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ﻤـﻘـﺩﻤــﺔ Introduction
The Numbers
اﻷﻋﺪاد اﻟﺤﻘﯿﻘﯿﺔ R: real numbers
اﻷﻋﺪاد اﻟﺘﺨﯿﻠﯿﺔ Imagine numbers ﺍﻟﺠﺫﻭﺭ ﺍﻟﺯﻭﺠﻴﺔ ﻟﻸﻋﺩﺍﺩ ﺍﻟﺴﺎﻟﺒﺔ 4 ,
5 , 4 16 , .......
** ﻻ ﯾﻨﺘﻤﻲ ﻷي ﺷﻲء ﻣﻦ اﻟﻤﻘﺮر
ﺍﻷﻋﺩﺍﺩ ﺍﻟﻐﻴﺭ ﻨﺴﺒﻴﺔ
اﻷﻋﺪاد اﻟﻨﺴﺒﯿﺔ Q: rational numbers
II: irrational numbers
a : a, b z , b 0 b
o
, e , 2 , 5 , 3 7 , sin 1, ......
∗ * 2 1.4142135 .....
∗
* 3.14159..........
2 = 0.6666. . = 0. 6 3
157 = 0.31717 … = 0.317 495
ﺟﺰء ﻣﻦ اﻷرﻗﺎم ﺑﻌﺪ اﻟﻔﺎﺻﻠﺔ ﻣﻜﺮر
.ﻧﻼﺣﻆ أن اﻷرﻗﺎم ﺑﻌﺪ اﻟﻔﺎﺻﻠﺔ ﻏﯿﺮ ﻣﻜﺮرة
R Q
z w N
* N W Z Q R
اﻷﻋﺪاد اﻟﺼﺤﯿﺤﺔ Z: Integers {...., -2 , -1 , 0 , 1 , 2 , ....}
اﻷﻋﺪاد اﻟﻜﻠﯿﺔ
* II R P=II
W: whole numbers
{0 , 1 , 2 , .......}
*P∪Q=R
Zero
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ﺍﻷﻋﺩﺍﺩ ﺍﻟﻁﺒﻴﻌﻴﺔ N: natural numbers {1 , 2 , 3 , .....}
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Example: 1 Whole number = w is: a)
3
b) 12
8 2
c) 0.5
d)
2 Whole number = w is : a) -2
b)
c) -3.2
d)
25 5
3 Integer = z is : a) 4
b)
25 5
2 d) 3
c) 5.3
2
Integer = z is : a)
b)
c)
2
d) 5.3
3 8 2
5 Irrational number = II is: a)
2 3
b)
c) 0
2
d)
5
25
5 1
d)
2
5
6 Rational number = Q is: a)
5
5
b)
c)
2
3
d)
4
7 Rational number = Q is: a)
5
5
2 3
b) 4
14 3
c)
3
4
8 Natural number = N is : a) 4
b)
c)
3
4
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1.1
Basics of sets A set is a collection of objects are called members or elements : ھﻲ ﺗﺠﻤﻊ ﻣﻦ اﻷﺷﯿﺎء ﺗﺴﻤﻰ أﻋﻀﺎء أو ﻋﻨﺎﺻﺮ.
A L S A A D I
ﺷﺮط أﺳﺎﺳﻲ ﻟﻜﻲ ﺗﻜﻮن اﻟﻌﻨﺎﺻﺮ ﻣﺠﻤﻮﻋﺔ أن ﺗﻜﻮن ﻣﻌﺮوﻓﺔ وﻣﺤﺪودة ﺗﺤﺪﯾﺪاً ﺗﺎﻣﺎً أي ﻻ ﯾﺨﺘﻠﻒ ﻋﻠﯿﮭﺎ.
