫א‬‫א‬(1)‫א‬ (1)(10)‫א‬

0 1  –0

(1)‫א‬



1 0

      1 ‫א‬

W‫א‬    

0



1 1

‫א‬ 





   

 ‫א‬‫א‬‫א‬‫א‬‫א‬٧ J١ One's and Two's Complements of Binary Numbers ‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

K ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬(1)(0)(0)(1)‫א‬‫א‬

 W‫א‬

-١١-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0

 ‫א‬‫א‬

‫א‬‫א‬







 W‫א‬‫א‬‫א‬‫א‬

‫א‬ (1)‫א‬K ‫א‬‫א‬W‫א‬‫א‬  W‫א‬‫א‬‫א‬‫א‬ 1 +‫א‬‫א‬Z‫א‬‫א‬

K10110011‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬(1)‫א‬‫א‬‫א‬ 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 + 1 0 1 0 0 1 1 0 1

 ‫א‬‫א‬





‫א‬‫א‬





 ‫א‬‫א‬





(1) 





  ‫א‬‫א‬(LSB)‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬

‫א‬‫א‬(0) ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ F‫א‬

،E ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬(10101101)2‫א‬‫א‬

‫א‬‫א‬

1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1

 ‫א‬‫א‬

 ‫א‬‫א‬



  



 

Representation of Signed Numbers‫א‬‫א‬‫א‬‫א‬٨ J١ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬

(1)،‫א‬(0)‫א‬،‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬K‫א‬ -١٢-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



‫א‬‫א‬ (Sign Bit)‫א‬‫א‬‫א‬‫א‬   .(Magnitude) ‫א‬‫א‬W   ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  .(2's Complement)‫א‬‫א‬ (1's Complement) ‫א‬‫א‬(Sign-Magnitude)  (Sign-Magnitude System) ‫א‬‫א‬١ J٨ J١

‫א‬‫א‬‫א‬(Bit)‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬

‫א‬K ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

(+23)‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬

0 0 0 1 0 1 1 1

‫א‬ (Sign Bit)



 ‫א‬‫א‬



(Magnitude Bits)

      

 W ‫–( ﻓﺈﻧﻨﺎ‬23)‫א‬‫א‬      1 0 0 1 0 0 1 1 1  K‫א‬(–23) , (+23)‫א‬‫א‬‫א‬  (1's Complement System) ‫א‬‫א‬٢ J٨ J١

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

K  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬K ‫א‬‫א‬

 W‫א‬‫א‬‫א‬(–23) ‫א‬‫א‬ 00010111

 1 1 1 0 1 0 0 0

(+23)‫א‬

(–23)‫א‬





 

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬  

-١٣-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



 (2's Complement)‫א‬‫א‬٣ J٨ J١

‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬‫א‬‫א‬

K  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬

 W(+23)‫א‬‫א‬‫א‬(–23)‫א‬‫א‬ 00010111

 1 1 1 0 1 0 0 1

(+23)‫א‬

(–23)‫א‬





 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬   Arithmetic Operations with Signed ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬٩ J١ Numbers ، ‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬KE٦ J١F‫א‬‫א‬، ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬   ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬

 W‫א‬

 K‫א‬‫א‬‫א‬‫א‬11111010‫א‬‫א‬00001110‫א‬‫א‬‫א‬WE٩ J١F  

 W‫א‬W‫א‬

14 – (– 6) = 14 + 6 = 20

 W‫א‬ 00001110 + 00000110 00010100

‫א‬(+14)

‫א‬‫א‬(+6) ‫א‬(+20)

 W‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬WE١٠ J١F 

(00001000)2 – (00000100)2

 W‫א‬W‫א‬

8 – 4 = 8 + (– 4) = 4

 W

 

-١٤-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



00001000 +11111100 100000100  ‫א‬

 

‫א‬(+8)

‫א‬‫א‬(– 4) ‫א‬(+4)

 

(Discard carry)

 K‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬WE١١ J١F 

(11100111)2 – (00001001)2

 W‫א‬W‫א‬

– 25 – (+9) = – 25 – 9 = – 34 11100111 +11110111 111011110  ‫א‬

 W

‫א‬(– 25)

‫א‬‫א‬(– 9)

‫ –( א‬34)

 

(Discard carry)

 The Octal Numbering System‫א‬‫א‬‫א‬١٠ J١ (8)(8)‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E7‫و‬6‫و‬5‫و‬4‫و‬3‫و‬2‫و‬1‫و‬0F

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 Octal-to-Decimal Conversion ‫א‬‫א‬‫א‬‫א‬١ J١٠ J١

(8)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬  ،‫א‬(……83

80)

‫א‬‫א‬‫א‬‫א‬‫א‬K،‫א‬(…... 512 64 8 1)

  ‫א‬‫א‬(2275)8‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬  W

‫א‬‫א‬: 83 82 81 80

‫א‬‫ א‬: 2

7 5

∴ (2275)8 = (2 × 83) + (2 × 82) + (7 × 81) + (5 × 80) -١٥-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



= (2 × 512) + (2 × 64) + (7 × 8) + (5 × 1) = 1024 + 128 + 56 + 5 = (1213)10  Decimal–to–Octal Conversion ‫א‬‫א‬‫א‬‫א‬٢ J١٠ J١

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،(8)‫א‬

 K(2)(8)

 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬١ J٢ J١٠ J١

(8)150‫א‬‫א‬‫א‬(150)10‫א‬‫א‬

.(0)‫א‬ ‫א‬(8)‫א‬‫א‬

K  ‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬

‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬ {Most Significant Digit} ‫א‬‫א‬‫א‬‫א‬{Least Significant Digit} (LSD)  W‫א‬‫א‬(MSD)

  ‫א‬ 150 ÷ 8 = 18 18 ÷ 8 = 2 2 ÷8=0

6 2 2

(LSD) (MSD)

 W‫א‬

(150)10 = (226)8

 K‫א‬‫א‬(624)10‫א‬‫א‬WE١٢ J١F 

 ‫א‬  624 ÷ 8 = 78 78 ÷ 8 = 9 9 ÷8=1 1 ÷8=0

0 6 1 1

W‫א‬

(LSD) (MSD)

 W‫א‬

(624)10 = (1160)8

 ‫א‬‫א‬‫א‬‫א‬‫א‬٢ J٢ J١٠ J١

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

  ‫א‬‫א‬(0.265)‫א‬‫א‬K(8)‫א‬‫א‬ -١٦-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



‫א‬‫א‬‫א‬ ،(8)0.265‫א‬‫א‬

‫א‬‫א‬‫א‬(0)‫א‬‫א‬‫א‬‫א‬(8)

K  ‫א‬‫א‬‫א‬‫א‬‫( א‬Carried Digits)‫א‬‫א‬K‫א‬ ‫א‬(MSD)  ‫א‬‫א‬(LSD)‫א‬‫א‬‫א‬

 W  

‫اﻟﺤﺎﻣﻞ‬ 0.265 × 8 = 2.12 0.12 × 8 = 0.96 0.96 × 8 = 7.68 0.68 × 8 = 5.44 0.44 × 8 = 3.52 0.52 × 8 = 4.16

2 (MSD) 0 7 5 3 4 (LSD)  W‫א‬‫א‬(6)‫א‬‫א‬‫א‬‫א‬‫א‬ (0.625)10 = (0.207534)8

 K‫א‬‫א‬(44.5625)10‫א‬‫א‬WE١٣ J١F 

 K(8)‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬   ‫א‬

44 ÷ 8 = 5 5 ÷8=0

4 5

(LSD) (MSD)

 W‫א‬

(44)10 = (54)8

 W(8)‫א‬‫א‬‫א‬‫א‬  

‫اﻟﺤﺎﻣﻞ‬ 0.5625 × 8 = 4.5 0.5 × 8 = 4.00

4 4

 W

(0.5625)10 = (0.44)8

 W‫א‬‫א‬‫א‬

(44.5625)10 = (54.44)8 -١٧-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



 Octal-to-Decimal Conversion

‫א‬‫א‬‫א‬‫א‬‫א‬٣ J١٠ J١

(8)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬‫א‬   ‫א‬K‫א‬ 1‫ و‬8‫ و‬64‫ و‬512‫و‬4096‫א‬‫א‬ (Digit)‫א‬‫א‬‫א‬‫א‬

K  ‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬

 K‫א‬‫א‬(324)8‫א‬‫א‬WE١٤ J١F 

 W‫א‬

‫א‬‫א‬: 82 81 80

‫א‬‫א‬: 3 2 4

∴ (324)8 = (3 × 82) + (2 × 81) + (4 × 80) = (3 × 64) + (2 × 8) + (4 × 1) = 192 + 16 + 4 = (212)10

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬   (Octal Point)  ‫א‬‫א‬‫א‬ ……84

 W‫א‬‫א‬‫א‬‫א‬

80 • 8-1

8-2

8-3

8-4…….

 ‫א‬‫א‬

 K‫א‬‫א‬(567.14)8‫א‬‫א‬WE١٥ J١F  ‫א‬‫א‬: 82 81 80 • 8-1 8-2

‫א‬‫א‬: 5 6 7 • 1 4

∴ (567.14)8 = (5 × 82) + (6 × 81) + (7 × 80) + (1 × 8-1) + (4 × 8-2) = (5 × 64) + (6 × 8) + (7 × 1) + (1 × 0.125) + (4 × 0.015625) = 320 + 48 + 7 + 0.125 + 0.0625 = (375.1875)10

-١٨-

 W‫א‬

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



 Octal-to-Binary Conversion ‫א‬‫א‬‫א‬‫א‬‫א‬٤ J١٠ J١

‫א‬‫א‬ (Digit)

‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬،(3-bits) KE١ J١F

 ‫א‬‫א‬

 ‫א‬‫א‬

000

001

010

011

100

101

110

111

 K‫א‬‫א‬‫א‬E١ J١F ‫א‬

‫א‬‫א‬‫א‬

 K‫א‬

 K‫א‬‫א‬(357)8‫א‬‫א‬WE١٦ J١F  (357)8 =

3 011

 W‫א‬

101 111

= (011101111)2

 K‫א‬(1276.543)8‫א‬‫א‬WE١٧ J١F  (1276.543)8 =

1 001

6 • 5

 W‫א‬

010 111 110 • 101 100 011

= (1010111110.101100011)2

 K‫א‬‫א‬‫א‬

 Binary-to-Octal Conversion ‫א‬‫א‬‫א‬‫א‬‫א‬٥ J١٠ J١

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 J  –‫א‬‫א‬K‫א‬

‫א‬‫א‬‫א‬‫א‬

  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ -١٩-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬(1011001011100.00101)2‫א‬‫א‬WE١٨ J١F  001 1

011 3

001 1

011 3

100 4

• 001 010 •

 W‫א‬  

‫א‬‫א‬‫א‬‫א‬‫א‬

 W

(1011001011100.00101)2 = (13134.12)8   Arithmetic Operations in Octal System ‫א‬‫א‬‫א‬‫א‬٦ J١٠ J١

 K‫א‬‫א‬‫א‬‫א‬

Octal Addition ‫א‬‫א‬١ J٦ J١٠ J١

‫א‬(9‫و‬0)‫א‬–‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬(10)‫א‬(9)‫א‬‫א‬‫א‬،(9)

‫א‬  ‫א‬‫א‬K ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬(10)‫א‬‫א‬(1‫و‬0)‫א‬‫א‬‫א‬‫א‬

‫א‬K‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬(7)‫א‬‫א‬‫א‬–‫א‬‫א‬‫א‬(7) ‫ و‬16‫ و‬15‫ و‬14‫ و‬13‫ و‬12‫و‬11)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬(10)

( 37‫ و‬..... ‫ و‬31‫و‬30)‫א‬‫א‬‫א‬( 27‫ و‬...... ‫ و‬22‫و‬21‫و‬20) ‫א‬‫א‬‫א‬(17

‫א‬‫א‬‫א‬‫א‬‫א‬E٢ J١F ‫א‬K‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬K  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 W‫א‬‫א‬

-٢٠-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ •  K(7)

(2)‫א‬‫א‬(7)‫א‬‫א‬‫א‬ •

‫א‬(7)‫א‬(8)‫א‬‫א‬(7)‫א‬‫א‬،‫א‬ ‫א‬‫א‬(2)‫א‬(10)‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬F‫א‬‫א‬‫א‬ KE‫א‬‫א‬‫א‬

7 7 10 11 12 13 14 15 16

6 6 7 10 11 12 13 14 15

5 5 6 7 10 11 12 13 14

4 4 5 6 7 10 11 12 13

3 3 4 5 6 7 10 11 12

2 2 3 4 5 6 7 10 11

1 1 2 3 4 5 6 7 10

0 0 1 2 3 4 5 6 7

+ 0 1 2 3 4 5 6 7

 K‫א‬‫א‬‫א‬E٢ J١F ‫א‬  

 K(42)8، (34)8‫א‬‫א‬‫א‬WE١٩ J١F  34 + 42 76

 W‫א‬‫א‬W‫א‬

∴(34)8 + (42)8 = (76)8

 

(7)E4‫و‬3)E2‫و‬4F‫א‬‫א‬

 K‫א‬

-٢١-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



 K(63)8(56)8‫א‬‫א‬‫א‬WE٢٠ J١F

 W‫א‬

5 6 + 6 3 1 4 1 ‫א‬(2)(7)‫א‬‫א‬‫א‬

 K‫א‬‫א‬(Carry)‫א‬

Subtraction in Octal System‫א‬‫א‬‫א‬٢ J٦ J١٠ J١

 W‫א‬‫א‬‫א‬

 K ‫א‬‫א‬‫א‬‫א‬‫א‬ •

‫א‬‫א‬‫א‬–‫א‬‫א‬(1)‫א‬‫א‬‫א‬ •

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬(8)

K‫א‬‫א‬

(657)8 – (346)8 6 5 7 – 3 4 6 3 1 1

‫א‬

W‫א‬‫א‬ ‫א‬WE٢١ J١F 

 W‫א‬W‫א‬

‫א‬

∴(657)8 – (346)8 = (311)8

‫א‬‫א‬‫א‬

 K ‫א‬‫א‬

(732)8 – (634)8W‫א‬‫א‬ ‫א‬WE٢٢ J١F

 W‫א‬

6 2 1 7 3 2 – 6 3 4 0 7 6

‫א‬ ‫א‬

∴(732)8 – (634)8 = (76)8 -٢٢-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



‫א‬‫א‬(2)(4)‫א‬‫א‬

‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬(1)‫א‬  K‫א‬‫א‬(2)(3)‫א‬‫א‬‫א‬

 Hexadecimal Numbering System‫א‬‫א‬‫א‬‫א‬١١ J١ (16)‫א‬‫א‬‫א‬‫א‬‫א‬  (F‫و‬E‫و‬D‫و‬C‫و‬B‫و‬A‫و‬9‫و‬8‫و‬7‫و‬6‫و‬5‫و‬4‫و‬3‫و‬2‫و‬1‫و‬0)   ‫א‬(16)

 K‫א‬E 15‫ و‬14‫ و‬13‫ و‬12‫ و‬11‫و‬10F‫א‬‫א‬(F‫و‬E‫و‬D‫و‬C‫و‬B‫و‬A)‫א‬

Hexadecimal–to–Decimal Conversion‫א‬‫א‬‫א‬‫א‬١ J١١ J١

16‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

(... 4096 256 16 1 )‫א‬‫א‬‫א‬‫א‬(……163 162 161 160)  W(522.39)16‫א‬‫א‬‫א‬

‫א‬‫א‬: 162 161 160 • 16-1 16-2

‫א‬‫א‬: 5

• 3

 

∴ (522.39)16 = (5 × 162) + (2 × 161) + (2 × 160) + (3 × 16-1) + (9 × 16-2) = (5 × 256) + (2 × 16) + (2 × 1) + (3 × 0.0625) + (9 × 0.0039062) = 1280 + 32 + 2 + 0.1875 + 0.0351558 = (1314.222655)10

‫א‬K‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 K ‫א‬‫א‬(16)‫א‬‫א‬‫א‬

Decimal-to-Hexadecimal Conversion ‫א‬‫א‬‫א‬‫א‬٢ J١١ J١

(16)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 .(2)(8)(16)‫א‬‫א‬

 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬١ J٢ J١١ J١

(16)97‫א‬‫א‬‫א‬(97)10‫א‬‫א‬

‫א‬ ‫א‬(16)‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬K(0) -٢٣-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬K‫א‬‫א‬

 W‫א‬‫א‬(MSD)‫א‬‫א‬(LSD)

  ‫א‬ 97 ÷ 16 = 6 6 ÷ 16 = 0

1 6

(LSD) (MSD)

 W‫א‬

(97)10 = (61)16

 K‫א‬‫א‬‫א‬(314)10‫א‬‫א‬WE٢٣ J١F

W‫א‬

  ‫א‬

314 ÷ 16 = 19 19 ÷ 16 = 1 1 ÷ 16 = 0

A 3 1

(LSD) (MSD)

 W‫א‬

(314)10 = (13A)16

 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬٢ J٢ J١١ J١

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

(0.78125)10‫א‬‫א‬K(16)‫א‬‫א‬‫א‬

‫א‬‫א‬(16)‫א‬‫א‬‫א‬‫א‬‫א‬

(0)‫א‬‫א‬‫א‬‫א‬‫א‬(16)‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬

(MSD)‫א‬‫א‬‫א‬(LSD)‫א‬‫א‬‫א‬K‫א‬‫א‬   ‫اﻟﺤﺎﻣﻞ‬ 0.78125 × 16 = 12.5 0.5 × 16 = 8.00

 W‫א‬ 

C 8

 W

∴(0.78125)10 = (0.C8)16

-٢٤-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



 K‫א‬‫א‬(329.52)10‫א‬‫א‬WE٢٤ J١F 

 W16‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬   ‫א‬

329 ÷ 16 = 20 20 ÷ 16 = 1 1 ÷ 16 = 0

9 4 1

(LSD) (MSD)

 W‫א‬

∴(329)10 = (149)16

 W‫א‬‫א‬(16)‫א‬‫א‬  

‫א‬ 0.52 × 16 = 8.32 0.32 × 16 = 5.12 0.12 × 16 = 1.92 0.92 × 16 = 14.72 0.72 × 16 = 11.52 0.52 × 16 = 8.32

8 (MSD) 5 1 E B 8 (LSD)

 

 W‫א‬(6)‫א‬‫א‬‫א‬‫א‬‫א‬ (0.52)10 = (0.851EB8)16

 W‫א‬‫א‬‫א‬

(329.52)10 = (149.851EB8)16 Hexadecimal-to-Decimal Conversion

‫א‬‫א‬‫א‬‫א‬٣ J١١ J١

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬K(16)‫א‬

 W‫א‬‫א‬K‫א‬‫א‬  K‫א‬‫א‬(F9B)16‫א‬‫א‬‫א‬WE٢٥ J١F 

 W‫א‬

‫א‬‫א‬: 162 161 160

‫א‬: F

B -٢٥-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



∴ (F9B)16 = (F × 162) + (9 × 161) + (B × 160) = (15 × 256) + (9 × 16) + (11 × 1) = 3840 + 144 + 11 = (3995)10

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 W‫א‬‫א‬‫א‬‫א‬‫א‬

……163

162

161

160 • 16-1

16-2

16-3 …….