أﯾﺎم اﻷﺳﺒﻮع ) :(Days of weekﺗﻤﺜﻞ ﻣﺠﻤﻮﻋﺔ ﻷﻧﮭﺎ ﻣﺤﺪدة ﺗﺤﺪﯾﺪاً ﺗﺎﻣﺎً وﻻ ﯾﺨﺘﻠﻒ ﻋﻠﯿﮭﺎ. اﻷﻃﻌﻤﺔ اﻟﻠﺬﯾﺬة ) : (the best foodsﻻ ﺗﻤﺜﻞ ﻣﺠﻤﻮﻋﺔ ﻷﻧﮭﺎ ﺗﺨﺘﻠﻒ ﻣﻦ ﺷﺨﺺ ﻵﺧﺮ. اﻟﻤﺠﻤﻮﻋﺔ ﯾﺮﻣﺰ ﻟﮭﺎ ﺑﺎﻟﺤﺮوف اﻟﻜﺒﯿﺮة )ﻛﺎﺑﺘﻞ(. A, B, X, Y ………. اﻟﻌﻨﺎﺻﺮ ﯾﺮﻣﺰ ﻟﮭﺎ ﺑﺎﻟﺤﺮوف اﻟﺼﻐﯿﺮة )ﺻﻤﻮل( a, b, x, y………. } X = { Saturday, Sunday, …. , Friday } X = { x : x is a day of the week
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, … ,
}
ﻋﻼﻗﺔ ﻋﻨﺼﺭ ﺒﻤﺠﻤﻭﻋﺔ
:ﺍﻟﻌﻨﺼﺭ ﻴﻨﺘﻤﻲ ﺇﻟﻰ ﺍﻟﻤﺠﻤﻭﻋﺔ
∈X
∉X
Example: Saturday
:ﺍﻟﻌﻨﺼﺭ ﻻﻴﻨﺘﻤﻲ ﺇﻟﻰ ﺍﻟﻤﺠﻤﻭﻋﺔ
Example: Summer ∉ X
∈X
ﻋﻼﻗﺔ ﻤﺠﻤﻭﻋﺔ ﺒﻤﺠﻤﻭﻋﺔ ﺁﺨﺭﻯ
Y ﻣﻮﺟﻮده ﻓﻲ اﻟﻤﺠﻤﻮﻋﺔX ﻛﻞ ﻋﻨﺎﺻﺮ اﻟﻤﺠﻤﻮﻋﺔ
⊆Y
Subset ⊂ ﺠﺯﺌﻴﺔ Example: Natural number ⊂ lntegers N = {1,2,3,…..} ⊂ Z= {…,-1,0,1,…}
Y ﻏﻴﺭ ﻤﻭﺠﻭﺩ ﻓﻲX ﻋﻨﺼﺭ ﻋﻠﻰ ﺍﻻﻗل ﻓﻲ
⊄Y Not subset ⊄ ﻟﻴﺴﺕ ﺠﺯﺌﻴﺔ Example: Prime numbers ⊄ odd numbers {2,3,5,…..} ⊄ {1,3,5,…..}
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Example: }X = {1,3,5,6,7,8 }Y = {2,3,4,6,7
If :
Find : ∩
1
∪
2
A 3 / L S Solution A } = {3,6,7ﺍﻟﻌﻨﺼﺭ ﺍﻟﻤﺸﺘﺭﻜﺔ ∩ 1 1. A D ﺍﻟﻜل ﺒﺩﻭﻥ ﺘﻜﺭﺍﺭ ∪ 2 2. }={1,3,5,6,7,8,2.4 I /
ﺍﻟﻤﻭﺠﻭﺩ ﻓﻲ Xﻭﻏﻴﺭ ﻤﻭﺠﻭﺩ ﻓﻲ Y }= {1,5,8
33.