 

 ‫א‬‫א‬

 

 K‫א‬(A15.C3)16‫א‬‫א‬‫א‬WE٢٦ J١F ‫א‬‫א‬: 162 161 160 • 16-1 16-2

‫א‬‫א‬: A

5 • C

 W‫א‬

∴ (A15.C3)16 = (A × 162) + (1 × 161) + (5 × 160) + (C × 16-1) + (3 × 16-2) = (10 × 256) + (1 × 16) + (5 × 1) + (12 × 0.0625) + (3 × 0.0039062) = 2560 + 16 + 5 + 0.75 + 0.0117186 = (2581.7617)10

Hexadecimal-to-Binary

  ‫א‬‫א‬‫א‬‫א‬‫א‬٤ J١١ J١

Conversion

EF‫و‬E‫و‬D‫و‬C‫و‬B‫و‬A‫و‬9‫و……و‬2‫و‬1‫و‬0F‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬EF‫و‬E‫و‬D‫و‬C‫و‬B‫و‬AF‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬KE15‫و‬14‫و‬13‫و‬12‫و‬11‫و‬10F (4-bits)‫א‬‫א‬‫א‬،‫א‬‫א‬  WE٣ J١F‫א‬‫א‬

 K‫א‬(3A5)16‫א‬WE٢٧ J١F 

(3A5)16

3 0011

1010 0101

= (001110100101)2 -٢٦-

 W‫א‬

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



  ‫א‬‫א‬‫א‬

 ‫א‬‫א‬

0 1 2 3 4 5 6 7 8 9 A B C D E F

 ‫א‬‫א‬

0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

 K‫א‬‫א‬‫א‬E٣ J١F ‫א‬

 

 K‫א‬‫א‬(B35.D1)16‫א‬WE٢٨ J١F  (B35.D1)16 =

•

 W‫א‬

1011 0011 0101 • 1101 0001 = (101100110101.11010001)2 Binary-to-Hexadecimal Conversion‫א‬‫א‬‫א‬‫א‬‫ א‬٥ J١١ J١

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K

‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬ -٢٧-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



 

 K‫א‬‫א‬(110111101.101001)2‫א‬‫א‬WE٢٩ J١F 

 W‫א‬

0001 1011 1101 • 1010 0100 1

•

  A

 K‫א‬‫א‬‫א‬

∴(110111101.101001)2 = (1BD.A4)16  K‫א‬‫א‬‫א‬(11010010011.011001)2‫א‬‫א‬WE٣٠ J١F 

 W‫א‬

0001 1010 1011 • 0110 1000 1

 

•

∴(11010010011.011001)2 = (1AB.68)16

Hexadecimal-to-Octal Conversion‫א‬‫א‬‫א‬‫א‬‫א‬٦ J١١ J١

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬

 K‫א‬‫א‬(AB3E.87D)16‫א‬WE٣١ J١F

 W‫א‬‫א‬‫א‬‫א‬W‫א‬

(AB3E.87D)16 = (1010101100111110.100001111101)2

‫א‬‫א‬‫א‬‫א‬‫א‬

 W

001 010 101 001 111 110 • 100 001 111 101 1

•

 

  4

 K‫א‬‫א‬ -٢٨-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



∴(AB3E.87D)16 = (125476.4175)8 Octal-to-Hexadecimal Conversion‫א‬‫א‬‫א‬‫א‬‫ א‬٧ J١١ J١

‫א‬‫א‬‫א‬‫א‬

‫א‬،

،‫א‬‫א‬‫א‬‫א‬

 W‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬(25.342)8‫א‬‫א‬WE٣٢ J١F  ∴(25.342)8 = (010101.011100010)2

 W‫א‬‫א‬W‫א‬  

 W‫א‬‫א‬‫א‬‫א‬‫א‬

0001 0101 • 0111 0001 1

•

  7

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ∴(25.342)8 = (12.71)16

 ‫א‬‫א‬‫א‬‫א‬‫א‬٨ J١١ J١

Arithmetic Operations in Hexadecimal System

 K‫א‬‫א‬‫א‬‫א‬

Hexadecimal Addition‫א‬‫א‬‫א‬‫א‬١ J٨ J١١ J١

(F)‫א‬‫א‬‫א‬‫א‬Ef‫و‬0F‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬(10)‫א‬‫א‬،(10) ‫א‬  K  ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

(B)16(9)16‫א‬‫א‬(A)16‫א‬(9)16‫א‬‫א‬

 K(F)16‫א‬

-٢٩-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



‫א‬‫א‬(10)16‫א‬(F)16‫א‬‫א‬

‫א‬‫א‬‫א‬(11)16‫א‬(F)16‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬

،‫א‬‫א‬‫א‬‫א‬E٤ J١F‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬

F F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E

E D C B E D C B F E D C 10 F E D 11 10 F E 12 11 10 F 13 12 11 10 14 13 12 11 15 14 13 12 16 15 14 13 17 16 15 14 18 17 16 15 19 18 17 16 1A 19 18 17 1B 1A 19 18 1C 1B 1A 19 1D 1C 1B 1A

A A B C D E F 10 11 12 13 14 15 16 17 18 19

9 9 A B C D E F 10 11 12 13 14 15 16 17 18

8 8 9 A B C D E F 10 11 12 13 14 15 16 17

7 7 8 9 A B C D E F 10 11 12 13 14 15 16

6 6 7 8 9 A B C D E F 10 11 12 13 14 15

5 5 6 7 8 9 A B C D E F 10 11 12 13 14

4 4 5 6 7 8 9 A B C D E F 10 11 12 13

3 3 4 5 6 7 8 9 A B C D E F 10 11 12

2 2 3 4 5 6 7 8 9 A B C D E F 10 11

1 1 2 3 4 5 6 7 8 9 A B C D E F 10

0 0 1 2 3 4 5 6 7 8 9 A B C D E F

+ 0 1 2 3 4 5 6 7 8 9 A B C D E F

 K‫א‬‫א‬‫א‬‫א‬E٤ J١F‫א‬

 W‫א‬‫א‬WE٣٣ J١F 

(35AB2)16 + (1A675)16

 

 K‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬W‫א‬ 1 3 + 1 5

1 5 A 0

1 A 6 1

B 7 2

∴(35AB2)16 + (1A675)16 = (50127)16 -٣٠-

2 5 7

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



Hexadecimal Subtraction‫א‬‫א‬‫א‬‫א‬٢ J٨ J١١ J١

 W‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ •  K‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬(1)‫א‬‫א‬‫א‬ •

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

W‫א‬‫א‬

 W‫א‬‫א‬ ‫א‬WE٣٤ J١F 

(F2ABD)16 – (EF4CE)16

 W‫א‬ E F – E

2 F 3

9 A 4 5

1A B C E

D E D

 

 K‫א‬‫א‬‫א‬

                   

  -٣١-

 

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



 

 

a) 64 e) 77.0625

a) 11011 e) 10101.1101

b) 112 f) 47.875

 W‫א‬‫א‬‫א‬‫א‬‫א‬  E١ c) 257 g) 33.125

 W‫א‬‫א‬‫א‬‫א‬‫א‬  E٢

b) 1110101 f) 1100001.11011

d) 1110.11

b) 1110.11 + 11.10 d) 1001.101 + 1101.11

 W‫א‬‫א‬‫א‬‫א‬‫א‬ E٤

a) 1101 – 0100 c) 11010 – 10111

a) 11110110

c) 111111

 W‫א‬‫א‬‫א‬‫א‬E٣

a) 100 + 111 c) 1111 + 1101

a) 00110101

d) 27.26

b) 1001 – 0111 d) 1100 – 1001

b) 11100100

b) 01011101

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٥ c) 00010101

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ E٦

c) 00110011

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٧ a) +28

b) – 83

 W(8-bits) ‫א‬‫א‬

c) +99

d) – 120

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٨ a) +14

b) – 63

 W(8-bits) ‫א‬‫א‬

c) +107

d) – 122

  -٣٢-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



 K‫א‬‫א‬‫א‬‫א‬E٨F‫א‬‫א‬E٩  

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E١٠

a) 10111000

b) 01100100

c) 10110011

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E١١

a) 10011101

b) 01100110

c) 10101101

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E١٢

a) 10101011

b) 000111101

c) 10111011

 

 W‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬E١٤

a) 00010110 – 00110011 c) 10001100 – 00111001

a) 50 e) 120.515625

a) 42 e) 96.11

a) 72 e) 122.775

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  E١٥

b) 100 f) 144.5625

b) 254 f) 115.3

c) 6391 g) 915.141

d) 77.375

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E١٦

b) 113 f) 417.632

a) 110101.1101 d) 10001001011.1001

a) (15)8 + (17)8 c) (123)8 + (321)8

b) 01110000 – 10101111 d) 11011001 – 11100111

c) 1057 g) 14367.12

d) 37.5

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E١٧ c) 16.3 g) 276.621

d) 37.6

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E١٨

b) 11110100.110101 e) 1010111.11101

c) 110110111.10101

 W‫א‬‫א‬‫א‬‫א‬E١٩

b) (44)8 + (66)8 d) (272)8 + (456)8 -٣٣-

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬

 ‫א‬



 W‫א‬‫א‬‫א‬‫א‬E٢٠

a) (32)8 – (25)8 c) (315)8 – (222)8

b) (147)8 – (74)8 d) (437)8 – (340)8

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٢١

a) 14 e) 62500

b) 80 f) 204.125

b) D52 f) B3.E

a) 37 e) 1600.524

c) 67F g) 1111.1

d) ABCD h) 888.8

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٢٣

b) 1C

a) 1001.1111 d) 10100111.111011

a) 13A e) 78.D3

d) 3000 h) 631.25

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٢٢

a) 9F e) F.4

a) 8

c) 560 g) 255.875

c) A64

d) 1F.C

e) 239.4

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٢٤ b) 10000.1 e) 1000000.000111

b) 25E6 f) 2659.F41

b) 725 f) 3000.6125

a) (41)16 + (36)16 c) (9B)16 + (65)16 f) (77CB5)16 + (A5F72)16

c) 110101.11001 f) 1111100.1000011

 W‫א‬‫א‬‫א‬‫א‬‫א‬E٢٥

c) 3016

d) B4.C

 W‫א‬‫א‬‫א‬‫א‬‫א‬E٢٦

c) 2476.2

d) 1117.16

 W‫א‬‫א‬‫א‬‫א‬E٢٧ b) (C8)16 + (3A)16 d) (11D)16 + (2E1)16 g) (13EFD)16 + (21BB3)16

 

-٣٤-

‫א‬‫א‬‫א‬

 ‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 ‫א‬‫א‬‫א‬

 

 W‫א‬‫א‬‫א‬

K‫א‬‫א‬‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫• א‬

- ٣٥ -

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 Introduction١ J٢ ‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬ J‫א‬ ‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬ ،‫א‬‫א‬‫א‬‫א‬‫א‬



 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬?‫א‬‫א‬?  K‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬



‫א‬K(INVERTER)‫א‬NOT‫א‬OR‫א‬AND‫א‬‫א‬

،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 AND Gate AND‫א‬٢ J٢ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬AND‫א‬‫א‬  ‫א‬‫א‬،‫א‬ AND‫א‬‫א‬K(Logic Functions)  ‫א‬‫א‬‫א‬‫א‬،(Logical Multiplication)‫א‬ ‫א‬A, B‫א‬،E١ J٢F ‫א‬‫א‬‫א‬‫א‬

‫א‬(0)(Two Binary Variables)‫א‬‫א‬‫א‬

 K(Closed) ‫א‬‫א‬(1)(Open) ‫א‬  

(A)    (B)



Voltage Source

 

(L)

       

 K‫א‬‫א‬AND‫א‬‫א‬E١ J٢F‫א‬ - ٣٦ -

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



  (1)‫א‬‫א‬‫א‬"L"‫א‬

‫א‬‫א‬K(OFF)‫א‬(0)(ON)‫א‬

‫א‬‫א‬E١ J٢F، ‫א‬،

(L)‫א‬‫א‬K ‫א‬(L)‫א‬

 K(Truth Table)‫א‬‫א‬‫א‬‫א‬، ‫א‬  A

 

L 

            

KE١ J٢F ‫א‬‫א‬E١ J٢F ‫א‬

‫א‬،AND‫א‬(Standard)‫א‬‫א‬‫א‬E٢ J٢F‫א‬

‫א‬‫א‬E٢ J٢F‫א‬K AND‫א‬‫א‬،Y‫א‬ BA  KAND

 ‫א‬

    

A 0 0 1 1

   

B 0 1 0 1

‫א‬ Y 0 0 0 1

A B

AND‫א‬‫א‬E٢ J٢F‫א‬KAND‫א‬‫א‬E٢ J٢F  ‫א‬

 K  

‫א‬(1)‫א‬،(bits)‫א‬



،‫א‬‫א‬AND‫א‬  ،‫א‬(1) BA‫א‬  K(1)‫א‬(1)‫א‬

 W‫א‬‫א‬‫א‬E‫א‬F‫א‬‫א‬ N = 2n

- ٣٧ -



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 K‫א‬‫א‬N W  K‫א‬‫א‬n

 W

  N = 2 2 = 4 ‫א‬‫א‬‫א‬   N = 2 3 = 8 ‫א‬‫א‬

  N = 2 4 = 16 ‫א‬‫א‬



 WE١ J٢F 

 KAND‫א‬‫א‬‫• א‬  ‫؟‬AND‫א‬‫א‬ •

E٣ J٢F،‫א‬‫א‬‫א‬AND‫א‬W‫א‬  K‫א‬‫א‬‫א‬

 

 ‫א‬

‫א‬

 A  B C Y 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0  1  0 0 0 1 0 1 0 1 1 0 0 1 1 1 1  KAND‫א‬‫א‬E٣ J٢F‫א‬

 

 W‫א‬‫א‬‫א‬• 

N = 2 n = 2 5 = 32

 

‫א‬‫א‬‫א‬‫א‬(Boolean Algebra)‫א‬‫א‬

‫א‬‫א‬(Boolean Expression)‫א‬‫א‬،‫א‬

 W‫א‬AND‫א‬‫א‬‫א‬K

Y = A•B - ٣٨ -

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



‫א‬،EAND • FA AND BY‫א‬W‫א‬

 W‫א‬‫א‬

Y = AB

 KA AND BY‫א‬

 ‫א‬‫א‬‫א‬



‫א‬K(LOW)‫א‬(HIGH)‫א‬‫א‬(Pulses)

‫א‬‫א‬،‫א‬‫א‬AND‫א‬  K‫א‬

‫א‬(1) BA‫א‬E٣ J٢F،



A‫א‬،t2‫א‬‫א‬،(1)‫א‬ Y‫א‬‫א‬t1‫א‬

‫א‬‫א‬‫א‬،(0)Y‫א‬B‫א‬(0)

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K  ‫א‬‫א‬

 K(Timing Diagram)

 

A B

  t1

t2 t3

t6 t7

 

A B

       

 KAND‫א‬‫א‬‫א‬E٣ J٢F ‫א‬  OR Gate OR‫א‬ ٣ J٢ ‫א‬‫א‬K ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬OR‫א‬‫א‬ 

‫א‬‫א‬‫א‬،‫א‬OR ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ،(Logical Addition)  BA‫א‬AND‫א‬‫א‬KE٤ J٢F   ‫א‬(1)(Open) ‫א‬(0)

 K(Closed)

- ٣٩ -

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 

 (A)

 



 



 

 (B)

Voltage Source

 

 K‫א‬‫א‬OR‫א‬‫א‬E٤ J٢F‫א‬

‫א‬‫א‬،‫א‬‫א‬‫א‬E٤ J٢F



 K ‫א‬(L)‫א‬‫א‬

 

                KE٤ J٢F‫א‬‫א‬E٤ J٢F‫א‬

‫א‬A, B‫א‬،OR‫א‬‫א‬‫א‬‫א‬E٥ J٢F‫א‬

 KOR‫א‬‫א‬E٥ J٢F‫א‬KY  

 ‫א‬

‫א‬

 A  B  Y 0  0 0 0 1 1 A Y 1 0  1 B  1 1 1 OR‫א‬‫א‬E٥ J٢F‫א‬KOR‫א‬‫א‬E٥ J٢F‫א‬

 K

‫א‬ (1)‫א‬E٥ J٢F‫א‬



‫א‬(0)‫א‬،(1)‫א‬

 W‫א‬OR‫א‬‫א‬‫א‬K‫א‬(0) - ٤٠ -

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



Y=A+B

 K(OR+)A OR BY‫א‬W‫א‬

،‫א‬‫א‬OR‫א‬‫א‬



‫א‬‫א‬‫א‬AND‫א‬  K‫א‬

‫א‬t1‫א‬‫א‬(1) BA‫א‬ E٦ J٢F



A‫א‬،t2‫א‬ ‫א‬،(1)‫א‬  Y‫א‬

‫א‬‫א‬‫א‬‫א‬،(1)Y‫א‬B‫א‬(0)

 K‫א‬

 

B t1

A B

 

 KOR‫א‬‫א‬‫א‬E٦ J٢F‫א‬  NOT Gate (INVERTER) E‫א‬F NOT ‫א‬٤ J٢ ‫א‬(Inversion)‫א‬NOT‫א‬‫א‬ 

(1)‫א‬،‫א‬‫א‬‫א‬ K(Complementation)  K(1)(0)‫א‬،(0)‫א‬

‫א‬‫א‬E٧ J٢FK  ‫א‬‫א‬ ‫א‬‫א‬NOT‫א‬‫א‬



 K‫א‬‫א‬‫א‬E٦ J٢F‫א‬،‫א‬‫א‬‫א‬

 

‫א‬

 A 0 1

‫א‬

 Y 1 0

 K‫א‬NOT‫א‬‫א‬E٦ J٢F‫א‬KNOT‫א‬‫א‬E٧ J٢F‫א‬  

- ٤١ -



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



‫א‬‫א‬،‫א‬‫א‬‫א‬

 W‫א‬

Y= A ‫א‬  barA‫א‬not AY‫א‬W‫א‬‫א‬

 KE A FA barY‫א‬،‫א‬

 NAND Gate  NAND ‫א‬٥ J٢ ‫א‬‫א‬،AND(NOT AND)‫א‬(NAND) ،E٨ J٢FAND‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬AND‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 KNAND‫א‬‫א‬E٧ J٢FK‫א‬‫א‬‫א‬

 

 ‫א‬

‫א‬

 A  B Y 0  0 1 0 1 1 A 1 0 1 Y  1 1 0 B  KNAND‫א‬‫א‬E٧ J٢F‫א‬KNAND‫א‬‫א‬E٨ J٢F‫א‬

 

‫א‬‫א‬‫א‬(0)‫א‬‫א‬



(0)‫א‬‫א‬‫א‬ (1) ‫א‬،‫א‬(1)

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬NAND‫א‬‫א‬KAND‫א‬‫א‬‫א‬،‫א‬  ‫א‬‫א‬‫א‬،‫א‬

NAND‫א‬‫א‬،‫א‬‫א‬، AND ORNOT‫א‬

 W‫א‬

Y = AB

،‫א‬‫א‬NAND‫א‬‫א‬

 K(1)‫א‬(0) NAND‫א‬‫א‬

- ٤٢ -

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



‫א‬t1‫א‬‫א‬(1) BA‫א‬ E٩ J٢F



A‫א‬،t2‫א‬‫א‬،(0)‫א‬ Y‫א‬

‫א‬‫א‬‫א‬،(1)Y‫א‬(1)B‫א‬(0)  K‫א‬‫א‬

 

   

 

B t1

 

 KNAND‫א‬‫א‬‫א‬E٩ J٢F‫א‬  

 NOR Gate  NOR ‫א‬٦ J٢ ‫א‬‫א‬،OR(NOT OR)‫א‬(NOR)  OR‫א‬‫א‬(NOT gate)‫א‬‫א‬‫א‬

NOR‫א‬‫א‬KNOR‫א‬‫א‬‫א‬‫א‬،E١٠ J٢F  KE٨ J٢F 

 

 ‫א‬

‫א‬

 A  B Y 0  0 1 0 1 0 A 1 0 0 Y B  1 1 0  KNOR‫א‬‫א‬E٨ J٢F‫א‬KNOR‫א‬‫א‬E١٠ J٢F‫א‬  

‫א‬‫א‬(0)(Y)‫א‬‫א‬

‫א‬‫א‬(1) ‫א‬،‫א‬ (1)‫א‬

 K‫א‬(0)

- ٤٣ -

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬NAND‫א‬‫א‬‫א‬NOR‫א‬‫א‬

‫א‬K، AND ORNOT‫א‬،‫א‬

 WNOR‫א‬‫א‬

Y= A+B

،‫א‬‫א‬ BA‫א‬NOR‫א‬E١١ J٢F



 K‫א‬(Y)‫א‬‫א‬NOR‫א‬‫א‬

 

  t1

t5 

 

 KNOR‫א‬‫א‬‫א‬E١١ J٢F‫א‬  

 Exclusive-OR Gate E‫א‬F‫ א‬OR ‫א‬٧ J٢ ،XOR-gate??‫א‬‫א‬OR‫א‬‫א‬ 

 K‫א‬‫א‬‫א‬E١٢ J٢F  

 ‫א‬

 A 0 0 1  1

 B  0 1 0 1

‫א‬

 Y 0 1 1 0

A B

 KXOR‫א‬‫א‬E٩ J٢F‫א‬KXOR‫א‬‫א‬E١٢ J٢F‫א‬

  (Y)‫א‬‫א‬،(9-2)XOR‫א‬‫א‬ ،‫א‬(0)‫א‬(1)، BA‫א‬‫א‬(1)  K‫א‬(0)  - ٤٤ -

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



‫א‬‫א‬‫א‬OR‫א‬‫א‬XOR‫א‬‫א‬ (1)XOR‫א‬‫א‬  ،A = B = 1  ، ‫א‬ ‫א‬‫א‬(1) (1)‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ Y = AB + AB

 W‫א‬‫א‬‫א‬‫א‬

Y=A⊕B XOR‫א‬‫א‬‫א‬‫א‬KBA⊕‫א‬

‫א‬‫א‬E١٣ J٢F‫א‬‫א‬، NOT ORAND‫א‬‫א‬‫א‬‫א‬

 K‫א‬XOR‫א‬‫א‬‫א‬

 

A B

   

 K NOT ORAND‫א‬XOR‫א‬‫א‬E١٣ J٢F‫א‬  

‫א‬XOR‫א‬‫א‬E١٤ J٢F



‫א‬‫א‬‫א‬،‫א‬

 K‫א‬

A B

B t1

t6 t7

 

 Y  

 KXOR‫א‬‫א‬‫א‬E١٤ J٢F‫א‬

   

 

 Exclusive-NOR Gate E‫א‬F‫ א‬NOR ‫א‬٨ J٢ - ٤٥ -

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 K‫א‬‫א‬‫א‬E١٥ J٢F،XNOR-gate‫א‬NOR‫א‬‫א‬ (Y)‫א‬‫א‬،E١٠ J٢F XNOR‫א‬‫א‬



(0) A = B = 1A = B = 0 BA‫א‬‫א‬(1)

،‫א‬(0)‫א‬(1)‫א‬

‫א‬‫א‬، ‫א‬ ‫א‬‫א‬(1)   K‫א‬‫א‬‫א‬‫א‬  

 ‫א‬

‫א‬

 A  B Y 0  0 1 0 1 0 A 1 0 0 Y B  1 1 1  KXNOR‫א‬‫א‬E١٠ J٢F‫א‬KXNOR‫א‬‫א‬E١٥ J٢F‫א‬

 

Y = AB + A B

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬‫א‬

Y=A~B XNOR‫א‬‫א‬‫א‬‫א‬K ‫א‬~‫א‬

‫א‬‫א‬E١٦ J٢F‫א‬ ‫א‬، NOT ORAND‫א‬‫א‬‫א‬‫א‬

 K‫א‬XNOR‫א‬‫א‬‫א‬

 

A B

 

 KAND, OR, NOT‫א‬XNOR‫א‬‫א‬E١٦ J٢F‫א‬

،‫א‬ BA‫א‬XNOR‫א‬E١٧ J٢F



  K(Y)‫א‬‫א‬XNOR‫א‬‫א‬  

- ٤٦ -

A B

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 

       

 KXNOR‫א‬‫א‬‫א‬E١٧ J٢F‫א‬  

 The Boolean Expression for a Logic Circuit‫א‬‫א‬‫א‬‫א‬٩ J٢ ‫א‬‫א‬، ‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬،K‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬‫א‬‫א‬KE١٨ J٢F‫א‬

 K AB A, B ‫א‬‫א‬AND‫א‬‫א‬‫א‬K١

 K AC  A ,C ‫א‬‫א‬AND‫א‬‫א‬‫א‬K٢

 K AB + AC  AB, AC ‫א‬‫א‬OR‫א‬‫א‬‫א‬K٣  W‫א‬‫א‬‫א‬

Y = AB + AC A B

B A

 



 K‫א‬‫א‬‫א‬‫א‬E١٨ J٢F‫א‬



   

 KE١٩ J٢F‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬WE٢ J٢F 

 W‫א‬

A B

A+B B

- ٤٧ -

 

D( A + B ) Y

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



   

     

 K‫א‬‫א‬‫א‬E٢ J٢F‫א‬‫א‬‫א‬E١٩ J٢F‫א‬ Y = D( A + B ) + ( B + C)

 W‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬١٠ J٢ Implementation of a Logic Circuit Using a Boolean Expression ‫א‬‫א‬‫א‬‫א‬   W‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬

 

Y = AB(CD + EF)

 (CD + EF)  BA‫א‬‫א‬‫א‬‫א‬‫א‬

FE،AND‫א‬ C , D  (CD + EF) ‫א‬،AND‫א‬

KOR‫א‬AND‫א‬‫א‬  ،AND‫א‬  W‫א‬‫א‬

- ٤٨ -

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 

AND

 

NOT OR

 Y = A B (CD + EF)

 

AND

 

‫( ؛‬CD + EF) ‫א‬‫א‬ ‫א‬‫א‬‫א‬  ‫א‬‫א‬‫ ؛‬CD, EF ‫א‬‫א‬‫א‬‫א‬‫א‬

K  ‫א‬‫א‬‫א‬، D

 W AB(CD + EF) ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K D ‫א‬NOT‫א‬K١

 K CD, EF ‫א‬AND‫א‬K٢

 K (CD + EF) ‫א‬‫א‬OR‫א‬K٣

 KY‫א‬‫א‬AND‫א‬K٤

 KE٢٠ J٢F ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 

A B

C D

 

E F

 

   

 K AB(CD + EF) ‫א‬‫א‬‫א‬‫א‬E٢٠ J٢F‫א‬

 ‫א‬‫א‬‫א‬‫א‬١١ J٢ Implementation of a Logic Circuit via a Truth Table ‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬ - ٤٩ -

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬E١٢ J٢FK‫א‬  W‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬

‫א‬‫א‬،Y = 1‫א‬‫א‬‫א‬‫א‬ K١

،A = 0, B = 1, C = 0‫א‬Y = 1‫א‬‫א‬

،(1)‫א‬‫א‬ ABC ‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬(1)‫א‬،(0)‫א‬  K ABC ‫א‬‫א‬

 ‫א‬

 A 0 0 0 0  1 1 1 1

 B 0 0 1 1  0 0 1 1

‫א‬ C 0 1 0 1 0 1 0 1

Y 0 0 1 0 0 0 1 0

 K‫א‬‫א‬‫א‬E١٢ J٢F‫א‬  

 WOR‫א‬Y = 1‫א‬‫א‬‫א‬‫א‬‫א‬ K٢ Y = AB C + ABC

‫א‬‫א‬‫א‬ ABC ‫א‬‫א‬‫א‬‫א‬‫א‬



 ABC ‫א‬  ‫א‬‫א‬‫א‬،AND‫א‬ A, B, C

OR‫א‬‫א‬‫א‬‫א‬،AND‫א‬ A, B, C ‫א‬‫א‬‫א‬  KY‫א‬‫א‬‫א‬

  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ NOT‫א‬W



OR‫א‬، ABC ، ABC ‫א‬‫א‬AND‫א‬‫ ؛‬A, C ‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬، ABC + ABC ‫א‬‫א‬‫א‬

 KE٢١ J٢F 

- ٥٠ -

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 

A B C

 

     K AB C + AB C ‫א‬‫א‬‫א‬‫א‬E٢١ J٢F‫א‬

 KE١٣ J٢F ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬WE٣ J٢F 

 

 ‫א‬

 A 0 0 0 0  1 1 1 1

 B 0 0 1 1  0 0 1 1

‫א‬ C 0 1 0 1 0 1 0 1

Y 0 1 0 1 0 1 0 0

 K‫א‬‫א‬‫א‬‫א‬‫א‬E١٣ J٢F‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬  WOR‫א‬E‫א‬‫א‬FY = 1

Y = ABC + ABC + ABC   KE٢٢ J٢F ‫א‬‫א‬‫א‬

 

B C

       K A B C + ABC + AB C ‫א‬‫א‬‫א‬‫א‬E٢٢ J٢F‫א‬ - ٥١ -

     

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 ‫א‬‫א‬‫א‬١٢ J٢ Converting a Boolean Expression to a Truth Table ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

،(22 = 4)،‫א‬‫א‬K(1or 0)  K‫א‬،(23 = 8)،‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬



،‫א‬‫א‬(Y)‫א‬(1)‫א‬‫א‬

 K‫א‬‫א‬،‫א‬‫א‬(0)

 W‫א‬‫א‬‫א‬WE٤ J٢F 

 

Y = AB C + ABC + ABC + ABC

 

‫א‬،‫א‬‫א‬‫א‬ (C BAF  ‫א‬W‫א‬

KE١٤ J٢F  ‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

A B C = 000, AB C = 010, AB C = 110, ABC = 111

،(Y)‫א‬(1)‫א‬‫א‬

 K(Y)‫א‬(0)‫א‬‫א‬‫א‬

 

 ‫א‬ A  0 0 0 0  1 1  1 1

 B 0 0 1 1  0 0 1 1

‫א‬ C 0 1 0 1 0 1 0 1

 

Y 1 0 1 0 0 0 1 1

 K Y = AB C + AB C + AB C + ABC ‫א‬‫א‬E١٤ J٢F‫א‬ - ٥٢ -

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 

 

‫א‬،BA‫א‬‫א‬AND‫א‬X‫א‬‫א‬‫א‬E١   A

 

 K١ J ‫א‬‫א‬

   

 ١ J‫א‬

 

‫א‬،BA‫א‬‫א‬OR‫א‬X‫א‬‫א‬‫א‬E٢  K١ J ‫א‬‫א‬

 

‫א‬،A,B‫א‬‫א‬NAND‫א‬X‫א‬‫א‬‫א‬ E٣  K٢ J ‫א‬‫א‬

   

 

   

 ٢ J‫א‬ - ٥٣ -

 

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



‫א‬‫א‬،A,B‫א‬‫א‬NOR‫א‬X‫א‬‫א‬‫א‬ E٤  K٣ J ‫א‬‫א‬

  A

 

   

     

   

 ٣ J‫א‬  

‫א‬،BA‫א‬‫א‬XOR‫א‬X‫א‬‫א‬‫א‬ E٥  K٣ J ‫א‬‫א‬

 

‫א‬،A,B‫א‬‫א‬XNOR‫א‬X‫א‬‫א‬‫א‬ E٦  K٣ J ‫א‬‫א‬

 

 K٤ J‫א‬‫א‬‫א‬‫א‬‫א‬E٧ A B C

       

 ٤ J‫א‬

- ٥٤ -

   

‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٨

a) AB + AB c) AB(C + D)

b) AB + AB + ABC d) A + B[C + D(B + C )]

 

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٩  

 ‫א‬ A  0 0 0 0  1 1 1 1

B 0 0 1 1  0 0 1 1

‫א‬ C 0 1 0 1 0 1 0 1

Y 0 1 0 1 0 1 0 1

 

a) (A + B)C c) A(AC + AB)

 W‫א‬‫א‬‫א‬‫א‬‫א‬E١٠

b) (A + B)( B + C) d) A(A + AB)  

- ٥٥ -

‫א‬‫א‬‫א‬

 ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 ‫א‬‫א‬‫א‬  

 W‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬ •

K •

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ •

K(POS)(SOP)‫א‬‫א‬‫א‬ •

K‫א‬(POS)(SOP)‫א‬‫א‬‫א‬ •

K‫א‬(POS)(SOP)‫א‬‫א‬ •

K‫א‬(POS)(SOP)‫א‬‫א‬‫• א‬ K NOR NAND‫א‬‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬ •



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 Introduction١ J٣ ‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ 

‫א‬ ‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬K‫א‬‫א‬،‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬

E‫א‬‫א‬‫א‬F‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬ K‫א‬‫א‬‫א‬‫א‬

 K(K–map)K –‫א‬‫א‬ (Karnaugh-Map)‫א‬  Rules of Boolean Algebra ‫א‬‫א‬‫א‬٢ J٣ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E١ J٣F   K‫א‬

 

1. 3. 5. 7. 9.

A A A A A

+ 0 =A • 0 =0 + A=A • A=A =A

2. A + 1 = 1 4. A • 1 = A 6. A + A = 1 8. A • A = 0 10. A + AB = A

 K‫א‬‫א‬‫א‬‫א‬E١ J٣F‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬



 K‫א‬

‫א‬‫א‬‫א‬A + 0 = A :(1)‫א‬

‫א‬K(0)(1)‫א‬‫א‬،A،‫א‬‫א‬(0) ‫א‬OR‫א‬

(0)‫א‬A=0‫א‬KA‫א‬(1)‫א‬A=1

 K‫א‬‫א‬‫א‬(0)OR‫א‬KA

‫א‬(1)‫א‬OR‫א‬‫א‬‫א‬‫א‬A + 1 = 1:(2)‫א‬

‫א‬OR‫א‬‫א‬(1)K(0)‫א‬(1)‫א‬‫א‬،A،‫א‬

  K ‫א‬‫א‬‫א‬‫א‬‫א‬(1)    .(1) ‫א‬‫א‬(1)OR‫א‬



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



(0)‫א‬AND‫א‬‫א‬‫א‬‫א‬A • 0 = 0:(3)‫א‬

K ‫א‬‫א‬‫א‬‫א‬‫א‬(0)‫א‬‫א‬،A،‫א‬‫א‬

.(0)‫א‬‫א‬(0)AND‫א‬

(1)‫א‬AND‫א‬‫א‬‫א‬‫א‬A • 1 = A:(4)‫א‬

AND‫א‬‫א‬A=0‫א‬‫א‬،(A)‫א‬‫א‬،A،‫א‬‫א‬

‫א‬‫א‬(1)AND‫א‬‫א‬A=1‫א‬‫א‬،(0)

 K‫א‬‫א‬‫א‬(1)AND‫א‬K(1) OR‫א‬‫א‬‫א‬‫א‬A + A = A:(5)‫א‬

‫א‬،0 + 0 = 0A = 0‫א‬‫א‬K  ‫א‬‫א‬‫א‬،A‫א‬  K1 + 1 = 1‫א‬A = 1‫א‬

‫א‬A‫א‬W  ‫א‬ A + A = 1 W(6)‫א‬

A=0‫א‬K(1)‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬ A ‫א‬OR  K1 + 1 = 1 + 0 = 1A = 1‫א‬K 0 + 0 = 0 + 1 = 1

‫א‬‫א‬AND‫א‬‫א‬A‫א‬‫א‬A • A = A:(7)‫א‬

،1 • 1 = 1‫א‬A = 1‫א‬‫א‬،0 • 0 = 0A = 0‫א‬‫א‬K‫א‬  KA‫א‬AND‫א‬‫א‬‫א‬

‫א‬‫א‬ A ‫א‬AND‫א‬A‫א‬ A • A = 0 :(8)‫א‬  A A‫א‬‫א‬‫א‬،(0) ‫א‬‫א‬‫א‬‫א‬

 K (0)‫א‬‫א‬AND‫א‬(0)، ‫א‬(0)

‫א‬‫א‬K  ‫א‬‫א‬‫א‬‫א‬ A = A :(9)‫א‬ (0)(1)‫א‬،(1)A = 0  K‫א‬‫א‬

 W(4)‫א‬(2)‫א‬‫א‬‫א‬:(10)‫א‬ A + AB = A (1 + B) = A (1) =A



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 Demorgan's Theorems٣ J٣ ‫א‬‫א‬‫א‬،‫א‬‫א‬  ‫א‬ ‫א‬‫א‬KOR‫א‬AND‫א‬

 W‫א‬‫א‬،‫א‬‫א‬‫א‬(bars)  

A+B=A•B

  A • B = A + B 

 W‫א‬

 W‫א‬ 

 

AND‫א‬OR‫א‬‫א‬

AND‫א‬‫א‬‫א‬‫א‬NOR‫א‬‫א‬E١ J٣F  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

KE٢ J٣F‫א‬‫א‬‫א‬

 K(negative AND)‫א‬AND‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  

A+B

 

A B

 

 KANDOR‫א‬E١ J٣F‫א‬  ‫א‬

 ‫א‬

 + B A

 • B A

0 0 1 1

0 1 0 1

1 0 0 0

 K‫א‬‫א‬E٢ J٣F‫א‬ OR‫א‬AND‫א‬‫א‬

‫א‬OR‫א‬‫א‬‫א‬‫א‬NAND‫א‬‫א‬E٢ J٣F ‫א‬،E‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬F‫א‬



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



OR‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬KE٣ J٣F‫א‬‫א‬‫א‬

 K(negative OR)‫א‬

≡

 

A+B

 KORAND‫א‬E٢ J٣F‫א‬  ‫א‬

 ‫א‬

 A

 • B A

 + B A

0 0 1 1

0 1 0 1

1 1 1 0

 K‫א‬E٣ J٣F‫א‬ K ‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬

 W‫א‬‫א‬‫א‬WE١ J٣F

 

Y = (A + B + C) • (A + B + C )

 W‫א‬ Y = (A + B + C) • ( A + B + C) = (A + B + C ) + (A + B + C ) = A BC + A BC= AB C+ A BC

 

 W‫א‬‫א‬‫ א‬ WE٢ J٣F Y = (A + B) + CD

 W‫א‬



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



Y = (A + B) + CD = (A + B).CD = (A.B)(C + D) = A B(C + D)

 W‫א‬‫א‬‫ א‬ WE٣ J٣F

 

Y = ( A + BC) + B(A + C )

 W‫א‬  

Y = ( A + BC) + B(A + C )

) = A(BC) • (B + (A + C ) ) (

= (A + BC) • B(A + C ) = A(B + C )(B + A + C )

 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬٤ J٣ Simplification of Boolean Expressions Using Boolean algebra Rules E ‫א‬‫א‬‫א‬F‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ 

  ،‫א‬،‫א‬‫א‬‫א‬

‫א‬،‫א‬ ، ‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬WE٤ J٣F 

Y = AB + A(A + C) + B(A + C)

 



 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬ Y = AB + AA + AC + AB + BC

 

 W‫א‬‫א‬E‫א‬‫א‬‫א‬7‫א‬‫א‬FAAA‫א‬ Y = AB + A + AC + AB + BC

 



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 W‫א‬‫א‬،AB + AB = AB،A + A = A5‫א‬ Y = AB + A + AC + BC

 

 W‫א‬‫א‬‫א‬‫א‬   A‫א‬



Y = A(B + 1 + C) + BC Y = A (1) + BC Y = A + BC

 W،A + 1 = 12‫א‬  W،A(1) = A4‫א‬‫א‬

K‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬

 K‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬E٣ J٣F

‫א‬‫א‬‫א‬‫א‬،EEF‫א‬F‫א‬‫א‬ A B

 

   

 KEEF‫א‬F‫א‬‫א‬

    E F

 

B C

EF

 KE٤ J٣F‫א‬‫א‬‫א‬E٣ J٣F‫א‬

 

‫א‬، ‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬، C BA

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬WE٥ J٣F Y = AB C + ABC + ABC + ABC

 K‫א‬



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 W،‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬W‫א‬  

Y = ( AB C + ABC) + ( ABC + ABC)

= AB( C + C) + BC( A + A)

 W6‫א‬ Y = AB • 1 + BC • 1

 

 W‫א‬‫א‬‫א‬4‫א‬

 

Y = A B + B C 

 

 K‫א‬‫א‬‫א‬‫א‬ E٤ J٣F  

 

B C

          E F

 

A Y

B Y C

EF

 KE٥ J٣F‫א‬‫א‬‫א‬E٤ J٣F‫א‬ Standard Forms of Boolean Expressions ‫א‬‫א‬‫א‬‫א‬٥ J٣ ،،‫א‬ ،‫א‬‫א‬‫א‬ ،(SOP)‫א‬‫א‬(sum-of-products) ‫א‬‫א‬‫א‬‫א‬

‫א‬K(POS)‫א‬‫א‬(product-of-sums)‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 The Sum-of-Products (SOP) form (SOP)‫א‬١ J٥ J٣

 ‫א‬‫א‬ ‫א‬‫א‬K(product term)‫א‬‫א‬

K‫א‬K ABCD , AB AB,‫א‬‫א‬‫א‬

 ‫א‬‫א‬   ‫א‬‫א‬  W(Sum-of-Products)

A BC+ABC+ABC

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬

‫א‬،AND‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬

 KEAND‫א‬‫א‬‫א‬F(0)(1)

 The Product-of-Sums (POS) form (POS)‫א‬٢ J٥ J٣

K(sum term)‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬

K‫א‬K A + B + C , A + B  ‫א‬‫א‬‫א‬‫א‬

   ‫א‬‫א‬  ‫א‬‫א‬ ( A + B + C )(A + B + C )(A + B + C)

 W(Product-of-Sums)

‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬،‫א‬‫א‬ ‫א‬‫א‬

،OR‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬  KEOR‫א‬‫א‬‫א‬F(1)(0)‫א‬

 (POS) ‫א‬‫א‬(SOP)‫א‬‫א‬‫א‬٦ J٣

Converting Standard (SOP) to Standard (POS) ‫א‬ (binary values)‫א‬‫א‬

‫א‬‫א‬‫א‬،K(POS)‫א‬‫א‬‫א‬(SOP)

، K(POS)‫א‬‫א‬‫א‬(SOP)‫א‬‫א‬  W‫א‬‫א‬‫א‬،(POS)‫א‬‫א‬(SOP)‫א‬‫א‬



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



‫א‬‫א‬‫א‬‫א‬،(SOP)‫א‬‫א‬W‫א‬‫א‬  K‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬W‫א‬‫א‬  K(POS)‫א‬

‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬

 K(SOP)‫א‬‫א‬(POS)

 K‫א‬(POS)‫א‬‫א‬‫א‬(SOP)‫א‬WE٦ J٣F Y = A BC+ A BC+ A BC + A BC

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬

 W،(0)‫א‬‫א‬‫א‬،(1)‫א‬

Y = 001 + 011 + 100 + 110 +111

K(23)‫א‬‫א‬ ،‫א‬‫א‬‫א‬ 

‫א‬،‫א‬(SOP)‫א‬

‫א‬،000, 010, 101‫א‬‫א‬‫א‬(POS)  W

Y = (A + B + C)(A + B + C)(A + B + C )

‫א‬‫א‬‫א‬ ،(0)‫א‬‫א‬‫א‬

 K(1)

 K‫א‬(POS)‫א‬‫א‬‫א‬(SOP)‫א‬WE٧ J٣F

 

Y=A BC+A BC+ABC+ABC Y = 000 + 001 + 101 + 110

010, 011, 100, 111

 W‫א‬‫א‬‫א‬W‫א‬  W‫א‬‫א‬‫א‬  W‫א‬‫א‬(POS)‫א‬‫א‬

Y = (A + B + C)(A + B + C )(A + B + C)(A + B + C )

 



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 (SOP) ‫א‬‫א‬(POS)‫א‬‫א‬‫א‬٧ J٣

Converting Standard (POS) to Standard (SOP) ،‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬

‫א‬‫א‬‫א‬K(SOP)‫א‬‫א‬(POS)‫א‬‫א‬‫א‬

 K‫א‬

 K‫א‬(SOP)‫א‬‫א‬‫א‬(POS)‫א‬WE٨ J٣F

Y = (A + B + C)(A + B + C )(A + B + C )(A + B + C )(A + B + C)

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬

 W،(1)‫א‬‫א‬‫א‬،(0)‫א‬

Y = (000)(001)(011)(101)(110) (SOP)‫א‬،‫א‬ (POS)‫א‬

 W‫א‬،010, 100, 111‫א‬‫א‬‫א‬

Y = ABC + AB C + ABC  K‫א‬(SOP)‫א‬‫א‬‫א‬(POS)‫א‬WE٩ J٣F Y = (A + B + C)(A + B + C )(A + B + C )(A + B + C )

 W‫א‬‫א‬‫א‬W‫א‬

Y = (010)(011)(101)(111) Y = 000 001 100 110

 W‫א‬‫א‬‫א‬  W‫א‬‫א‬(SOP)‫א‬‫א‬

Y=A BC+A BC+ABC+ABC

  ‫א‬‫א‬(SOP)‫א‬‫א‬٨ J٣

Converting Standard (SOP) Expressions to Truth Table Format ‫א‬،‫א‬(SOP)‫א‬

،K  ‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬

‫א‬،(23 = 8)‫א‬‫א‬

(1)‫א‬K(24 = 16)‫א‬‫א‬

‫א‬‫א‬(0)،‫א‬‫א‬‫א‬‫א‬(Y)‫א‬  K‫א‬‫א‬K‫א‬



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 

 W‫א‬(SOP)‫א‬‫א‬‫א‬WE١٠ J٣F

 

Y = ABC + AB C + ABC

،‫א‬  ‫א‬‫א‬W‫א‬ ‫א‬‫א‬‫א‬KE٤ J٣F‫א‬‫א‬‫א‬‫א‬ ABC ⇒ 001

AB C ⇒ 100

ABC ⇒ 111

 W‫א‬‫א‬  

،‫א‬(Y)‫א‬(1)،‫א‬‫א‬

 K‫א‬(0)‫א‬‫א‬‫א‬  

 ‫א‬ A  0 0 0 0  1 1 1 1

B 0 0 1 1  0 0 1 1

‫א‬ C 0 1 0 1 0 1 0 1

Y 0 1 0 0 1 0 0 1

 KE١٠ J٣F‫א‬E٤ J٣F‫א‬  W‫א‬(SOP)‫א‬‫א‬‫א‬WE١١ J٣F Y = ABC + ABC + ABC + ABC  W‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬

ABC ⇒ 010

ABC ⇒ 011

ABC ⇒ 101

ABC ⇒ 110

 

(Y)‫א‬(1)،‫א‬‫א‬

 K‫א‬(0)‫א‬‫א‬‫א‬،E٥ J٣F‫א‬



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



   ‫א‬ A  0 0 0 0  1 1 1 1

B 0 0 1 1  0 0 1 1

‫א‬ C 0 1 0 1 0 1 0 1

Y 0 0 1 1 0 1 1 0

 KE١١ J٣F‫א‬E٥ J٣F‫א‬   ‫א‬‫א‬(POS)‫א‬‫א‬٩ J٣

Converting Standard (POS) Expressions to Truth Table Format ‫א‬‫א‬،‫א‬‫א‬،

‫א‬‫א‬ ‫א‬‫א‬‫א‬،‫א‬(POS)  K‫א‬‫א‬

 W‫א‬(POS)‫א‬‫א‬‫א‬WE١٢ J٣F

 

Y = (A + B + C)(A + B + C)(A + B + C)

،‫א‬‫א‬‫א‬‫א‬W‫א‬

‫א‬‫א‬‫א‬KE٦ J٣F‫א‬‫א‬‫א‬‫א‬

 W‫א‬(POS)‫א‬

A + B + C ⇒ 000

A + B + C ⇒ 010

 

A + B + C ⇒ 100

 

،‫א‬(Y)‫א‬(0)،‫א‬‫א‬

 K‫א‬(1)‫א‬‫א‬‫א‬    



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



   ‫א‬ A  0 0 0 0  1 1 1 1

B 0 0 1 1  0 0 1 1

‫א‬ C 0 1 0 1 0 1 0 1

Y 0 1 0 1 0 1 1 1

 KE١٢ J٣F‫א‬E٥ J٣F‫א‬  W‫א‬(POS)‫א‬‫א‬‫א‬WE١٣ J٣F

 

Y = (A + B + C )(A + B + C )(A + B + C)(A + B + C )

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬  

A + B + C ⇒ 001 , A + B + C ⇒ 011 , A + B + C ⇒ 110 , A + B + C ⇒ 111

(Y)‫א‬(0)،‫א‬‫א‬  

 K‫א‬(1)‫א‬‫א‬‫א‬،E٦ J٣F‫א‬  

 ‫א‬ A  0 0 0 0  1 1 1 1

B 0 0 1 1  0 0 1 1

‫א‬ C 0 1 0 1 0 1 0 1

Y 0 1 1 0 1 1 0 0

 KE١٣ J٣F‫א‬E٦ J٣F‫א‬



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



  ‫א‬‫א‬‫א‬‫א‬‫א‬١٠ J٣

Determining Standard Expressions from a Truth Table ‫א‬‫א‬،‫א‬‫א‬(SOP)‫א‬‫א‬

‫א‬(1)‫א‬،‫א‬‫א‬‫א‬K(1)

0101‫א‬‫א‬،K‫א‬‫א‬(0)،  W

0101 ⇒ ABC D

 

  ‫א‬‫א‬،‫א‬‫א‬(POS)‫א‬‫א‬

‫א‬(0)‫א‬،‫א‬‫א‬‫א‬K(0)

1010‫א‬‫א‬،K ‫א‬‫א‬(1)،  W

1010 ⇒ A + B + C + D

 

 W(POS)،(SOP)‫א‬‫א‬‫א‬،E٧ J٣F‫א‬WE١٤ J٣F

 

 ‫א‬ A  0 0 0 0  1 1 1 1

B 0 0 1 1  0 0 1 1

‫א‬ C 0 1 0 1 0 1 0 1

Y 0 0 0 1 1 0 1 1

 KE١٤ J٣F‫א‬E٧ J٣F‫א‬ K011, 100, 110, and 111W  ‫א‬‫א‬‫א‬‫א‬1'sW ‫א‬ 011⇒ ABC 

 W‫א‬‫א‬

100 ⇒ AB C 

110 ⇒ ABC 

111⇒ ABC



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 W(Y)(SOP)‫א‬‫א‬

 

Y = ABC + AB C + ABC + ABC

‫א‬K and 101 010 001000‫א‬‫א‬(0)‫א‬،(POS)

 W‫א‬  

000 ⇒ A + B + C  001 ⇒ A + B + C  010 ⇒ A + B + C  101 ⇒ A + B + C

 W(Y)(POS)‫א‬‫א‬  

  Y = (A + B + C)(A + B + C )(A + B + C)(A + B + C )  

  NOR ‫ و‬NAND‫א‬‫א‬‫א‬‫א‬١١ J٣

The Universal Property of NAND and NOR Gates ‫א‬،AND‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬NOR‫א‬NAND‫א‬‫א‬‫א‬K‫א‬‫א‬،OR

‫א‬‫א‬‫א‬K   (Universal Gates)

KNOR،AND‫א‬‫א‬‫א‬NAND‫א‬، NOR‫א‬‫א‬‫א‬NOR‫א‬

 KNANDORAND‫א‬‫א‬‫א‬  

 NAND‫א‬‫א‬١ J١١ J٣   NAND gate as a Universal Logic Element  ،AND،‫א‬‫א‬‫א‬‫א‬NAND‫א‬‫א‬

NAND‫א‬‫א‬‫א‬KNOR،OR

K  ‫א‬NAND‫א‬EEF٥ J٣F ‫א‬‫א‬‫א‬

KEEF٥ J٣F  NAND‫א‬‫א‬AND



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



NOR  ‫א‬‫א‬‫א‬KEEF٥ J٣FNAND‫א‬ ‫א‬OR‫א‬‫א‬  KEEF٥ J٣F

≡

 

A

E F  

  A

AB = AB

 

≡

EF

 

  A .B = A + B

 

≡

A+B

E F  A B

A B

 

A .B = A + B

 

A+B

≡

A B

A+B

  EF

 

 KNAND‫א‬‫א‬‫א‬E٥ J٣F‫א‬ NOR Gate as a Universal Logic Element NOR‫א‬‫א‬٢ J١١ J٣

،ORAND،  ‫א‬‫א‬‫א‬NOR  ‫א‬‫א‬،NAND  ‫א‬

NOT‫א‬NOR‫א‬‫א‬E٦ J٣FKNAND‫א‬

 KNAND‫א‬OR‫א‬  



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



   

 

≡

 

E F

 

A+B A

A+B = A+B

 

≡

A+B

 

EF

  A

 

A + B = A.B

 

B B

≡

 EF   A

A + B = A.B

   

B B

≡

A B

 

EF

 KNOR‫א‬‫א‬‫א‬E٦ J٣F‫א‬ NOR , NAND‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬١٢ J٣ Design of Combinational Logic Circuits using NAND and NOR Gates



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



‫א‬‫א‬NOR‫א‬،NAND‫א‬‫א‬‫א‬

،(Negative - OR)‫א‬OR‫א‬‫א‬NAND‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬K(Negative - AND)‫א‬AND‫א‬‫א‬NOR‫א‬‫א‬

 K‫א‬ (Logic diagram) ‫א‬‫א‬‫א‬‫א‬AND،OR‫א‬ NAND LogicNAND‫א‬‫א‬‫א‬١ J١٢ J٣

‫א‬،‫א‬OR ‫א‬NAND  ‫א‬NAND‫א‬،

 W‫א‬

 A • B = A + B Negative-OR

NAND

   

 KE٧ J٣F‫א‬‫א‬‫א‬‫א‬‫א‬

 

 

Y = AB + CD

C D

 

   

 KNAND‫א‬‫א‬‫א‬‫א‬‫א‬E٧ J٣F‫א‬  

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬(Y)‫א‬‫א‬ Y = (AB)(CD)

 W‫א‬ Y = AB + CD

 W(bars)‫א‬‫א‬‫א‬ Y = AB + CD

AND‫א‬  ،AB+CD ،(Y)‫א‬‫א‬

‫א‬NAND‫א‬‫א‬(Y)‫א‬‫א‬‫א‬.OR‫א‬

KOR‫א‬‫א‬NAND‫א‬AND‫א‬E٧ J٣F



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



NAND‫א‬‫א‬‫א‬‫א‬EEF٨ J٣F‫א‬(Y)‫א‬‫א‬

 ‫א‬‫א‬K ‫א‬OR ‫א‬‫א‬

‫א‬‫א‬E٧ J٣F‫א‬‫א‬،EEF٨ J٣F‫א‬

 W‫א‬،EEF٨ J٣F

 (NAND-NAND-NAND)(AND-AND-OR) A B

≡

Y = AB + CD

C D

 

E F



 

Y = AB + CD

EF

   

  .(٧-٣) ‫ ﺗﻜﺎﻓﺊ اﻟﺪاﺋﺮة ﻓﻲ ﺷﻜﻞ‬AND-AND-OR ‫( إﺛﺒﺎت أن‬٨-٣) ‫اﻟﺸﻜﻞ‬ ‫א‬‫א‬‫א‬NAND‫א‬‫א‬‫א‬‫א‬E٩ J٣F  K‫א‬ JOR‫א‬‫א‬‫א‬

 

B C

ABC

 

DE F

 

K‫א‬ JOR‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬E٩ J٣F‫א‬

   WE٩ J٣F‫א‬(Y)‫א‬ 

Y = [(AB)C] • [(DE)F] = [(A + B)C] • [( D + E )F] = ( A + B )C + ( D + E ) F ∴ Y = ( A + B )C + ( D + E ) F

 

  ‫א‬ ‫א‬‫א‬   NAND‫א‬ ‫א‬ ‫א‬ JOR ‫א‬ ‫א‬

 K‫א‬‫א‬‫א‬(Y)‫א‬،E١٠ J٣F  



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



A+B

 

( A + B )C

   

Y = ( A + B )C + ( D + E )F

 

D+E

 

E ( D + E )F

 

K‫א‬ JOR‫א‬‫א‬E٩ J٣F ‫א‬‫א‬‫ א‬E١٠ J٣F‫א‬

 

 WNAND‫א‬ ‫א‬‫א‬‫א‬WE١٥ J٣F

(a ) Y = ABC + DE

(b) Y = ABC + D + E

 KE١١ J٣F‫א‬‫א‬W‫א‬

ABC

C D E

 

B Y = ABC + DE

E F

D E

 

ABC

Y = ABC + D + E

 

EF

 

KE١٥ J٣F‫א‬‫א‬‫א‬‫ א‬E١١ J٣F ‫א‬

   

 NOR Logic NOR‫א‬‫א‬‫א‬٢ J١٢ J٣

‫א‬ JAND‫א‬NOR‫א‬NOR‫א‬‫א‬

 W‫א‬  

 A + B = A • B NOR

Negative-AND 

 

 KE١٢ J٣F‫א‬‫א‬‫א‬‫א‬

     



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 

A+B

 

(A + B) (C + D)

 

C D

 

C+D

 

NOR‫א‬‫א‬‫א‬‫א‬‫א‬E١٢ J٣F‫א‬

 

 W‫א‬‫א‬‫א‬‫א‬‫א‬ Y = (A + B) + (C + D)

 W‫א‬

Y = (A + B) • (C + D)

 W‫א‬‫א‬‫א‬

Y = (A + B) • (C + D)

 

‫א‬،AND‫א‬OR‫א‬(A + B)(C + D)‫א‬

AND‫א‬‫א‬‫א‬‫א‬OR‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬EEF١٣ J٣F ‫א‬‫א‬‫א‬KEEF١٣ J٣F

 K‫א‬ JAND  

A+B

(A + B) (C + D)

C D

C+D

E F

≡    

B C

   

A+B

(A + B)  (C  + D)

 

EF

 K‫א‬ JAND‫א‬‫א‬E١٢ J٣F‫א‬‫א‬‫א‬E١٣ J٣F‫א‬

 

‫א‬‫א‬‫א‬،NOR‫א‬‫א‬‫א‬ ‫א‬E١٤ J٣F

 W‫א‬(Y)‫א‬ K‫א‬ JAND‫א‬‫א‬



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



  Y = [(A + B) + C] + [(D + E) + F]

 

= [ AB + C] + [ D E + F] = ( AB + C)( D E + F) A+B

A B

 

   

 

( A + B) + C

 

D+E

 

E ( D + E) + F

KNOR‫א‬‫א‬‫א‬‫א‬E١٤ J٣F‫א‬

       

 KE١٥ J٣F‫א‬‫א‬NOR‫א‬‫א‬‫א‬ JAND‫א‬‫א‬

E F

 

A B + C

 

C D

 

  DE

Y = ( A B + C) (D E + F)

   

DE + F

 

 KE١٤ J٣F‫א‬‫א‬‫א‬‫א‬E١٥ J٣F‫א‬



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 WNOR‫א‬‫א‬‫א‬‫א‬‫א‬WE١٦ J٣F Y = A B C + (D + E)

  A B C D E

 KE١٦ J٣F‫א‬‫א‬W‫א‬

 

A+B+C=A BC

   

Y = A B C + (D + E)

   

 KNOR‫א‬‫א‬‫א‬‫א‬‫א‬E١٦ J٣F‫א‬  Karnaugh Map١٣ J٣ ‫א‬‫א‬،‫א‬‫א‬‫א‬K- ‫א‬K‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬   ‫א‬،‫א‬‫א‬‫א‬  KK‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬

،‫א‬K‫א‬

‫א‬‫א‬  ،(cells)‫א‬ (array)

‫א‬‫א‬‫א‬‫א‬K‫א‬  K‫א‬

،‫א‬‫א‬‫א‬‫א‬

K  ‫א‬‫א‬،‫א‬

‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬ (Quine - McClusky) ‫א‬‫א‬‫א‬K‫א‬



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



‫א‬‫א‬K ‫א‬‫א‬،

 K 2 4 = 16 ‫א‬‫א‬ 2 3 = 8   ‫א‬١ J١٣ J٣

Karnaugh Map for Two, Three, and Four Variables

KE‫א‬F‫א‬‫א‬‫א‬

  ( A, B )‫א‬(A‫ و‬B)‫א‬،E١٧ J٣F  KE11100100FE‫א‬F  

AB AB

   K‫א‬E١٧ J٣F‫א‬  K ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬

E١٨ J٣F‫א‬(Input Labels)‫א‬

 ‫א‬‫א‬‫א‬،K‫א‬‫א‬ ،‫א‬‫א‬ A 

‫א‬‫א‬ B ‫א‬‫א‬K‫א‬‫א‬A‫א‬ ‫א‬‫א‬‫א‬‫א‬،K‫א‬‫א‬‫א‬B‫א‬،‫א‬

 K AB ‫א‬

 B

 

 KE22 = 4FE١٨ J٣F‫א‬



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



،E F‫א‬EEF١٩ J٣F،EEF١٩ J٣F

 KEF‫א‬  

AB BC

B C

 

 AB

  AB

  E F

 

EF

 K‫א‬E١٩ J٣F‫א‬ Karnaugh Map (SOP) Minimization

(SOP)‫א‬‫א‬٢ J١٣ J٣

،‫א‬

‫א‬،K(SOP)‫א‬‫א‬‫א‬  KEEF٢٠ J٣F‫א‬‫א‬

،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

(SOP)‫א‬‫א‬‫א‬‫א‬(1)‫א‬‫א‬

 KEEF٢٠ J٣F

‫א‬‫א‬KEEF٢٠ J٣F ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 KEEF٢٠ J٣F‫א‬‫א‬‫א‬

‫א‬‫א‬

‫א‬(1)‫א‬K ‫א‬‫א‬‫א‬‫א‬

(0)‫א‬(0)‫א‬،‫א‬‫א‬(1)

 K  ‫א‬‫א‬ ‫א‬‫א‬(1)



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



‫א‬‫א‬KEABF‫א‬‫א‬‫א‬،E A B F‫א‬

 K‫א‬‫א‬(0)،‫א‬(0)( AB, A B )  

A A B B

 ‫א‬

‫א‬ Y

Y=AB+AB

EF

EF

E F B

B 

 

1 

 

A A B B Y=A

 EFEFEF  K‫א‬‫א‬‫א‬‫א‬E٢٠ J٣F‫א‬ ‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬K A + A = 1 ‫א‬،EComplements)

 K‫א‬‫א‬‫א‬‫א‬‫א‬،EEF٢٠ J٣F  ‫א‬‫א‬EEF٢٠ J٣F ‫א‬

‫א‬ ‫א‬K‫א‬(adjacent cells)

‫א‬‫א‬‫א‬K  ‫א‬،‫א‬

‫א‬‫א‬‫א‬EEF٢٠ J٣F‫א‬(1)



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 B  B  AB, A B ‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬  WA،‫א‬،‫א‬

E‫א‬‫א‬‫א‬F Y = A B + AB  Y = A(B + B) = A •1 = A

EEF٢٠ J٣F‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬K(A)‫א‬(Y)‫א‬‫א‬

 KEEF٢٠ J٣F

 EEF٢١ J٣F‫א‬‫א‬‫א‬WE١٧ J٣F  K‫א‬

،‫א‬‫א‬‫א‬،‫א‬W‫א‬  KEEF٢١ J٣F

EEF٢١ J٣F  ‫א‬(1)‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬ ‫א‬‫א‬،‫א‬‫א‬‫א‬(0)KEEF٢١ J٣F

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،EEF٢١ J٣F

 ‫א‬E‫א‬‫א‬F  A , A ‫א‬‫א‬‫א‬K  K A B ‫א‬ C , C ‫א‬‫א‬‫א‬، BC ‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬KEF٢١ J٣،‫א‬ ‫א‬OR‫א‬‫א‬AND‫א‬

‫א‬‫א‬‫א‬،١٦‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬،OR‫א‬AND‫א‬

 KEEF٢١ J٣F‫א‬٦‫א‬‫א‬      



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



‫א‬ A 0 0 0 0 1 1 1 1

B C 0 0 0 1 1 0 1 1 0 0 0 1 1 0 1 1   E F

‫א‬ Y 0 0 1 0 1 1 1 0

A A B B C C

BC 1

   

 

A 1 1 1  

 

Y = AB + B C A B  BC   EF EF

 K‫א‬‫א‬‫א‬‫א‬E٢١ J٣F‫א‬

E‫א‬F‫א‬(1's)‫א‬

،‫א‬E٢٢ J٣FK2 ‫א‬‫א‬،، ‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬E1'sF‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬ K‫א‬

‫א‬‫א‬F‫א‬‫א‬،‫א‬ ‫א‬،‫א‬‫א‬، KE‫א‬‫א‬‫א‬    



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



  AB

C D CD

 

AB C

 

AD Y = A B C D + A B C D + A B CD + A B C D + AB C D + ABC D + AB C D + AB C D + ABCD + A B C D + A B CD Y = AB C + AD + AB D + A B

E F

Y = AC + B C + D

E‫א‬F

EF

E‫א‬F

E‫א‬F

     

C D CD

E‫א‬F

     

   

Y = A B C D + ABC D + ABC D + ABCD

+ AB C D + ABCD + A B CD + ABCD Y = B+D

EF

Y = A B C D + A B C D + A B CD + A B C D + A B C D + A B C D + AB C D + A B C D

+ A B C D + A B CD + A B C D

D 1

+ ABCD + ABC D + AB C D + ABC D

Y = A B C D + A B CD + A B C D + AB C D

E‫א‬F E‫א‬F

C D C D CD

 

   

+ ABC D + ABC D + ABCD + AB C D  + AB C D + ABCD + ABCD E‫א‬F Y = C D + AB + B D

E‫א‬F

EF

K‫א‬‫א‬E٢٢ J٣F‫א‬



     



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



‫א‬‫א‬‫א‬(SOP)‫א‬‫א‬‫א‬‫א‬‫א‬W١٨ J٣

 K‫א‬،E٨ J٣F  

 ‫א‬ A 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

B 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1

C 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

‫א‬ D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

Y 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 1

 KE١٨ J٣F‫א‬‫א‬‫א‬‫א‬E٨ J٣F‫א‬  ‫א‬‫א‬‫א‬‫א‬W‫א‬ (Y)‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬‫א‬،(1)‫א‬‫א‬

 

Y = A BCD + A BCD + ABCD + ABCD + A BCD + ABCD

 

،E٢٣ J٣F‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬(Y)‫א‬‫א‬‫א‬      



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 

  CD

 KE١٨ J٣F‫א‬E٢٣ J٣F‫א‬

 

‫א‬E٢٣ J٣F   

‫א‬‫א‬‫א‬‫א‬  K(1's)‫א‬

K AD ‫א‬ C ‫א‬ C   ‫א‬ B ‫א‬ B ‫א‬ ‫א‬‫א‬‫א‬

 W‫א‬‫א‬‫א‬K CD ‫א‬ A  A  B  B ‫א‬‫א‬

Y = AD + CD Karnaugh Map (POS) Minimization

(POS)‫א‬‫א‬٣ J١٣ J٣

‫א‬،(SOP)‫א‬‫א‬‫א‬ ‫א‬

 K(POS)‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬(POS)‫א‬‫א‬‫א‬‫א‬‫א‬W١٩ J٣

 K‫א‬،E٩ J٣F

 ‫א‬‫א‬‫א‬‫א‬W‫א‬ ‫א‬‫א‬،(POS)

(POS)‫א‬،(0)‫א‬‫א‬(Y)‫א‬‫א‬

 W‫א‬‫א‬



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



Y = (A + B + C + D)(A + B + C + D )(A + B + C + D)(A + B + C + D ) (A + B + C + D )(A + B + C + D)(A + B + C + D)