ﺍﻟﺘﻘﺎﻁﻊ
∩
1
ﺘﻘﺎﻁﻊ ﻤﺠﻤﻭﻋﺘﻴﻥ ﻫﻲ ﺍﻟﻌﻨﺎﺼﺭ ﺍﻟﻤﺸﺘﺭﻜﺔ ﺒﻴﻥ ﺍﻟﻤﺠﻤﻭﻋﺘﻴﻥ
} ∈
∈ ∩ ={ :
ﺍﻻﺘﺤﺎﺩ
∪
2
ﺍﺘﺤﺎﺩ ﻤﺠﻤﻭﻋﺘﻴﻥ ﻫﻲ ﻜل ﻋﻨﺎﺼﺭ ﺍﻟﻤﺠﻤﻭﻋﺘﻴﻥ ﺒﺩﻭﻥ ﺘﻜﺭﺍﺭ } ∈ ∈ ∪ = { :
ﺍﻟﻤﻜﻤﻠﺔ /
3
ﻤﻜﻤﻠﺔ Yﺒﺎﻟﻨﺴﺒﺔ ﻟـX Complement of Y with respect to X ﺍﻟﻌﻨﺎﺼﺭ ﺍﻟﻤﻭﺠﻭﺩﺓ ﻓﻲ X ﻭﻏﻴﺭ ﻤﻭﺠﻭﺩﺓ ﻓﻲ Y
} ∉
∈ / ={ :
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Cartesian Product X × Y : all ordered pairs obtained from the element of X as the first coordinate and element of Y as the second coordinate. وﻣﺴﺎﻗﻄﮭﺎX ﻋﺒﺎرة ﻋﻦ ﻛﻞ اﻷزواج اﻟﻤﺮﺗﺒﺔ اﻟﺘﻲ ﻣﺴﺎﻗﻄﮭﺎ اﻷوﻟﻰ ﻋﻨﺎﺻﺮ ﻣﻦ .Y اﻟﺜﺎﻧﯿﺔ ﻋﻨﺎﺻﺮ ﻣﻦ
Example: If : = {2 , 3 , 5} ,
= { , }
Find : (1) ×
(2) ×
(3) ×
Solution (1)
×
= {(2, ), (2, ), (3, ), (3, ), (5, ), (5, )}
(2)
×
= {( , 2), ( , 3), ( , 5), ( , 2), ( , 3), ( , 5)}
(3)
×
= {( , ), ( , ), ( , ), ( , )}
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Types of intervals
1) Finite intervals: ﻓﺘﺮات ﻣﺤﺪودة
Notation
Set description
Type
(a , b)
{ x ∈ R / a < x < b}
Open
[a , b]
{ x ∈ R / a ≤ x ≤ b}
Closed
[a , b)
{ x ∈ R / a ≤ x < b}
Half - open
(a , b]
{ x ∈ R / a < x ≤ b}
Half - open
Picture a
b
a
b
a
b
a
b
2) Infinite intervals: ﻓﺘﺮات ﻏﯿﺮ ﻣﺤﺪودة
Notation
Set description
Type
(a ,∞)
{ x ∈ R / x > a}
Open
Picture
a
(-∞, b]
{ x ∈ R / x ≤ b}
Half - Closed
(-∞,∞)
R: set of all real numbers
open
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Example: Write the sets as the intervals and show on the real line.
1
{ ∈ /− ≤
< 3}
= [ -3, 3 ) -3
2
{ ∈ /− <
3
A L S A A D I
< 5} -2
5
= (-2 , 5) 3
{ ∈ / ≤ − } = (−∞, − ]
4
{ ∈ / > −2}
-2
= (-2 ,∞)
5
-2
{ ∈ }
= (-∞,∞)
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Example : 1
7
](1) (1,7] ∖(4,8) = (1,4
4
8
](2) (1,7] ∩ (4,8) = (4,7
A L S A A D I
ﻤﻥ ﺍﻟﺭﺴﻡ ﻓﻲ ) (1ﺍﻟﺠﺯﺀ ﺍﻟﻤﺸﺘﺭﻙ ﻓﻲ ﺍﻟﺘﻅﻠﻴل
)(3) (1,7] ∪ (4,8) = (1,8 ﻤﻥ ﺍﻟﺭﺴﻡ ﻓﻲ ) (1ﺒﺩﺍﻴﺔ ﺍﻟﺭﺴﻡ ﻟﻨﻬﺎﻴﺘﻪ 5
-2 0
-4
](4) [-2,5] ∖(-4,0] = (0,5
](5) [-2,5] ∩ (-4,0] = [-2,0 ﻤﻥ ﺍﻟﺭﺴﻡ ﻓﻲ ) (4ﺍﻟﺠﺯﺀ ﺍﻟﻤﺸﺘﺭﻙ ﻓﻲ ﺍﻟﺘﻅﻠﻴل
](6) [-2, 5] ∪ (-4,0] = (-4,5 ﻤﻥ ﺍﻟﺭﺴﻡ ﻓﻲ ) (4ﺒﺩﺍﻴﺔ ﺍﻟﺭﺴﻡ ﻟﻨﻬﺎﻴﺘﻪ
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إذا ﻛﺎﻧﺖ اﻟﻔﺘﺮﺗﯿﻦ ﻣﺘﺒﺎﻋﺪﺗﯿﻦ أي ﻟﯿﺲ ﺑﯿﻨﮭﻤﺎ ﺟﺰء ﻣﺸﺘﺮك ﻓﺈن
2 6
ﺍﻟﻤﻜﻤﻠﺔ ﻫﻲ ﺍﻟﻔﺘﺭﺓ ﺍﻷﻭﻟﻰ ﺩﺍﺌﻤﺎﹰ
-1
ﺍﻟﺘﻘﺎﻁﻊ ∅
ﺍﻻﺘﺤﺎﺩ ﻜﻤﺎ ﻫﻭ
4
ﺍﻟﻤﻭﺠﻭﺩ ﻓﻲ ﺍﻟﻔﺘﺭﺓ ﺍﻷﻭﻟﻰ ] [-1 , 2
A L S A A D I
ﻭﻏﻴﺭ ﻤﻭﺠﻭﺩ ﻓﻲ ﺍﻟﻔﺘﺭﺓ ﺍﻟﺜﺎﻨﻴﺔ ] ( 4 , 6
ﺍﻟﻤﻭﺠﻭﺩ ﻓﻲ ﺍﻟﻔﺘﺭﺓ ﺍﻷﻭﻟﻰ ] ( 4 , 6 ﻭﻏﻴﺭ ﻤﻭﺠﻭﺩ ﻓﻲ ﺍﻟﻔﺘﺭﺓ ﺍﻟﺜﺎﻨﻴﺔ ] [-1 , 2
ﺍﻟﺘﻘﺎﻁﻊ :ﻻﻴﻭﺠﺩ ﻤﺸﺘﺭﻙ ﺒﻴﻥ ﺍﻟﻔﺘﺭﺘﻴﻥ
∴ ﺍﻟﺘﻘﺎﻁﻊ ∅
] ∖ ( ,
11) – ,
اﻟﻔﺘﺮة اﻷوﻟﻰ ]= [−1, 2
] 22) ( , ] ∖ [− , اﻟﻔﺘﺮة اﻷوﻟﻰ ] = ( ,
] 33) [− , ] ∩ ( , اﻟﻤﺠﻤﻮﻋﺔ اﻟﺨﺎﻟﯿﺔ ∅ =
] 44) [− , ] ∪ ( ,
ﺍﻻﺘﺤﺎﺩ :ﻜﻤﺎ ﻫﻭ
ﺃﻭ ﻤﻤﻜﻥ ﻜﺘﺎﺒﺔ ﺍﻟﻨﺎﺘﺞ ﺒﺎﻟﺸﻜل
] = [− , ] − ( , ] Or [− , ] ∖ ( ,
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Exercises ﺤل ﺘﻤﺎﺭﻴﻥ ﺍﻟﻜﺘﺎﺏ
Evaluate the following:
0
7 10
-11-
1
Dream Team
][0,7) ∪ (1,10] = [0,10
1
اﻻﺗﺤﺎد ∪ :ﺑﺪاﯾﺔ اﻟﺮﺳﻢ ﻟﻨﮭﺎﯾﺘﮫ.