 ‫א‬ A 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

B 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1

C 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

‫א‬ D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

Y 0 1 0 1 0 0 1 0 0 1 1 1 0 1 1 1

 KE١٩ J٣F‫א‬‫א‬‫א‬‫א‬E٩ J٣F‫א‬

 

،E٢٤ J٣F ‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬(Y)‫א‬‫א‬‫א‬   ‫א‬     E٢٤ J٣F      

K   ‫א‬ ‫א‬ ،(0's) ‫א‬      ،   B ‫א‬ B ‫א‬     ‫א‬ ‫א‬ ‫א‬ ‫א‬   ‫א‬ ‫א‬KA+C ‫א‬ D  ‫א‬ D ‫א‬

KC+D‫א‬ A  A  B  B ‫א‬‫א‬  ‫א‬ ، A + B + D ‫א‬، C ‫א‬C‫א‬‫א‬

 W(POS)‫א‬‫א‬‫א‬

Y = (C + D)(A + C)(A + B + D )



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 A + C AB

 1

C+D

 

 1

  A+B+D

     

   

 KE١٩ J٣F‫א‬E٢٤ J٣F‫א‬                              



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



   

 W‫א‬‫א‬‫א‬E١ b) AB(CD + EF)

a) AB(C + D ) c) (A + B + C + D ) + ABCD

d) (A + B + C + D) (AB C D)

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ E٢

a) F = AB + A(B + C) + B(B + C) b) F = [AB(C + BD) + AB]CD

c) F = A B C + A B C + A B C + A B C d) F = A B + A C + A B C :‫( اﻟﻘﻴﺎﺳﻴﺔ‬POS) ‫( اﻵﺗﻴﺔ إﻟﻰ اﻟﺘﻌﺒﻴﺮات‬SOP) ‫ﺣﻮل اﻟﺘﻌﺒﻴﺮات اﻟﻘﻴﺎﺳﻴﺔ‬E٣ a) F = A B C + A B C + A B C + A B C b) F = A B C + A B C + A B C + A B C c) F = A B C + A B C + A B C + A B C + A B C

W‫א‬(SOP)‫א‬‫א‬‫א‬(POS)‫א‬‫א‬‫א‬E٤ a) F = (A + B + C)(A + B + C )( A + B + C )( A + B + C) b) F = (A + B + C)(A + B + C)(A + B + C )(A + B + C) c) F = (A + B + C )(A + B + C)( A + B + C )( A + B + C)( A + B + C )

W‫א‬(SOP)‫א‬‫א‬‫א‬‫א‬E٥ a) F = A B C + A B C + A B C + A B C + A B C b) F = A B C + A B C + A B C + A B C c) F = A B C + A B C + A B C + A B C + A B C



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



 W‫א‬(POS)‫א‬‫א‬‫א‬‫א‬E٦ a) F = (A + B + C)(A + B + C)(A + B + C )(A + B + C) b) F = (A + B + C )(A + B + C)(A + B + C )(A + B + C) c) F = (A + B + C)(A + B + C)(A + B + C )(A + B + C)(A + B + C )

W‫א‬‫א‬ (POS),(SOP) ‫א‬‫א‬‫א‬‫א‬E٧

A 0 0 0 0 1 1 1 1

B 0 0 1 1 0 0 1 1

C 0 1 0 1 0 1 0 1

F 1 1 0 0 1 0 1 0

   WNAND‫א‬ ‫א‬‫א‬‫א‬‫א‬  E٨ a) ABCD + DE b) ABC + AB + D c) AB C + D + E d) ABC + ABC + ABC + ABC   WNOR‫א‬ ‫א‬‫א‬‫א‬‫א‬E٩ a) (A + B + C) (A + B) b) ABC + (D + E ) c) (AB + C) (DE + F)

d) (A + B) + ( C + D)

،(SOP)‫א‬ ‫א‬E١٠  W‫א‬‫א‬

 



‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬



‫א‬

‫א‬

 

 W(SOP), (POS)‫א‬‫א‬‫א‬‫א‬  ‫א‬E١١ a) F1 = AB C D + ABCD + ABCD + ABC D + ABC D + ABCD b) F2 = ABCD + ABC D + AB C D + ABC D + ABCD + ABCD + ABCD c) F3 = ABC D + ABC D + ABC D + ABC D + AB C D

       

d) F4 = AB C D + + AB C D + ABC D + ABC D + ABCD + ABCD + AB C D + ABCD          

‫א‬‫א‬‫א‬

 ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 ‫א‬‫א‬‫א‬

 

 W‫א‬‫א‬‫א‬

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ •

K‫א‬‫א‬(Decoder)‫א‬‫א‬ •

K‫א‬‫א‬(Encoder)‫א‬‫א‬‫א‬ •

K‫א‬‫א‬(Multiplexer)‫א‬‫א‬‫א‬ •

K‫א‬‫א‬(Demultiplexer)‫א‬‫א‬ • K‫א‬‫א‬(Comparator)‫א‬‫א‬ •

                           

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 Introduction ١ J٤ ‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

،‫א‬‫א‬‫א‬‫א‬  ‫א‬

،‫א‬‫א‬‫א‬‫א‬‫א‬K ‫א‬‫א‬‫א‬‫א‬‫א‬،  K‫א‬‫א‬‫א‬(10)‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

،‫א‬ ‫א‬‫א‬  ‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬

 ‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬،‫א‬‫א‬‫א‬

 K‫א‬                                  

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 Binary Adders and Subtractors ‫א‬‫א‬‫א‬‫א‬٢ J٤   ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

The Half-Adder Circuit ‫א‬‫א‬‫א‬١ J٢ J٤

‫א‬‫א‬‫א‬E١ J٤F‫א‬،‫א‬‫א‬‫א‬‫א‬

 .[Carry (C)]‫א‬‫א‬‫א‬[Sum(S)]‫א‬‫א‬ A , B ‫א‬  

 ‫א‬

 ‫א‬

 

A 0

B 0

S 0

C 0

0 + 0 = 0 

0 + 1 = 1 

1 + 0 = 1 

1 + 1 = 102 or 210 1  0 ‫א‬

 K‫א‬‫א‬‫א‬‫א‬E١ J٤F‫א‬

‫א‬‫א‬K(XOR)‫א‬‫א‬‫א‬(S )‫א‬‫א‬

EEF١ J٤FKAND‫א‬‫א‬(C)‫א‬

K‫א‬‫א‬‫א‬ C, S ‫א‬‫א‬ A, B ‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬ A

A B

S(sum) ‫א‬

 

≡

 

C(carry) ‫א‬

 

S C  EF EF

 K‫א‬‫א‬‫א‬‫א‬E١ J٤F‫א‬

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬EEF١ J٤F‫א‬‫א‬‫א‬‫א‬‫א‬

CS‫א‬‫א‬‫א‬‫א‬K  ‫א‬‫א‬(Half Adder) HA

 W‫א‬،‫א‬‫א‬

 

 S = AB + A B 



C = AB





 The Full-Adder Circuit ‫א‬‫א‬‫א‬٢ J٢ J٤

   ‫א‬‫א‬‫א‬‫א‬ (2-bits)

،‫א‬‫א‬‫א‬(carry)‫א‬‫א‬‫א‬‫א‬

 ‫א‬‫א‬‫א‬   (3-bits) 

‫א‬،‫א‬ ‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬

(3-bits)‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬BA‫א‬‫א‬،،‫א‬

‫א‬‫א‬‫א‬‫א‬   ‫א‬EInput carryFCin‫א‬‫א‬

‫א‬‫א‬‫א‬.(Sum ٍ )‫א‬ ، (Carry)‫א‬K‫א‬

 KE٢ J٤F‫א‬‫א‬

 

‫א‬

‫א‬

A 0

B 0

Cin 0

S 0

C 0

0+0+0=0



0+0+1=1



0+1+0=1



0 + 1 + 1 = 102 or 210 1  0‫א‬

1+0+0=1



1 + 0 + 1 = 102 or 210 1  0 ‫א‬ 1 + 1 + 0 = 102 or 210 1  0 ‫א‬ 1 + 1 + 1 = 112 or 310 1  1 ‫א‬

 K‫א‬‫א‬‫א‬‫א‬E٢ J٤F‫א‬

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬ A, B, C ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬ C, S ‫א‬‫א‬K‫א‬E 2 3 = 8 F‫א‬

K‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬ S , C ‫א‬‫א‬‫א‬‫א‬  

S = A BC in + ABC in + A BC in + ABC in C = ABC in + A BC in + ABC in + ABC in

 

‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 WS‫א‬‫א‬‫א‬‫א‬  

S = A BC in + ABC in + A BC in + ABC in = (AB + A B)C in + (A B + AB)C in

 

XNOR

  A B + AB  ‫א‬‫א‬،XOR AB + A B ‫א‬‫א‬

 W‫א‬‫א‬‫א‬‫א‬  

S = (A ⊕ B)C in + (A ⊕ B)C in

   Cin ‫א‬ (A ⊕ B) XOR‫א‬

 

 WS‫א‬‫א‬

 S = (A ⊕ B) ⊕ C in = A ⊕ B ⊕ C in

 

‫א‬ A, B ‫א‬، XOR ‫א‬‫א‬ S 

 

 K Cin ‫א‬

 W‫א‬‫א‬ C ‫א‬

 

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



C = ABC in + A BC in + ABC in + ABC in = (AB + A B)C in + AB(C in + C in ) = (A ⊕ B)C in + AB ⇐ ( Cin + C in = 1)

 

‫א‬‫א‬‫א‬KEEF٢ J٤F‫א‬CS

‫א‬FA‫א‬EEF٢ J٤F‫א‬‫א‬

 K‫א‬‫א‬ (Full Adder)

  A B

 

B Cin

S(sum)

≡

 

  Cin

C(carry)

 

E F

EF

 K‫א‬‫א‬‫א‬‫א‬E٢ J٤F‫א‬

‫א‬‫א‬‫א‬EEF٢ J٤F‫א‬‫א‬

OR‫א‬2‫א‬‫א‬ ‫א‬‫א‬OR‫א‬‫א‬  KE٣ J٤F‫א‬

Cin S

HA B

 

HA C

 

  K‫א‬‫א‬‫א‬E٣ J٤F‫א‬

 

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 Half Subtractor Circuit ‫א‬‫א‬‫א‬٣ J٢ J٤

‫א‬‫א‬‫א‬

K ‫א‬‫א‬‫א‬    ‫א‬‫א‬K K  ‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬(bit)،‫א‬

،‫א‬‫א‬‫א‬K(difference)‫א‬‫א‬(bit)

K ‫א‬‫א‬(Borrowed)(1)‫א‬  K،

‫א‬(2-bits)‫א‬‫א‬‫א‬‫א‬

A  K  ‫א‬‫א‬(1)

 KB‫א‬

، A ≥ B KB, A  ‫א‬(A – B)

 ‫א‬ ‫א‬.(Difference bit) ‫א‬‫א‬0 – 0 = 0, 1 – 0 = 1, 1 – 1 = 0W

‫א‬‫א‬‫א‬K ‫א‬‫א‬(1)‫א‬‫א‬‫א‬،0 – 1A < B (10)‫א‬،‫א‬‫א‬،‫א‬2  K2 – 1 = 1‫א‬،(2)‫א‬،‫א‬

‫א‬‫א‬ (D)‫א‬،‫א‬‫א‬

 K(B0)‫א‬‫א‬

‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬( B 0 )‫א‬،(D)‫א‬‫א‬KE٣ J٤F  W‫א‬

D = AB + A B B 0 = AB

‫א‬‫א‬(S)‫א‬(D)‫א‬

A ‫א‬  ‫א‬ ‫א‬  (C) ‫א‬   ( B 0 ) ‫א‬  ، XOR ‫א‬   K B  A ‫א‬ AND ‫א‬( B 0 )‫א‬

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 ‫א‬ A 0 0 1 1

 ‫א‬

B 0 1 0 1

D 0 1 1 0

B0 0 1 0 0

 K‫א‬‫א‬‫א‬‫א‬E٣ J٤F‫א‬ ‫א‬EEF٤ J٤F‫א‬،‫א‬‫א‬EEF٤ J٤F‫א‬  K(Half Subtractor)‫א‬ HS ‫א‬،‫א‬ A A B

D(difference) ‫א‬

   

≡

E F

 

B0(borrow) ‫א‬

  K‫א‬‫א‬‫א‬‫ א‬E٤ J٤F ‫א‬

 

EF D

 

 The Full-Subtractor Circuit ‫א‬‫א‬‫א‬٤ J٢ J٤

‫א‬ (2-bits)‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬K ‫א‬‫א‬K ‫א‬‫א‬(1)‫א‬

K  ‫א‬( B i n )  ‫א‬‫א‬(B)‫א‬(A)‫א‬ A, B, B in ‫א‬

 KE٤ J٤F‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬ D, B 0 ‫א‬ ‫א‬ 0's,1's ‫א‬‫א‬‫א‬‫א‬‫א‬

K A − B − B in ‫א‬‫א‬‫א‬ 0's,1's K‫א‬‫א‬

K‫א‬‫א‬‫א‬‫א‬ B in = 0 ‫א‬‫א‬

(2) B 0 = 1‫א‬‫א‬‫א‬(1) A = 0, B = 0, B in = 1  K D = 1 ،2 – 0 – 1 = 1،A

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬

A 0 0 0 0 1 1 1 1

B 0 0 1 1 0 0 1 1

‫א‬

Bin 0 1 0 1 0 1 0 1

D 0 1 1 0 1 0 0 1

B0 0 1 1 1 0 0 0 1

 K‫א‬‫א‬‫א‬‫א‬E٤ J٤F‫א‬ ‫א‬‫א‬‫א‬(1) A = 0, B = 1, B in = 1 

 K D = 0 ،2 – 1 – 1 = 0،A = 2 B 0 = 1

K D = 0 ، B 0 = 0 ‫א‬A – B – Bin = 0، A = 1, B = 0, B in = 1 

‫א‬‫א‬‫א‬(1) A = 1, B = 1, B in = 1 ‫א‬  K D = 1 ،3 – 1 – 1 = 1،A=3، B 0 = 1

 W‫א‬‫א‬‫א‬‫א‬‫א‬ D = ABB in + ABBin + AB Bin + ABBin

   

‫א‬‫א‬،‫א‬‫א‬(S)  D = (A ⊕ B) ⊕ Bin = A ⊕ B ⊕ Bin

 W‫א‬

 W‫א‬‫א‬،(B0)‫א‬ B 0 = ABB in + ABBin + ABB in + ABBin = B in ( AB + AB) + AB( Bin + B in ) B 0 = B in (A ⊕ B) + AB ⇐ ( Bin + B in = 1)



     

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬‫א‬‫א‬‫א‬،EEF٥ J٤F (B0), (D)‫א‬

(Full Subtractor)‫א‬FS‫א‬،EEF٥ J٤F‫א‬

 K‫א‬‫א‬

‫א‬‫א‬‫א‬EEF٥ J٤F ‫א‬‫א‬‫א‬

‫א‬2‫א‬‫א‬‫א‬‫א‬،OR‫א‬‫א‬

 KE٦ J٤F‫א‬OR  

A B

≡

 K‫א‬‫א‬‫א‬‫א‬E٥ J٤F‫א‬

A D

HS B

 

  D

 

HS A

 

EF  

E F

Bin



 

Bin

Bin

   

K‫א‬‫א‬‫א‬ E٦ J٤F‫א‬  

       

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 Decoder‫א‬٣ J٤ (bits)‫א‬‫א‬‫א‬‫א‬‫א‬   K‫א‬‫א‬‫א‬،‫א‬

،(n)‫א‬‫א‬



‫א‬،‫א‬‫א‬‫א‬



 K(2n)‫א‬،(n)

،AND‫א‬‫א‬‫א‬K(1001)‫א‬‫א‬

،(1)‫א‬‫א‬(1) AND‫א‬‫א‬

‫א‬،(1001)‫א‬(1)AND‫א‬‫א‬‫א‬  KE٧ J٤F (0's)‫א‬‫א‬

1 0

0 1

E F

A0 A1

(LSB)

 

A 1

A2 A3

 

Y=A A A A 3 2 1 0

 

EF

 

(MSB)

 K(1001)AND‫א‬‫א‬E٧ J٤F‫א‬

EEF٧ J٤F‫א‬‫א‬‫א‬‫א‬

    

‫א‬ ‫א‬(0)AND‫א‬‫א‬KEEF٧ J٤F  KA0 = 1, A1 = 0, A2 = 0, and A3 = 1W‫א‬

 K‫א‬‫א‬E٨ J٤F A0 A1

2-to-4 line decoder

D0 D1 D2 D3

K‫א‬‫ א‬E٨ J٤F‫א‬

         



 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬،(2-bits)‫א‬‫א‬

E ‫א‬FE ‫א‬F ‫א‬‫א‬K(22 = 4)AND  K‫א‬‫א‬‫א‬‫א‬E٥ J٤FK(2-to-4 line decoder)  

 ‫א‬ A1 0 0 1 1

 ‫א‬

A0 0 1 0 1

D0 D1 D2 D3 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

 K‫א‬E٥ J٤F‫א‬

AND‫א‬‫א‬‫א‬‫א‬‫א‬  D 0 = A1 A 0

D1 = A 1 A 0

 W

D 3 = A1 A 0

D 2 = A1A 0



‫א‬   ‫א‬‫א‬‫א‬E٩ J٤F

 KE٥ J٤F‫א‬

 

  D0 D1 D2 D3

 K‫א‬‫א‬‫א‬E٩ J٤F‫א‬

 

         

‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬،‫א‬

 K‫א‬‫א‬‫א‬،‫א‬  

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 Encoder‫א‬٤ J٤ K ‫א‬‫א‬‫א‬‫א‬‫א‬  ،(digit)‫א‬‫א‬

K ‫א‬‫א‬‫א‬‫א‬‫א‬K  ‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬E١٠ J٤F



 K‫א‬‫א‬‫א‬‫א‬K‫א‬

 

 

D1 Octal input digits

D2 D3

 8-to-3 

line encoder

D4 D5 D6 D7

 

Binary outputs

   

  ‫א‬‫א‬‫א‬ E١٠ J٤F‫א‬

‫א‬‫א‬،E١٠ J٤F‫א‬‫א‬‫א‬E٦ J٤F

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  

‫א‬

 ‫א‬

‫اﻷرﻗﺎم اﻟﺜﻤﺎﻧﻴﺔ‬

D0 D1 D2 D3 D4 D5 D6 D7

0 0 0 0 1 1 1 1

0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1

 K‫א‬‫א‬‫א‬‫א‬E٦ J٤F‫א‬  

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



(1)A2(MSB)‫א‬‫א‬‫א‬‫א‬

‫א‬A2‫א‬OR‫א‬KD7D4‫א‬‫א‬

 W‫א‬

A2 = D4 + D5 + D6 + D7

 WA1, A0‫א‬OR‫א‬

A1 = D2 + D3 + D6 + D7 A0 = D1 + D3 + D5 + D7

‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬‫א‬ OR‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬  KE١١ J٤F‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K  

D1 D2 D3 D4 D5 D6

A0 (LSB)

 

A1 A2 (MSB)

K‫א‬‫א‬‫א‬E١١ J٤F‫א‬ (1)W E١١ J٤F‫א‬‫א‬‫א‬

       

D6‫א‬‫א‬،K ‫א‬‫א‬،‫א‬‫א‬

‫א‬(1)‫א‬‫א‬E(0)‫א‬‫א‬F(1)

‫א‬‫א‬(110)‫א‬‫א‬‫א‬،A0‫א‬(0)A2,A3  K(6)

 Multiplexer (MUX)‫א‬٥ J٤ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬   K‫א‬‫א‬‫א‬

‫א‬K‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



KE١٢ J٤F‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬  

 ‫א‬

 

4-to-1 MUX

Y ‫א‬‫א‬

     

‫א‬‫א‬

 

K‫א‬‫א‬‫א‬‫ א‬E١٢ J٤F ‫א‬

   

(S0, S1)‫א‬‫א‬‫א‬‫א‬،E١٢ J٤F 

(S0, S1)‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬

‫א‬K ‫א‬D0‫א‬‫א‬‫א‬،S1 = 0, S0 = 0 D1‫א‬‫א‬‫א‬،S1 = 0, S0 = 1(S0, S1)‫א‬‫א‬

(S0, S1)‫א‬‫א‬‫א‬K ‫א‬

‫א‬‫א‬‫א‬K ‫א‬D2‫א‬‫א‬‫א‬،S1 = 1, S0 = 0 D3‫א‬‫א‬‫א‬، S0 = 1 S1 = 1(S0, S1)‫א‬‫א‬  KE٧ J٤F‫א‬‫א‬‫א‬K‫א‬  

‫א‬

S1 0 0 1 1

S0 0 1 0 1

 ‫א‬

Y D0 D1 D2 D3

 K‫א‬‫א‬‫א‬‫א‬‫א‬E٧ J٤F‫א‬

‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬ K‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



(Y)‫א‬‫א‬K ‫א‬‫א‬‫א‬

 WS1 = 0, S0 = 0‫א‬(D0)‫א‬‫א‬

∴ Y = D 0 S1 S0

 WS1 = 0, S0 = 1‫א‬(D1)‫א‬‫א‬(Y)‫א‬‫א‬ ∴ Y = D1 S1 S0

 WS1 = 1, S0 = 0‫א‬(D2)‫א‬‫א‬(Y)‫א‬‫א‬ ∴ Y = D 2S1 S0

 WS1 = 1, S0 = 1‫א‬(D3)‫א‬‫א‬(Y)‫א‬‫א‬ ∴ Y = D 3S1 S0

 W‫א‬‫א‬‫א‬،OR‫א‬‫א‬‫א‬

 

Y = D 0 S1 S0 + D1 S1S0 + D 2S1 S0 + D 3S1S0

،‫א‬(AND)‫א‬‫א‬‫א‬‫א‬

(NOT)‫א‬،‫א‬(OR)‫א‬

 KE١٣ J٤F(S0, S1)

 

     

 

D1 Y D2

 

 

 K‫א‬‫א‬‫א‬‫א‬E١٣ J٤F‫א‬      

 

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 Demultiplexer (DMUX)‫א‬٦ J٤ ‫א‬‫א‬‫א‬‫א‬   K‫א‬‫א‬‫א‬،‫א‬

‫א‬،‫א‬‫א‬‫א‬‫א‬E١٤ J٤F

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬  K‫א‬‫א‬

 



‫א‬

1-to-4 DMUX

‫א‬



‫א‬

       

K‫א‬‫א‬‫א‬‫ א‬E١٤ J٤F ‫א‬ (S0, S1)‫א‬‫א‬‫א‬‫א‬،E١٤ J٤F

   



(S0, S1)‫א‬‫א‬‫א‬K‫א‬(I)‫א‬

‫א‬KD0‫א‬(I)‫א‬‫א‬‫א‬،S1 = 0, S0 = 0

‫א‬‫א‬،S1 = 0, S0 = 1(S0, S1)‫א‬‫א‬

(S0, S1)‫א‬‫א‬‫א‬KD1‫א‬(I)‫א‬

‫א‬‫א‬‫א‬KD2 ‫א‬(I) ‫א‬‫א‬‫א‬،S1 = 1, S0 = 0

‫א‬‫א‬،S1 = 1, S0 = 1(S0, S1)‫א‬‫א‬

 KE٨ J٤F‫א‬‫א‬‫א‬KD3‫א‬(I)‫א‬  

‫א‬

S1 0 0 1 1

S0 0 1 0 1

 ‫א‬ D0 D1 D2 D3 I 0 0 0 0 I 0 0 0 0 I 0 0 0 0 I

 K‫א‬‫א‬‫א‬‫א‬E٨ J٤F‫א‬

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬‫א‬K ‫א‬‫א‬‫א‬‫א‬‫א‬

  (D0)‫א‬‫א‬K‫א‬‫א‬‫א‬

 W S0 = 0 S1 = 0‫א‬(I)‫א‬‫א‬

∴ D 0 = I S1 S0

 WS0 = 1S1 = 0,‫א‬(I)‫א‬‫א‬(D1)‫א‬‫א‬ ∴ D1 = I S1 S0

 W S0 = 0S1 = 1‫א‬(I)‫א‬‫א‬(D2)‫א‬‫א‬ ∴ D 2 = IS1 S0

 W S0 = 1S1 = 1‫א‬(I)‫א‬‫א‬(D3)‫א‬‫א‬ ∴ D 3 = IS1 S0

‫א‬‫א‬‫א‬‫א‬E١٥ J٤F

 KE٨ J٤F‫א‬‫א‬

‫א‬



 

S 0 

 

 

 

‫א‬

D1   D2

 

‫א‬

     

 K‫א‬‫א‬‫א‬‫א‬E١٥ J٤F‫א‬  Comparators‫א‬٧ J٤ K  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ 

K‫א‬ ‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬

‫א‬‫א‬(1)‫א‬‫א‬(XOR)‫א‬‫א‬،‫א‬‫א‬

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬‫א‬‫א‬‫א‬E١٦ J٤FK‫א‬‫א‬(0)،  K(XOR)

0 0

0 ‫א‬

0 1

1 ‫א‬

 

1 0

1 ‫א‬

1 1

0 ‫א‬

       

K‫א‬‫ א‬E١٦ J٤F ‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

  

‫א‬ ،EBAF

 K‫א‬‫א‬‫א‬‫א‬E١٧ J٤FKE A > B A < BA = BF‫א‬  

 

X(A=B)

 ‫א‬ Comparator

 

Y(A<B)

 

Z(A>B)

 

K‫א‬‫א‬‫ א‬E١٧ J٤F ‫א‬  KE١٧ J٤F‫א‬‫א‬‫א‬E٩ J٤F

 



 

 ‫א‬

 ‫א‬

0 0 1 1

0 1 0 1

X A=B 1 0 0 1

Y A<B 0 1 0 0

Z A>B 0 0 1 0

 KE١٧ J٤F‫א‬‫א‬‫א‬E٩ J٤F‫א‬

 



 

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 W‫א‬‫א‬

X = A B + AB ⇒ (A = B) Y = A B ⇒ (A < B) Z = A B ⇒ (A > B)

(Y)‫א‬   ،(XNOR)‫א‬(X)‫א‬‫א‬‫א‬



 K‫א‬‫א‬‫א‬‫א‬‫א‬E١٨ J٤FK(AND)‫א‬،(Z)‫א‬  

X (A=B)

 

Z (A>B)

 

Y (A<B)

 K‫א‬‫א‬‫א‬E١٨ J٤F‫א‬

 

       

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



   

(1 or 0)‫א‬‫א‬،E٢ J٤F ‫א‬‫א‬‫א‬‫א‬  E١  W‫א‬‫א‬

a) A = 1, B = 1, Cin = 1 c) A = 0, B = 1, Cin = 0

b) A = 0, B = 1, Cin = 1 d) A = 1, B = 1, Cin = 0

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٢ a) S = 0, Cout = 0 c) S = 1, Cout = 1

b) S = 1, Cout = 0 d) S = 0, Cout = 1

(1 or 0)‫א‬‫א‬،E٥ J٤F ‫א‬‫א‬‫א‬‫א‬  E٣  W‫א‬‫א‬

a) A = 1, B = 1, Bin = 1 c) A = 1, B = 1, Bin = 0

b) A = 1, B = 0, Bin = 1 d) A = 0, B = 1, Bin = 1

‫א‬‫א‬‫א‬،‫א‬(AND)‫א‬(1)‫א‬E٤  ‫؟‬‫א‬‫א‬

 

A0 A1

   

A0 A1

 

E F

   

EF

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،(NOT)‫א‬(AND)‫א‬E٥ a) 1101 e) 101010

b) 1000 f) 111110

c) 11011 g) 000101

d) 11100 h) 1110110

‫א‬E ‫א‬‫א‬F ‫א‬،E١١ J٤F‫א‬‫א‬E٦  ‫؟‬D5 = 1‫א‬

 W‫א‬‫א‬‫א‬،E١٣ J٤F‫א‬‫א‬E٧

D0 = 0, D1 = 1, D2 = 1, D3 = 0, S0 = 1, S1 = 0

‫א‬‫א‬‫א‬

 ‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬

٣٥

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 ‫א‬‫א‬‫א‬

 

 W‫א‬‫א‬‫א‬

K‫א‬‫א‬‫א‬‫א‬‫א‬ •

K‫א‬‫א‬S-R‫א‬‫א‬‫א‬ •

K‫א‬‫א‬D‫א‬‫א‬‫א‬ •

K‫א‬‫א‬T‫א‬‫א‬‫א‬ •

K‫א‬‫א‬J-K‫א‬‫א‬‫א‬ •

‫א‬‫א‬‫א‬ J ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ • K‫א‬

                                - ١١٤ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 Introduction١ J٥ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬(Combinational Logic Circuits) ‫א‬‫א‬،‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬ (Sequential Logic Circuits)   ‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬(Memory)

‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬ ،(Flip-Flop Circuit)‫א‬‫א‬‫א‬‫א‬‫א‬ (1)‫א‬(0)‫א‬‫א‬‫א‬

،(1)‫א‬(1)‫א‬‫א‬‫א‬‫א‬K

‫א‬‫א‬‫א‬‫א‬K(0)‫א‬(0)‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬

 ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬.(Bistable Multivibrator)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

K(Digital Integrated Circuits)‫א‬‫א‬NOR‫א‬NAND

‫א‬‫א‬،(Counters)‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬(Shift Registers)

 Latches‫א‬٢ J٥ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬K‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬

 K‫א‬‫א‬

K(Bistable Multivibrator)‫א‬‫א‬‫א‬‫א‬‫א‬(Latch)‫א‬

S-R‫א‬‫א‬‫א‬‫א‬‫א‬E١ J٥F (Set Input)"1" ‫א‬‫א‬‫א‬(S) 

- ١١٥ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



(Reset Input)"0"‫א‬‫א‬‫א‬(R)

 K‫א‬ Q ‫א‬Q  

 

SET INPUT

Q output

RESET INPUT

Q output

 KS-R‫א‬‫א‬‫א‬‫א‬‫א‬E١ J٥F‫א‬

 

Q =1,  Q =0(Set Condition)‫א‬‫א‬

‫א‬ .Q =0, Q = 1(Reset Condition)

‫א‬‫א‬(1)‫א‬S‫א‬‫א‬

‫א‬‫א‬‫א‬،‫א‬Q‫א‬E‫א‬‫א‬FQ = 1

‫א‬F Q = 0‫א‬‫א‬(1)‫א‬R‫א‬‫א‬K Q = 0

‫א‬RS‫א‬، Q = 1‫א‬‫א‬E‫א‬

،(unpredictable)‫א‬‫א‬(1)‫א‬  K‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬NOR‫א‬S-R‫א‬‫א‬

 KE٢ J٥F‫א‬‫א‬‫א‬‫א‬‫א‬  

   K‫א‬‫א‬‫א‬‫א‬S-R‫א‬‫א‬E٢ J٥F‫א‬

‫א‬‫א‬F(1)NOR‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،E ‫א‬ - ١١٦ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،E١ J٥F

 K(Active High Inputs)  

‫א‬ S R

‫א‬ Q

‫א‬ (Mode of Operation) E‫א‬F‫א‬ No Change ‫א‬‫א‬ Latch RESETS ‫א‬‫א‬ Latch SETS ‫א‬‫א‬‫א‬ Invalid condition

 K‫א‬‫א‬‫א‬S-R‫א‬‫א‬‫א‬E١ J٥F‫א‬

 

 W‫א‬‫א‬‫א‬

‫א‬‫א‬RS‫א‬(0)‫א‬‫א‬ -١ ‫א‬‫א‬‫א‬E‫א‬‫א‬‫א‬F(Q)‫א‬  K‫א‬

Q‫א‬‫א‬(1)(0)R‫א‬‫א‬‫א‬ -٢

‫א‬‫א‬،‫א‬‫א‬  ‫א‬E ‫א‬‫א‬FQ = 0 (0) KQ = 0

‫א‬‫א‬‫א‬(1)(0)S ‫א‬‫א‬‫א‬ -٣

‫א‬،‫א‬‫א‬‫א‬E‫א‬‫א‬FQ = 1(1)(0)Q KQ = 1‫א‬

‫א‬‫א‬‫א‬RS‫א‬(1)‫א‬‫א‬ -٤ ‫א‬  ‫א‬‫א‬‫א‬،NOR‫א‬‫א‬

K‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬ -٥

،‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬

K‫א‬

- ١١٧ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬‫א‬‫א‬E٣ J٥FNAND‫א‬‫א‬‫א‬

E٢ J٥F‫א‬‫א‬‫א‬‫א‬‫א‬(0)NAND‫א‬

 K(Active Low Inputs)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  

   

 

 K‫א‬‫א‬‫א‬‫א‬S-R‫א‬‫א‬E٣ J٥F‫א‬

  ‫א‬

‫א‬ Q

‫א‬ (Mode of Operation) ‫א‬‫א‬‫א‬ Invalid condition ‫א‬‫א‬ Latch SETS ‫א‬‫א‬ Latch RESETS E‫א‬F‫א‬ No Change

 K‫א‬‫א‬‫א‬S-R‫א‬‫א‬‫א‬E٢ J٥F‫א‬  W‫א‬‫א‬‫א‬

Q‫א‬‫א‬‫א‬‫א‬‫א‬ (1)‫א‬‫א‬ -١ KE‫א‬‫א‬F

‫א‬‫א‬ R = 1‫א‬، S = 0‫א‬‫א‬‫א‬ -٢ Q = 1‫א‬‫א‬،‫א‬‫א‬  ‫א‬(1) K

‫א‬‫א‬ R = 0‫א‬، S = 1 ‫א‬‫א‬‫א‬ -٣

Q = 0‫א‬‫א‬،‫א‬‫א‬  ‫א‬‫א‬،(0) K

- ١١٨ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬(0)‫א‬ -٤ K‫א‬NAND

‫א‬‫א‬‫א‬‫א‬‫א‬(Logic Symbol)‫א‬‫א‬E٤ J٥F‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  

 

  Q

 

 EFEF  ‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬ E٤ J٥F‫א‬  

‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

، S = 0, R = 0 K(Q)‫א‬ S, R 

 K‫א‬‫א‬

‫א‬KE٥ J٥F S, R ‫א‬‫א‬W١ J٥

 KQ = 0‫א‬Q‫א‬‫א‬‫א‬(Q)‫א‬  W‫א‬

S R

     

 

 K‫א‬‫א‬‫א‬‫א‬E٥ J٥F‫א‬     - ١١٩ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 Clocked S-R Flip–Flop ‫א‬‫א‬S-R‫א‬٣ J٥ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ S − R S-R‫א‬ ‫א‬‫א‬ ‫א‬‫א‬(Q)‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬ E‫א‬F‫א‬‫א‬‫א‬‫א‬‫א‬

  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 K ‫א‬‫א‬‫א‬S-R‫א‬‫א‬،‫א‬‫א‬

‫א‬‫א‬‫א‬

،(CK)‫א‬‫א‬(Clock Pulses)‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬S-R‫א‬‫א‬E٦ J٥F

 K(CK)‫א‬‫א‬‫א‬  

 

CK R

S CK

  Q

 

 EFEF

 K‫א‬‫א‬S-R‫א‬‫א‬E٦ J٥F‫א‬

‫א‬‫א‬‫א‬EEF٦ J٥F‫א‬

(Positive Edge Trigger) ‫א‬‫א‬‫א‬S-R‫א‬

‫א‬‫א‬‫א‬EEF٦ J٥F‫א‬،(1)(0)‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K(0)(1)‫א‬‫א‬ (Negative Edge Trigger)

‫א‬،NAND‫א‬‫א‬‫א‬‫א‬S-R‫א‬‫א‬E٧ J٥F

‫א‬‫א‬K ‫א‬‫א‬‫א‬‫א‬NAND ‫א‬‫א‬‫א‬‫א‬(Q)‫א‬R S‫א‬  K‫א‬‫א‬‫א‬

- ١٢٠ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 

   

 

 K‫א‬‫א‬S-R‫א‬‫א‬E٧ J٥F‫א‬

 W‫א‬‫א‬‫א‬‫א‬S-R‫א‬E٣ J٥F‫א‬

(0)‫א‬‫א‬RS‫א‬‫א‬،‫א‬CK‫א‬‫א‬  J١  K‫א‬‫א‬‫א‬‫א‬‫א‬

(0)‫א‬‫א‬(S = 0,R = 1)‫א‬R‫א‬‫א‬  J٢ .(Reset)‫א‬‫א‬‫א‬(0) ‫א‬(1)

(0)‫א‬‫א‬(R = 0‫ و‬S = 1)‫א‬‫א‬S‫א‬‫א‬  J٣ .(Set)‫א‬‫א‬‫א‬Q = 1‫א‬(1)

‫א‬‫א‬  R = 1S = 1‫א‬‫א‬

 K

‫א‬ S R CK

‫א‬ (Mode of Operation) ‫א‬ Q0 X X No Change E‫א‬F‫א‬ Q0 0 0 X No Change ‫א‬‫א‬ 0 1 0 Latch RESETS ‫א‬‫א‬ 1 0 1 Latch SETS ‫א‬‫א‬‫א‬ 1 1 ? Invalid condition  ↑Z(1)‫א‬(0)‫א‬  

 XZ

‫א‬ Q







 Q0Z‫א‬‫א‬‫א‬

 K‫א‬‫א‬S-R‫א‬‫א‬‫א‬E٣ J٥F‫א‬ - ١٢١ -



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



(1)‫א‬]‫א‬‫א‬‫א‬S-R‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬[(0)  K(0)(1)‫א‬‫א‬

‫א‬،E٦ J٥F‫א‬S-R‫א‬‫א‬ (Q)‫א‬‫א‬W٢ J٥

‫א‬‫א‬‫א‬KE٧ J٥FCKRS‫א‬

 K‫א‬‫א‬Q = 0    

 3

  S

 

 K‫א‬‫א‬S-R‫א‬‫א‬‫א‬‫א‬E٧ J٥F‫א‬  W‫א‬

 KQ = 0(Q)‫א‬، R = 0S = 0‫א‬‫א‬‫א‬ -١

K(Reset) Q = 0‫א‬، R = 1S = 0‫א‬‫א‬‫א‬ -٢

K(Set) Q = 1(1)Q ‫א‬، R = 0S = 1‫א‬‫א‬‫א‬ -٣ .(Reset) Q = 0‫א‬، R = 1S = 0‫א‬‫א‬‫א‬‫א‬ -٤

K(Set)Q =1‫א‬، R = 0S = 1‫א‬‫א‬‫א‬ -٥

KQ = 1 (1)‫א‬، R = 0 S = 1‫א‬‫א‬‫א‬ -٦      

- ١٢٢ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 D-Type Flip-FlopD ‫א‬‫א‬‫א‬٤ J٥ (Single Bit)‫א‬‫א‬‫א‬D‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬S-R‫א‬‫א‬‫א‬K(10)‫א‬  KE٨ J٥FD‫א‬‫א‬ S

 

CK R

 

 KD‫א‬‫א‬‫א‬E٨ J٥F‫א‬

‫א‬‫א‬D‫א‬‫א‬D‫א‬‫א‬‫א‬

‫א‬،CK‫א‬‫א‬‫א‬(1)‫א‬‫א‬D‫א‬KCK

R = 0‫א‬،S = 1‫א‬‫א‬،[Set](1)‫א‬‫א‬‫א‬ ‫א‬KQ = 1‫א‬E٣ J٥F‫א‬‫א‬S-R‫א‬‫א‬‫א‬

‫א‬‫א‬،CK‫א‬ ‫א‬‫א‬(0)‫א‬‫א‬D

R = 1 ‫א‬،S = 0‫א‬‫א‬،[Reset](0)‫א‬‫א‬

‫א‬(1)(Set)‫א‬‫א‬KQ = 0‫א‬E٣ J٥F  K‫א‬‫א‬(0)(0)‫א‬‫א‬،‫א‬

‫א‬‫א‬‫א‬‫א‬D‫א‬‫א‬‫א‬‫א‬‫א‬

 KE٤ J٥F‫א‬(Positive Edge Trigger) ‫א‬‫א‬  

‫א‬ D CK

‫א‬ Q

‫א‬ (Mode of Operation) E‫א‬F‫א‬ No Change ‫א‬‫א‬(RESET)

‫א‬‫א‬(SET)

 ↑Z(1)‫א‬(0)‫א‬

 K‫א‬‫א‬D ‫א‬‫א‬‫א‬E٤ J٥F‫א‬ - ١٢٣ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬K ‫א‬‫א‬(D)‫א‬(Q)‫א‬‫א‬

(D)‫א‬‫א‬‫א‬D‫א‬‫א‬E٩ J٥F

E١٠ J٥F‫א‬K  ‫א‬‫א‬‫א‬(CK) ‫א‬‫א‬

D D

 

   

 KNAND‫א‬D‫א‬‫א‬

Q CK

 

 KD‫א‬‫א‬E٩ J٥F‫א‬KNAND‫א‬D‫א‬‫א‬E١٠ J٥F‫א‬  

‫א‬E٩ J٥F‫א‬D‫א‬‫א‬‫א‬EQF‫א‬‫א‬W٣ J٥

‫א‬‫א‬‫א‬KE١١ J٥FEDF‫א‬

 K‫א‬Q = 0    W‫א‬

D Q

     

 KD‫א‬‫א‬‫א‬‫א‬‫א‬E١١ J٥F‫א‬  ‫א‬(1)(0)‫א‬‫א‬‫א‬‫א‬(D)‫א‬(Q)‫א‬

 K‫א‬

- ١٢٤ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 J-K Flip Flop‫א‬‫א‬J-K‫א‬٥ J٥ ‫א‬‫א‬KJ‫א‬ ‫א‬K‫א‬‫א‬‫א‬‫א‬J-K‫א‬‫א‬ ‫א‬S-R‫א‬‫א‬‫א‬،‫א‬

‫א‬‫א‬‫א‬S-R‫א‬J-K‫א‬K‫א‬ J-K‫א‬‫א‬.(Reset)‫א‬‫א‬(Set)‫א‬‫א‬‫א‬‫א‬  KS-R‫א‬‫א‬

K ‫א‬‫א‬‫א‬‫א‬J-K‫א‬‫א‬E١٢ J٥F‫א‬

‫א‬‫א‬‫א‬S-R‫א‬‫א‬‫א‬   K‫א‬‫א‬(1)‫א‬‫א‬KJ‫א‬‫א‬‫א‬  

   

J CK

   

J CK

 

 K‫א‬‫א‬‫א‬‫א‬J-K‫א‬‫א‬E١٢ J٥F‫א‬ ‫א‬S-R‫א‬‫א‬‫א‬‫א‬‫א‬E١٢ J٥F

 K‫א‬ Q ،Q

‫א‬‫א‬‫א‬J-K‫א‬E٥ J٥F‫א‬

‫א‬‫א‬‫א‬‫א‬،(0)KJ‫א‬

‫א‬،‫א‬‫א‬ K = 1J = 0‫א‬‫א‬(0)‫א‬(Reset)‫א‬

 K = 0 J = 1‫א‬‫א‬J-K(Set)‫א‬‫א‬‫א‬‫א‬

 ‫א‬K‫א‬ ‫א‬ ‫א‬J-K‫א‬‫א‬‫א‬‫א‬

Q‫א‬(1)‫א‬‫א‬KJ‫א‬  ،(Toggle)  KCK‫א‬‫א‬‫א‬‫א‬‫א‬ - ١٢٥ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



  ‫א‬ J K CK

‫א‬ Q

‫א‬ (Mode of Operation) ‫א‬ Q0 X X No Change E‫א‬F‫א‬ Q0 0 0 No Change ‫א‬‫א‬ 0 1 0 (RESET) ‫א‬‫א‬ 1 0 1 (SET) ‫א‬ Q0 1 1 Toggle  ↓Z(0)‫א‬(1)‫א‬