)[0,7) ∩ (1,10) = (1,7
2
ﻧﻔﺲ اﻟﺮﺳﻢ ﻓﻲ )(1 اﻟﺘﻘﺎﻃﻊ ∩ :اﻟﺠﺰء اﻟﻤﺸﺘﺮك ﻓﻲ اﻟﺮﺳﻢ
][0,7) ∖ (1,10] = [0,1
3
ﻧﻔﺲ اﻟﺮﺳﻢ ﻓﻲ )(1 اﻟﻤﻜﻤﻠﺔ ∖ اﻟﻤﻮﺟﻮد ﻓﻲ اﻟﻔﺘﺮة اﻷوﻟﻰ وﻏﯿﺮ ﻣﻮﺟﻮد ﻓﻲ اﻟﻔﺘﺮة اﻟﺜﺎﻧﯿﺔ.
](1,10] ∖ [0,7) = [7,10
4
ﻧﻔﺲ اﻟﺮﺳﻢ ﻓﻲ )(1 اﻟﻤﻜﻤﻠﺔ ∖ اﻟﻤﻮﺟﻮد ﻓﻲ اﻟﻔﺘﺮة اﻷوﻟﻰ ][1,10 وﻏﯿﺮ ﻣﻮﺟﻮد ﻓﻲ اﻟﻔﺘﺮة اﻟﺜﺎﻧﯿﺔ )[0,7
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)∞ [0, ∞) ∖{0} = (0,
0
5
اﻟﻤﻮﺟﻮد ﻓﻲ اﻟﻔﺘﺮة وﻏﯿﺮ ﻣﻮﺟﻮد ﻓﻲ اﻟﻤﺠﻤﻮﻋﺔ. 0 2
A L S A A D I
}[0, ∞) ∖ {1,2
1
6
اﻟﻤﻜﻤﻠﺔ ∖ :اﻟﻤﻮﺟﻮد ﻓﻲ اﻟﻔﺘﺮة وﻏﯿﺮ ﻣﻮﺟﻮد ﻓﻲ اﻟﻤﺠﻤﻮﻋﺔ أو ﺑﻤﻌﻨﻰ أﺧﺮ اﻟﻐﺎء اﻟﻤﺠﻤﻮﻋﺔ ﻣﻦ اﻟﻔﺘﺮة
)∞ = [0,1) ∪ (1,2) ∪ (2, -2
3 0
)∞ (-2,3) ∪ (0,
7
اﻻﺗﺤﺎد ∪ :ﻛﻞ اﻟﻤﻨﺎﻃﻖ اﻟﻤﻈﻠﻠﺔ )ﺑﺪاﯾﺔ اﻟﺘﻈﻠﯿﻞ ﻟﻨﮭﺎﯾﺔ(
)∞ = (-2,
)∞ (-2,3) ∩ (0,
8
ﻧﻔﺲ اﻟﺮﺳﻢ ﻓﻲ )(7
اﻟﺘﻘﺎﻃﻊ ∩ :اﻟﺠﺰء اﻟﻤﺸﺘﺮك ﻓﻲ اﻟﺘﻈﻠﯿﻞ اﻟﻌﻠﻮي واﻟﺴﻔﻠﻲ ﻟﻠﺨﻂ.
)= (0,3
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9
State: the set of non – negative integers. . اﻷﻋﺪاد اﻟﺼﺤﯿﺤﺔ اﻟﻐﯿﺮ ﺳﺎﻟﺒﺔ Non – negative integers = {0,1,2,3,…}
10 State: the set of non – negative real numbers. .اﻷﻋﺪاد اﻟﺤﻘﯿﻘﯿﺔ اﻟﻐﯿﺮ ﺳﺎﻟﺒﺔ
Non – negative real numbers = [0, ∞) 0
11 State: the set of positive real numbers. . اﻷﻋﺪاد اﻟﺤﻘﯿﻘﯿﺔ اﻟﻤﻮﺟﺒﺔ Positive real numbers = (0, ∞) 0
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