Q0Z‫א‬‫א‬‫א‬

 K‫א‬‫א‬J-K‫א‬E٥ J٥F‫א‬

‫א‬E١٢ J٥F‫א‬J-K‫א‬‫א‬(Q)‫א‬‫א‬W٤ J٥ ‫א‬KE١٣ J٥FCKJ-K‫א‬

 K‫א‬Q = 0 ‫א‬

 W‫א‬

J K

 

 K‫א‬‫א‬J-K‫א‬‫א‬‫א‬‫א‬E١٣ J٥F‫א‬ Q ‫א‬‫א‬‫א‬(1)KJ،‫א‬‫א‬‫א‬ -١ K(1)‫א‬

KJ = K = 0‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ -٢

KQ = 0(Reset) K = 1J = 0،‫א‬‫א‬ -٣ - ١٢٦ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



KQ = 1(Set) K = 0J = 1،‫א‬‫א‬‫א‬ -٤

Q‫א‬KJ‫א‬‫א‬‫א‬(Set)‫ א‬-٥ K(1)‫א‬

 T-Type Flip-Flop T ‫א‬‫א‬‫א‬٦ J٥ ‫א‬‫א‬J-K‫א‬‫א‬T‫א‬‫א‬‫א‬ ‫א‬‫א‬،E١٤ J٥F‫א‬KJ‫א‬

(Toggle)‫א‬T‫א‬K‫א‬‫א‬T‫א‬‫א‬T  K‫א‬‫א‬

‫א‬‫א‬،‫א‬‫א‬CK‫א‬(1)‫א‬(T)‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬CK‫א‬‫א‬‫א‬  KE١٤ J٥FCK‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  

 

 

 K



 KT‫א‬‫א‬‫א‬‫א‬‫א‬E١٤ J٥F‫א‬

 KE٦ J٥FT‫א‬‫א‬‫א‬‫א‬

‫اﻟﻤﺪﺧﻼت‬ T CK

‫وﺿﻊ اﻟﺘﺸﻐﻴﻞ‬ (Mode of Operation) ‫ﻋﺪم اﻟﺘﻐﻴﺮ‬ Q0 X No Change (‫وﺿﻊ اﻹﻣﺴﺎك )ﻋﺪم اﻟﺘﻐﻴﺮ‬ Q0 0 No Change ‫وﺿﻊ اﻟﺘﺒﺪﻳﻞ‬ Q0 1 Toggle  ↓Z(0)‫א‬(1)‫א‬  

 

‫اﻟﺨﺮج‬ Q

Q0Z‫א‬‫א‬‫א‬





 KT‫א‬‫א‬E٦ J٥F‫א‬

- ١٢٧ -



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



E١٤ J٥F‫א‬(T)‫א‬‫א‬‫א‬Q‫א‬‫א‬W٥ J٥

‫א‬‫א‬E١٥ J٥FCK‫א‬T‫א‬‫א‬

 K‫א‬Q = 0    W‫א‬

  CK T Q

     

 KT‫א‬‫א‬‫א‬‫א‬‫א‬E١٥ J٤F‫א‬

 

،‫א‬‫א‬‫א‬T = 1‫א‬Q‫א‬‫א‬

T = 1‫א‬‫א‬،Q = 0QT = 0‫א‬‫א‬‫א‬

 K‫א‬(1)(0)Q‫א‬

Master-Slave Flip-Flop‫א‬–‫א‬٧ J٥ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K(Edge Triggered)‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬  ،(Master-Slave)‫א‬–‫א‬‫א‬(Pulse Triggered)

‫א‬(Complete Clock Pulse)‫א‬‫א‬‫א‬

 K‫א‬‫א‬

‫א‬،‫א‬ J‫א‬‫א‬J-K‫א‬EEF١٦ J٥F

(Master)‫א‬‫א‬،(Slave)‫א‬(Master)‫א‬‫א‬‫א‬J-K

(Slave)‫א‬‫א‬،(CK)‫א‬‫א‬‫א‬‫א‬

 K( CK )‫א‬‫א‬    

- ١٢٨ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

Master J

 ‫ א‬١٦٧

‫א‬

 ‫א‬



   

Slave Y

     

   

 K‫א‬ J‫א‬J-K‫א‬‫א‬EEF١٦ J٥F‫א‬

 

،EEF١٦ J٥FCK CK  ‫א‬‫א‬

، ‫א‬‫א‬(CK)‫א‬‫א‬‫א‬‫א‬(Master)‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬(Slave)‫א‬‫א‬

 K( CK )‫א‬‫א‬‫א‬‫א‬‫א‬

 WKJ‫א‬Q Q 

(Master)‫א‬‫א‬(CK)(High)‫א‬‫א‬W‫א‬‫א‬

 KKJ‫א‬‫א‬(Enabled)‫א‬

 (slave)  ‫א‬‫א‬(CK)(Low)‫א‬‫א‬W‫א‬‫א‬

 KY‫א‬‫א‬‫א‬‫א‬Q‫א‬(Enabled)‫א‬

J-K‫א‬‫א‬‫א‬EEF١٦ J٥F‫א‬‫א‬

K ‫א‬‫א‬J-K‫א‬‫א‬‫א‬‫א‬K‫א‬ J‫א‬

‫א‬‫א‬‫א‬  ‫א‬‫א‬(CK)‫א‬

 K‫א‬‫א‬(Low)‫א‬(High)

‫א‬‫א‬،‫א‬ J‫א‬J-K‫א‬‫א‬‫א‬EEF١٦ J٥F  KKJ‫א‬‫א‬‫א‬t5t0‫א‬(CK)

 

- ١٢٩ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬(Enabled)‫א‬ (Master)‫א‬‫א‬،t0‫א‬ •

‫א‬‫א‬ K = 0 J = 1 ‫א‬(CK)‫א‬‫א‬(High)‫א‬ K( Y = 0 )Y = 1‫א‬،‫א‬‫א‬(Set)

(Low)‫א‬‫א‬(Disabled)‫א‬‫א‬،t1‫א‬ • ‫א‬‫א‬(Enabled)‫א‬(Slave)‫א‬‫א‬،CK

‫א‬Q‫א‬،‫א‬‫א‬‫א‬ Y, Y K CK (High)

‫א‬‫א‬‫א‬‫א‬KQ = 1(Set )‫א‬ F‫א‬‫א‬‫א‬

KE CK = 1‫א‬‫א‬ Q = 1, Q = 0 ‫א‬ Y = 1, Y = 0

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  

‫א‬ J K CK 0 0 0 1

‫א‬ Q Q0

‫א‬ (Mode of Operation) E‫א‬F‫א‬

‫א‬‫( א‬RESET)

‫א‬‫(א‬SET)

‫א‬

 

 K‫א‬ J‫א‬J-K‫א‬‫א‬‫א‬EEF١٦ J٥F‫א‬                   - ١٣٠ -

 

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

‫א‬

 ‫א‬



Master Enable

Master Enable

 ‫ א‬١٦٧



t2   t3

 

Master Enable

‫ﻜﺎل‬ ‫أﺷ‬ ‫ﻧﺒﻀﺎت اﻟﺘﺎﺑﻊ‬ (Master)

   

Slave Enable

Slave

Enable 

Slave Enable

   

Y   Q

‫أﺷﻜﺎت ﻧﺒﻀﺎت‬ ‫اﻟﻤﺘﺒﻮع‬ (Slave)



 

 K‫א‬‫א‬J-K‫א‬‫א‬‫א‬‫א‬EEF١٦ J٥F‫א‬ CK(High)‫א‬‫א‬،‫א‬‫א‬‫א‬،t2‫א‬ •

‫א‬‫א‬ Y = 0, Y = 1 ‫א‬ K = 1J = 0‫א‬  .(Reset) ‫א‬،CK  (Low)‫א‬‫א‬‫א‬‫א‬،t3‫א‬ • (Reset)‫א‬‫א‬‫א‬‫א‬K ‫א‬‫א‬

 KQ = 0‫א‬

،Y‫א‬(Low)‫א‬KJ‫א‬،t4‫א‬ •

J‫א‬،t4‫א‬‫א‬K(Y = 0)‫א‬‫א‬‫א‬

KY = 1‫א‬‫א‬(High)‫א‬

- ١٣١ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬  ،‫א‬‫א‬‫א‬‫א‬‫א‬،t5‫א‬ • KQ = 1Q‫א‬Y = 1

‫א‬‫א‬‫א‬‫א‬

‫א‬ (PRE) (PRESET)‫א‬‫א‬‫א‬K‫א‬‫א‬

‫א‬‫א‬E١٧ J٥F‫א‬ (CLR ) (CLEAR)‫א‬‫א‬

‫א‬ ‫א‬K CLR ، PRE  S-R ‫א‬ ،Q = 1(SET)‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬

 ‫א‬،Q = 0 (RESET)‫א‬‫א‬

(RESET)‫א‬‫א‬‫א‬K‫א‬Q‫א‬

 (PRE ) ‫א‬K  ‫א‬‫א‬‫א‬ ‫א‬(CLEAR)‫א‬ ، PRE = 0 (1)‫א‬Q‫א‬،‫א‬

(0)‫א‬Q‫א‬   (CLR ) ‫א‬ S-R‫א‬‫א‬‫א‬  ‫א‬E٧ J٥FK CLR = 0

Q‫א‬E F PRE = 0 ‫א‬ CLR = 1‫א‬

 PRE = 1 ‫א‬‫א‬‫א‬KCK, S, R‫א‬‫א‬،(1)

‫א‬(0)Q‫א‬‫א‬EF CLR = 0 

 KCK, S, R‫א‬

  PRE  

 CK R



 

  CLR

 K CLR ، PRE S-R‫א‬‫א‬‫א‬‫א‬E١٧ J٥F‫א‬   - ١٣٢ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬ ‫א‬ PRE CLR CK S 0 1 X X 1 0 X X

 ‫ א‬١٦٧

‫א‬

 ‫א‬



R X

‫א‬ Q 1

‫א‬ (Mode of Operation) ‫א‬‫א‬(SET)

‫א‬‫( א‬RESET)

‫א‬

   KS-R‫א‬‫א‬ CLR  PRE ‫א‬E٧ J٥F‫א‬

- ١٣٣ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



   

‫א‬‫א‬‫א‬‫א‬S-R‫א‬‫א‬(Q)‫א‬‫א‬ E١

‫א‬‫א‬(negative edge trigger)‫א‬‫א‬

K‫א‬‫א‬Q=0 ‫א‬‫א‬‫א‬K CK

‫א‬  ‫א‬‫א‬‫א‬D‫א‬‫א‬‫א‬(Q)‫א‬‫א‬E٢

K‫א‬‫א‬(positive edge trigger)‫א‬‫א‬ K‫א‬‫א‬Q=0 ‫א‬‫א‬‫א‬

 

‫א‬‫א‬‫א‬‫א‬JK‫א‬‫א‬(Q)‫א‬‫א‬E٣ ‫א‬K‫א‬‫א‬(negative edge trigger)‫א‬‫א‬ K‫א‬‫א‬Q=0 ‫א‬‫א‬   - ١٣٤ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



  CK

   

     

 

‫א‬‫א‬‫א‬‫א‬T‫א‬‫א‬‫א‬(Q)‫א‬‫א‬E٤ ‫א‬‫א‬(negative edge trigger)‫א‬‫א‬

K‫א‬‫א‬Q=0 ‫א‬‫א‬‫א‬K

- ١٣٥ -

‫א‬‫א‬‫א‬

 ‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 ‫א‬‫א‬‫א‬

 

 W‫א‬‫א‬‫א‬ K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ • K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ •

                                      - ١٣٦ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 Introduction١ J٦ ،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ 

‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 Registers‫א‬٢ J٦ ،‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬(bit)‫א‬‫א‬‫א‬‫א‬ ‫א‬،‫א‬‫א‬‫א‬،

‫א‬‫א‬‫א‬

Shift )‫א‬ (Shift Left)‫א‬‫א‬‫א‬(Buffer Register)‫א‬‫א‬

‫א‬(Parallel Data)‫א‬(Serial Data)‫א‬‫א‬‫א‬(Right  K(Shift Registers)‫א‬‫א‬

 Buffer Registers‫א‬١ J٢ J٦

(Digital word)‫א‬

  ‫א‬EEF١ J٦FK(bits)‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬D ‫א‬‫א‬‫א‬‫א‬(4-stages)  K(Positive edge-triggered)

- ١٣٧ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



(4-bit word to be stored)‫א‬‫א‬‫א‬‫א‬ D1

D2 Q

CLR

CK CLR

(parallel data outputs)‫א‬‫א‬‫א‬

E F

 KD‫א‬‫א‬‫א‬‫א‬‫א‬EEF١ J٦Fhg    

Clock

‫א‬ Input data

 

   

 

‫א‬ Output data

 

   

Q3 Q4

E F

 KEEF١ J٦F‫א‬‫א‬‫א‬EEF١ J٦F‫א‬

- ١٣٨ -

     

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬‫א‬(4-bits)‫א‬‫א‬‫א‬

‫א‬Q1,Q2, Q3,Q4‫א‬D1,D2,D3,D4  K(CK)‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬EEF١ J٦F‫א‬‫א‬‫א‬

‫א‬‫א‬Q1,Q2, Q3,Q4‫א‬

 K‫א‬‫א‬K‫א‬‫א‬‫א‬

‫א‬،‫א‬‫א‬‫א‬

‫א‬‫א‬ J‫א‬‫א‬‫א‬،‫א‬‫א‬

‫א‬‫א‬‫א‬(Clear-input)‫א‬K(Parallel-in, Parallel-out Registers)  KE‫א‬F‫א‬‫א‬(active-low)  

 Shift Registers‫א‬‫א‬٢ J٢ J٦

‫א‬(Shift)‫א‬(move)‫א‬‫א‬‫א‬‫א‬

 WKE٢ J٦F‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K 

(Serial-in, Serial-out Shift Registers)‫א‬‫א‬–‫א‬‫א‬‫א‬ -١

K(SISO)‫א‬‫א‬

(Serial-in, Parallel-out Shift Registers)‫א‬‫א‬–‫א‬‫א‬‫א‬ -٢ .(SIPO)‫א‬‫א‬

(Parallel-in, Serial-out Shift Registers)  ‫א‬‫א‬–‫א‬‫א‬‫א‬ J٣ K(PISO)‫א‬‫א‬

             

- ١٣٩ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



  Serial-in, serial-out (SISO) Shift Registers

Shift Right Serial-In

 

  Serial-Out

Shift Left Serial-In

   

Rotate Right

 

Rotate Left

E F

  Serial-in, parallel-out (SIPO) Shift Registers

 

Serial-In

Parallel-in, Serial-out Shift (PISO) Registers Parallel Data In

  Serial-Out

  EF

Parallel Data Out

 K‫א‬‫א‬E٢ J٦F‫א‬ ‫א‬ ‫א‬‫א‬

 W‫א‬

 ‫א‬‫א‬‫א‬–‫א‬‫א‬‫א‬‫א‬‫א‬١ J٢ J٢ J٦

Serial-in, Serial-out (SISO) Shift registers

‫א‬‫א‬K  ‫א‬‫א‬‫א‬،E١ J٦F

1001‫א‬‫א‬‫א‬‫א‬E‫א‬‫א‬F0110‫א‬‫א‬

 K‫א‬‫א‬‫א‬  

 

- ١٤٠ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



  ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫א‬  Q0

 —

1st

2nd

3rd

4th

Clock

Input

 —

 K‫א‬‫א‬E١ J٦F‫א‬

 

‫א‬‫א‬‫( א‬1st Clock pulse)‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬

،(2nd Clock pulse)‫א‬‫א‬‫א‬K ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬ (0110)‫א‬‫א‬

K  ‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬K(1001)‫א‬‫א‬‫א‬ ‫א‬(0110)‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬(1001)‫א‬‫א‬‫א‬،‫א‬

 K

 ‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬(4-bits)‫א‬‫א‬EEF٣ J٦F

‫א‬،(FF0)‫א‬‫א‬‫א‬D‫א‬‫א‬‫א‬‫א‬KD‫א‬‫א‬

(Q1)‫א‬‫א‬‫א‬،(FF1)‫א‬‫א‬‫א‬D‫א‬(Q0)‫א‬‫א‬

‫א‬‫א‬(Q2)‫א‬‫א‬‫א‬،(FF2)‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،(FF3)‫א‬‫א‬‫א‬

 K‫א‬

- ١٤١ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 

  Serial Data Input

FF1

FF0 D

Q 1

D CK

FF2

 

Serial Data Out

D CK

 

 

FF3

 

Clock Input

SISO Shift Right

   

Serial Data Out

Q CK

Clock Input

E F

FF3

D CK

FF2

 

Q 1 CK

   

FF1

Serial Data Input

FF0

SISO Shift Left

 

E F

    SISO Rotate Right

 

SISO Rotate Left

EF

 K‫א‬‫א‬‫א‬‫א‬‫א‬E٣ J٦F‫א‬ ،‫א‬‫א‬(Clock input)‫א‬‫א‬

،‫א‬‫א‬(1-bit)‫א‬‫א‬‫א‬ (Positive edge) ‫א‬‫א‬‫א‬‫ א‬‫א‬–‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬

 K‫א‬

- ١٤٢ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



EEF٣ J٦F‫א‬‫א‬‫א‬،

‫א‬‫א‬‫א‬–‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬D‫א‬‫א‬

EEF٣ J٦F‫א‬‫א‬‫א‬ .(SISO Shift-Right Shift Register) ‫א‬–‫א‬‫א‬D‫א‬‫א‬‫א‬‫א‬

 K(SISO Shift- Left Shift Register)‫א‬

EEF٣ J٦F،EEF٣ J٦F‫א‬‫א‬‫א‬،‫א‬

‫א‬‫א‬،‫א‬‫א‬ ‫א‬

‫א‬‫א‬‫א‬–‫א‬‫א‬‫א‬

 (SISO Rotate-Left) ‫א‬‫א‬‫א‬–‫א‬‫א‬(SISO Rotate-Right)  KEEF٣ J٦F

 ‫א‬‫א‬–‫א‬‫א‬‫א‬٢ J٢ J٢ J٦

Serial-in, parallel out (SIPO) Shift registers

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٤ J٦F‫א‬

 K‫א‬‫א‬–‫א‬

(4-bits)‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬

‫א‬F‫א‬‫א‬‫א‬‫א‬‫א‬(Serial data input)‫א‬‫א‬‫א‬

 KE‫א‬‫א‬‫א‬

Serial Data Input

FF1

FF0 D

 

FF2

Q 1

FF3 Q

 

Clock Input

  Q

   

Q 1

 

Parallel data outputs

 K‫א‬‫א‬ J‫א‬‫א‬‫א‬E٤ J٦F‫א‬ - ١٤٣ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 ‫א‬‫א‬‫א‬‫א‬‫א‬(4-bits) 

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬

 K‫א‬‫א‬(4-bits)(Q3,Q2,Q1,Q0)

 ‫א‬‫א‬–‫א‬‫א‬‫א‬٣ J٢ J٢ J٦

Parallel-in, Serial-out (PISO) Shift registers

‫א‬‫א‬‫א‬‫א‬E٥ J٦F

‫א‬‫א‬‫א‬KD‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ –

،(Low)‫א‬ SHIFT / LOAD ‫א‬K SHIFT / LOAD ‫א‬

‫א‬(Enabled)‫א‬‫א‬AND ‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬K ‫א‬Inverter‫א‬

‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬(D3,D2,D1,D0)‫א‬

  .(Q3,Q2,Q1,Q0)‫א‬‫א‬‫א‬‫א‬ ،(Clock pulse)  

(SHIFT / LOAD) control

Parallel data inputs

(1 for shift , 0 for load )

D 1

FF1

FF0 D CK

 

FF2 Q 1

FF3 Q

Clock Input

 K‫א‬‫א‬ J‫א‬‫א‬‫א‬E٥ J٦F‫א‬ - ١٤٤ -

Serial Out 3

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



AND‫א‬‫א‬،(High)‫א‬ SHIFT / LOAD ‫א‬

Q0‫א‬‫א‬‫א‬‫א‬K(Enabled)‫א‬‫א‬

،(FF2)‫א‬‫א‬‫א‬‫א‬Q1‫א‬،(FF1)‫א‬‫א‬‫א‬D‫א‬

‫א‬‫א‬،‫א‬‫א‬K(FF3)‫א‬‫א‬‫א‬‫א‬Q2‫א‬

(1-bit)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  .(clock inputF‫א‬‫א‬‫א‬‫א‬

Shift Register Sequencer (Ring Counter)E‫א‬F‫א‬‫א‬‫א‬٤ J٢ J٢ J٦

‫א‬‫א‬‫א‬EEF٦ J٦F

 KED0Q3‫א‬FFF0‫א‬‫א‬‫א‬(FF3)‫א‬‫א‬‫א‬

  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬Q0‫א‬Low‫א‬ SRART ‫א‬K

E CLR = 0 FLow‫א‬ Q1,Q2,Q3‫א‬،( PRE = 0 )High  KEEF٦ J٦F‫א‬  

PRE

D 1

PRE

Q 1

FF0

CLR

Clock

Q 1 PRE

CLR

PRE

CLR

FF2

START

E F

K‫א‬‫א‬‫א‬EEF٦ J٦F‫א‬

- ١٤٥ -

CLR

FF1

 

FF3

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



Clock START

Q 1 Q

0 0

2 1

E F

 K‫א‬‫א‬‫א‬EEF٦ J٦F‫א‬ Clock Pulses 0 1 2 3

‫א‬‫א‬ Q0 1 0 0 0

Q1 0 1 0 0

0 0 1 0

0 0 0 1

Four flip-flops will have Four output states.

Repeat Sequence

 K‫א‬‫א‬‫א‬E٢ J٦F‫א‬ ،‫א‬‫א‬‫א‬(1000)‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬(1)‫א‬  KE٢ J٦F‫א‬

 Johnson Counter‫א‬٥ J٢ J٢ J٦ - ١٤٦ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬،‫א‬‫א‬‫א‬EEF٧ J٦F

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K(D0)‫א‬‫א‬‫א‬‫א‬E Q 3 F

‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬

Q3،1000،E٣ J٦F‫א‬‫א‬EEF٧ J٦F‫א‬

‫א‬‫א‬(High)‫א‬ Q 3 ،‫א‬‫א‬(Low)‫א‬ ‫א‬ (High inputs)‫א‬‫א‬‫א‬‫א‬،D0‫א‬ Q3K(High)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

  ،(Low)‫א‬ Q 3 ،E‫א‬‫א‬‫א‬F(High)‫א‬

Low )‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K(Low)D0

Q3K(Low)‫א‬‫א‬‫א‬‫א‬(inputs

  (High)‫א‬ Q 3 ،E‫א‬‫א‬‫א‬F(Low)‫א‬

 K‫א‬‫א‬‫א‬‫א‬(High)D0  

Clock Pulses

‫א‬‫א‬ Q0

Four flip-flops will have

eight output states.

 Repeat Sequence



 K‫א‬‫א‬E٣ J٦F‫א‬  

- ١٤٧ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



    Q

PRE

Q 1

D 1

Q 1

PRE

CLR

FF0

 

CLR

Clock

PRE

CLR



Q 3

 

Clock

Q 3

FF3 

FF2

E F 0

 

CLR

FF1

PRE

     

START

   

 

Q 1 Q

   

E F

 

 K‫א‬‫א‬‫א‬‫א‬‫א‬E٧ J٦F‫א‬ ،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

F‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬KEE٢ J٦F - ١٤٨ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



EEF٧ J٦F‫א‬‫א‬‫א‬،‫א‬

 KE٣ J٦F(2 × 4 flip-flops = 8)

- ١٤٩ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 Counters‫א‬‫א‬‫א‬٣ J٦ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ،(binary bits)‫א‬‫א‬‫א‬K‫א‬   ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K(clock input)‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

K(Synchronous Counters )  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬(Asynchronous Counters) ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬

K(Master Clock)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬

  ،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 Asynchronous Binary-Up Counters ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬١ J٣ J٦

K ‫א‬‫א‬‫א‬EEF٨ J٦F

‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬K ‫א‬‫א‬J-K

K ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬  ،(High)‫א‬KJ‫א‬ K ‫א‬‫א‬(Negative edge)(Toggle)

‫א‬(Q)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬(4-bit word)‫א‬‫א‬Q3,Q2,Q1,Q0‫א‬KEEF٨ J٦F

‫א‬‫א‬‫א‬0000‫א‬‫א‬

FF0‫א‬‫א‬KE٤ J٦F‫א‬‫א‬‫א‬‫א‬  K(MSB)‫א‬(Q3)FF3‫א‬‫א‬(LSB)(Q0)  

- ١٥٠ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

FF0 J Clock Input

 ‫ א‬١٦٧

‫א‬

 ‫א‬



Q 1

FF3

FF2

FF1

Q 1

CK K

 E F 1

9 10 11 12 13 14 15 16 17

E F

 

 K‫א‬‫א‬‫א‬‫א‬E٨ J٦F‫א‬ ،(Clock input)‫א‬‫א‬‫א‬(FF0)‫א‬‫א‬

،‫א‬‫א‬‫א‬ (Toggle)Q0‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬KEEF٨ J٦FQ0‫א‬

‫א‬K‫א‬"0""1"‫א‬‫א‬‫א‬"1""0"Q0

Q0،FF1‫א‬‫א‬‫א‬Q0‫א‬

‫א‬Q1K(Toggle)Q1‫א‬

 KQ3‫א‬Q2،Q2  

- ١٥١ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 ‫א‬‫א‬ Q3

‫א‬

0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1

0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Cycle Repeats

 

Binary Count

 K‫א‬‫א‬‫א‬‫א‬‫א‬E٤ J٦F‫א‬  The Maximum Count (N) of a Counter ‫א‬• ‫א‬‫א‬‫א‬،E٤ J٦F‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬،[‫א‬‫א‬(1)‫א‬]0001‫א‬‫א‬‫א‬

0011‫א‬‫א‬‫א‬‫א‬‫א‬،[  ‫א‬‫א‬(2)‫]א‬0010‫א‬

‫א‬‫א‬‫א‬K‫א‬،[‫א‬‫א‬(3)]  W‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬

  N = 2 n − 1 

 

W

(N = maximum count before cycle repeats) ‫א‬‫א‬‫א‬ = N

 (n = number of flip-flops in the counter circuit)‫א‬‫א‬‫א‬‫א‬‫א‬ = n  W‫א‬EF٣٢ J٤‫א‬‫א‬‫א‬‫א‬ - ١٥٢ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



N = 2n − 1 = 24 − 1 = 16 − 1 = 1510 (11112 ) The Modulus (MOD) of a counter‫א‬‫א‬‫א‬•

‫א‬(MOD)(Modulus of a counter)‫א‬‫א‬‫א‬

 K ‫א‬‫א‬‫א‬ MODEEF٨ J٦F‫א‬‫א‬‫א‬

‫א‬11110000    (16)‫א‬‫א‬(16)  W‫א‬‫א‬‫א‬MODKE٤ J٦F

MOD = 2n MOD = modulus of the counter n = number of flip-flops in the counter circuit

 

 W‫א‬‫א‬‫א‬‫א‬‫א‬EEF٨ J٦F‫א‬‫א‬‫א‬‫א‬

 

MOD = 2 n = 24 = 16 The Frequency Division of a counter ‫א‬‫א‬•

EEF٨ J٦F‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬(frequency divider)‫א‬‫א‬ ‫א‬‫א‬،2‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

، ‫א‬2‫א‬K2‫א‬‫א‬

،‫א‬2‫א‬2‫א‬

4‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬EEF٨ J٦F ،2‫א‬‫א‬‫א‬‫א‬‫א‬، KQ1  ‫א‬ - ١٥٣ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬،8‫א‬،4‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬16  

Division Factor = 2n E‫א‬F

n = number of flip-flops in the counter circuit

 The Propagation Delay Time (tp) of a counter‫א‬‫א‬• ‫א‬‫א‬،(Ripple counter)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ K‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬

‫א‬‫א‬، ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬

  E ‫א‬‫א‬‫א‬F‫א‬‫א‬‫א‬،

‫א‬‫א‬K10000111‫א‬‫א‬‫א‬‫א‬

0111‫א‬‫א‬ (4 Flip-Flops × 10ns) 40ns  10ns (tp)‫א‬

‫א‬‫א‬(counting speed)‫א‬K1000

‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬

1 × 10 9 f= n× tp

f = upper clock pulse frequency limit n = number of flip-flops in the counter circuit tp= propagation delay time of each flip-flop in nanoseconds

 W

Asynchronous Binary Down Counters‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬٢ J٣ J٦

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K"1"

‫א‬EEF٩ J٦F‫א‬K ‫א‬"1"‫א‬

- ١٥٤ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 Q ‫א‬KJ-K‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬Q‫א‬‫א‬

‫א‬KEEF٩ J٦F‫א‬‫א‬‫א‬Q‫א‬‫א‬‫א‬

Q3,Q2,Q1,Q0  (RESET)‫א‬‫א‬‫א‬‫א‬  Q ‫א‬LowQ‫א‬‫א‬‫א‬‫א‬K0000

FF1FF2FF3‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K1111

‫א‬High‫א‬‫א‬‫א‬KJ‫א‬‫א‬KHigh

 K‫א‬‫א‬‫א‬(Toggle)  Q Q Q Q   0 1 2 3

HIGH FF1

FF0 Q

J Clock Input

CK K

Q 1

FF2

CK Q

CK Q 1

 

FF3 Q

CK Q

E F 1

9 10 11 12 13 14 15 16 17

       

Clock Input 0

 

  E F

 

 K‫א‬‫א‬‫א‬‫א‬E٩ J٦F‫א‬ "0"Q0‫א‬،FF0‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬"0""1" Q 0 ‫א‬‫א‬،"1" - ١٥٥ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 Q1 ‫א‬"0""1"Q1‫א‬،FF1‫א‬‫א‬

،FF2‫א‬"0""1" Q1 ‫א‬‫א‬K"0""1"

 K‫א‬  

‫א‬

 ‫א‬‫א‬ Q3 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0

Q2 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0

Q1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0

Q0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

Binary Count

Cycle Repeats

 

 K‫א‬‫א‬‫א‬‫א‬‫א‬E٥ J٦F‫א‬ (15)10 = 1111Q3,Q2,Q1,Q0‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬KE٥ J٦F‫א‬‫א‬‫א‬   ‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬

 FF0‫א‬‫א‬،EEF٩ J٦F‫א‬

 ،‫א‬‫א‬ ‫א‬،‫א‬Q0‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬Q3,Q2,Q1

- ١٥٦ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 ‫א‬‫א‬‫א‬L‫א‬‫א‬‫א‬‫א‬‫א‬٣ J٣ J٦

Asynchronous Binary Up/Down Counters

‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

Q‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K Q ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

L‫א‬E١٠ J٦F

 K UP / DOWN ‫א‬‫א‬AND-OR  

HIGH FF0 Q

J Clock Input

CK K

UP/DOWN control

Q 1

FF1 Q 1 J

FF2 Q

Q 1

 

  FF3 Q

CK Q

   

   K‫א‬‫א‬‫א‬‫א‬E١٠ J٦F‫א‬ ‫א‬AND‫א‬‫א‬،High‫א‬ UP / DOWN ‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬Q،(Enabled)‫א‬ ‫א‬‫א‬،‫א‬‫א‬‫א‬،‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬،Low‫א‬ UP / DOWN

(Enabled)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬(Disabled)‫א‬ ‫א‬‫א‬، ‫א‬‫א‬‫א‬‫א‬‫א‬ Q 

 K‫א‬

- ١٥٧ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



Asynchronous Decade (MOD-10) Counters ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬٤ J٣ J٦

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬EEF١١ J٦F

 K(MOD-10)‫א‬‫א‬  

 

HIGH FF1

FF0 J Clock Input

Q 1

CK K

FF2 J

CLR

CK K

FF3

CLR

 

CK K

CLR

 

CLR

  Q

Q 1

E F 1

   

9 10

 

Clock Input 0

 

   

CLR

 

E F

 K‫א‬‫א‬‫א‬‫א‬E١١ J٦F‫א‬  

‫א‬E9F1001E0F0000‫א‬‫א‬‫א‬‫א‬

‫א‬EEF١١ J٦F‫א‬‫א‬  KE٦ J٦F‫א‬

- ١٥٨ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬1510F1111 1010‫א‬‫א‬‫א‬‫א‬ ‫א‬

‫א‬‫א‬E CLR F‫א‬‫א‬‫א‬‫א‬NAND ‫א‬E‫א‬

‫א‬‫א‬KQ3‫א‬‫א‬Q1‫א‬‫א‬‫א‬K‫א‬

،High‫א‬Q1Q3E‫א‬‫א‬10F1010‫א‬

‫א‬K‫א‬ (CLEAR)Low NAND‫א‬

‫א‬(inactive) CLR ‫א‬EEF١١ J٦F‫א‬

‫א‬KHigh‫א‬Q1Q3‫א‬‫א‬‫א‬‫א‬K10010000

‫א‬‫א‬‫א‬(CLEAR)،Q3‫ و‬Q1‫א‬ E٦ J٦F‫א‬‫א‬‫א‬‫א‬‫א‬K CLR ‫א‬‫א‬‫א‬

 K‫א‬9‫א‬0‫א‬‫א‬،‫א‬‫א‬   ‫א‬

 ‫א‬‫א‬ Q3 0 0 0 0 0 0 0 0 1 1

Q2 0 0 0 0 1 1 1 1 0 0

Q1 0 0 1 1 0 0 1 1 0 0

Q0 0 1 0 1 0 1 0 1 0 1

0 1 2 3 4 5 6 7 8 9

Cycle Repeats

Binary Count

 

 K‫א‬‫א‬‫א‬‫א‬‫א‬E٦ J٦F‫א‬ ‫א‬‫א‬(MOD-10)90‫א‬‫א‬‫א‬‫א‬ (

1 )Q3‫א‬،‫א‬ 10  .(Clock input)‫א‬‫א‬‫א‬

- ١٥٩ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬(Digital Voltmeter)‫א‬‫א‬ ،(Digital clocks)‫א‬‫א‬‫א‬

 K(Frequency Counter)‫א‬

Synchronous Binary Counters ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬٥ J٣ J٦

AND‫א‬J-K‫א‬‫א‬E١٢ J٦F

‫א‬‫א‬(MOD-16) (4-bit)‫א‬‫א‬‫א‬‫א‬

  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬EF‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬K  ‫א‬‫א‬‫א‬(Triggered)

 K‫א‬‫א‬‫א‬‫א‬، ‫א‬‫א‬‫א‬  

Q 1

HIGH

 

Clock Input

FF1

FF0 Q

Q 1

FF2 J

FF3 J

CK K

 K‫א‬‫א‬‫א‬E١٢ J٦F‫א‬

FF0‫א‬‫א‬KJ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬(Toggle)‫א‬،High‫א‬

Low‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬LowHigh High

‫א‬2‫א‬‫א‬‫א‬FF1‫א‬‫א‬KJ‫א‬

‫א‬‫א‬Q1‫א‬،Low‫א‬Q0‫א‬‫א‬KFF0‫א‬

Q1‫א‬،High‫א‬Q0‫א‬(No change)FF1 K(Toggle)

- ١٦٠ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



AND(A)‫א‬‫א‬FF2‫א‬‫א‬KJ‫א‬

،High AND(A)‫א‬Q0 = Q1 = High‫א‬KQ1Q0  K‫א‬‫א‬FF2‫א‬‫א‬(Enable)‫א‬‫א‬

‫א‬AND(B)‫א‬‫א‬FF3‫א‬‫א‬KJ‫א‬

AND(B)‫א‬High‫א‬Q2,Q1,Q0‫א‬KQ0Q1Q2  K‫א‬FF3‫א‬‫א‬‫א‬‫א‬High

Synchronous Counters Advantages ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ٦ J٣ J٦

(Ripple counters)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬

 KEEF٩ J٦F‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬EEF٨ J٦F

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬K‫א‬

‫א‬‫א‬K  ‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬ (Propagation-delay time)  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬

E‫א‬‫א‬F‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬K

‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬

‫א‬‫א‬‫א‬،‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ 

tp = Single (flip-flop) tp + Single (AND-gate) tp

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 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



     

‫א‬‫א‬،‫א‬‫א‬‫א‬E١ J٦F E١

K ‫א‬1101‫א‬ ‫א‬‫א‬‫א‬،E‫א‬‫א‬F1010‫א‬ K‫א‬‫א‬‫א‬

‫א‬، ‫א‬E‫א‬‫א‬‫א‬F‫א‬‫א‬‫א‬E٢ K‫א‬

 

‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٣ K‫א‬‫א‬‫א‬،‫א‬

 

 ‫؟‬‫א‬‫א‬‫א‬‫א‬‫؟‬‫א‬‫א‬‫א‬‫א‬‫א‬E٤ (16)‫א‬‫א‬‫א‬(f) ‫א‬‫א‬‫א‬E٥  K10ns(tp)‫א‬‫א‬‫א‬،‫א‬

‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٦  K5nsAND‫א‬‫א‬،10ns(tp)‫א‬‫א‬

 - ١٦٢ -

 

 ‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 ‫א‬    ‫א‬   ‫א‬‫א‬W‫א‬‫א‬ ١ ............................................................................................................................................... ‫א‬‫א‬‫א‬ ٢ .................................................................................................................................... Introduction١ J١ ٣ ...................................................................................... Decimal Numbering System‫א‬‫א‬‫א‬٢ J١ ٤ .......................................................................................... Binary Numbering System‫א‬‫א‬‫א‬٣ J١ ٦ ......................................................... Decimal-to-Binary Conversion‫א‬‫א‬‫א‬‫א‬‫א‬٤ J١ ٩ ........................................................ Binary-to-Decimal Conversion‫א‬‫א‬‫א‬‫א‬‫א‬٥ J١ ١٠ ...................................................................................... Binary Arithmetic‫א‬‫א‬‫א‬‫א‬٦ J١ ١١ ..................................................................................................................‫א‬‫א‬‫א‬‫א‬‫א‬٧ J١ ١٢ ........................................................................ Representation of Signed Numbers‫א‬‫א‬‫א‬‫א‬٨ J١ ١٤ ..................................... Arithmetic Operations with Signed Numbers‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬٩ J١ ١٥ ................................................................................. The Octal Numbering System‫א‬‫א‬‫א‬١٠ J١ ٢٣ ................................................................... Hexadecimal Numbering System‫א‬‫א‬‫א‬‫א‬١١ J١ ٣٢ .........................................................................................................................................................   ‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬ ٣٥ ............................................................................................................................................. ‫א‬‫א‬‫א‬ ٣٦ ................................................................................................................................ Introduction١ J٢ ٣٦ ........................................................................................................................ AND Gate AND‫א‬٢ J٢ ٣٩ ............................................................................................................................... OR Gate OR‫א‬ ٣ J٢ ٤١ ...................................................................................... NOT Gate (INVERTER) E‫א‬F NOT ‫א‬٤ J٢ ٤٢ ..................................................................................................................... NAND Gate NAND ‫א‬٥ J٢ ٤٣ ......................................................................................................................... NOR Gate NOR ‫א‬٦ J٢ ٤٤ ......................................................................................... Exclusive-OR GateE‫א‬F‫ א‬OR ‫א‬٧ J٢ ٤٥ ................................................................................... Exclusive-NOR GateE‫א‬F‫ א‬NOR ‫א‬٨ J٢ ٤٧ .................................................The Boolean Expression for a Logic Circuit‫א‬‫א‬‫א‬‫א‬‫א‬٩ J٢ ٤٨ ...................................................................................................... ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬١٠ J٢ ٤٩ .......................................................................................................... ‫א‬‫א‬‫א‬‫א‬١١ J٢ ٥٢ .............................................................................................................. ‫א‬‫א‬‫א‬١٢ J٢ ٥٣ ......................................................................................................................................................... 

 ‫א‬

 ‫ א‬١٦٧

‫א‬

 ‫א‬



 ‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬  J٥٦ J .................................................................................................................................... ‫א‬‫א‬‫א‬  J٥٧ J ....................................................................................................................... Introduction١ J٣  J٥٧ J ................................................................................. Rules of Boolean Algebra‫א‬‫א‬‫א‬٢ J٣  J٥٩ J ...........................................................................................Demorgan's Theorems٣ J٣  J٦١ J .................................................................................. ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬٤ J٣  J٦٣ J.......................................... Standard Forms of Boolean Expressions‫א‬‫א‬‫א‬‫א‬٥ J٣  J٦٤ J ............................................................................ (POS)‫א‬‫א‬(SOP)‫א‬‫א‬‫א‬٦ J٣  J٦٦ J ........................................................................... (SOP)‫א‬‫א‬(POS)‫א‬‫א‬‫א‬٧ J٣  J٦٦ J .......................................................................................‫א‬‫א‬(SOP)‫א‬‫א‬٨ J٣  J٦٨ J ....................................................................................... ‫א‬‫א‬(POS)‫א‬‫א‬٩ J٣  J٧٠ J ................................................................................................ ‫א‬‫א‬‫א‬‫א‬‫א‬١٠ J٣  J٧١ J ................................................................................................... NAND, NOR‫א‬‫א‬‫א‬‫א‬١١ J٣  J٧٣ J .................................................................. NOR,NAND‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬١٢ J٣  J٧٩ J ......................................................................................................... Karnaugh Map١٢ J٣  J٩٠ J ................................................................................................................................................   ‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬‫א‬  J٩٣ J .................................................................................................................................... ‫א‬‫א‬‫א‬  J٩٤ J .......................................................................................................................... Introduction١ J٤  J٩٥ J ....................................................................Binary Adders and Subtractors‫א‬‫א‬‫א‬‫א‬٢ J٤  ١٠٣ ................................................................................................................................. Decoder‫א‬٣ J٤  ١٠٥ ......................................................................................................................................... Encoder‫א‬٤ J٤  ١٠٦........................................................................................................................ Multiplexer‫א‬٥ J٤  ١٠٩ ...................................................................................................................... Demultiplexer‫א‬٦ J٤  ١١٠ .............................................................................................................................. Comparators‫א‬٧ J٤ ١١٣ ........................................................................................................................................................   ‫א‬‫א‬‫א‬W‫א‬‫א‬  J١١٤ J ................................................................................................................................... ‫א‬‫א‬‫א‬  J١١٥ J ........................................................................................................................Introduction١ J٥  J١١٥ J ........................................................................................................................... Latches‫א‬٢ J٥  J١٢٠ J .....................................................................................Clocked S-R Flip–Flop‫א‬‫א‬S-R‫א‬٣ J٥  J١٢٣ J ......................................................................................... D-Type Flip-FlopD ‫א‬‫א‬‫א‬٤ J٥  J١٢٥ J .................................................................................................... J-K Flip Flop‫א‬‫א‬J-K‫א‬٥ J٥  J١٢٧ J..........................................................................................T-Type Flip-FlopT ‫א‬‫א‬‫א‬٦ J٥

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

 J١٢٨ J................................................................................... Master-Slave Flip-Flop‫א‬–‫א‬٧ J٥ ١٣٦ .......................................................................................................................................................   ‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬ ١٣٩............................................................................................................................................ ‫א‬‫א‬‫א‬  ١٤٠  K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K Introduction١ J٦ ١٤٠ ................................................................................................................................. Registers‫א‬٢ J٦ ١٥٢ .................................................................................................................................. Counters‫א‬‫א‬‫א‬٣ J٦  J١٦٢ J .............................................................................................................................................. 

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