Lecture10

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

CS 21 - Computer Organization and Assembly Language Programming Lecture 10 Boolean Algebra 1

University of the Philippines - Diliman College of Engineering Department of Computer Science

Boolean Algebra

Huntington’s Axioms/Postulates

Outline

1

Boolean Algebra

2

Huntington’s Axioms/Postulates

3

Fundamental Theorems

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Outline

1

Boolean Algebra

2

Huntington’s Axioms/Postulates

3

Fundamental Theorems

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

What is an Algebra?

An algebra is a mathematical theory involving a set of elements K and a set of operations I that acts on members of K

Boolean Algebra

Huntington’s Axioms/Postulates

High School Algebra

K = R (infinite set of real numbers) I = +, - , x , /

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

Boolean Algebra

K = 0, 1 Named after English mathematician George Boole From Boole’s book The Laws of Thought(1854) Boole showed how to reduce human reasoning to symbolic form resembling ordinary numerical algebra

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

Boolean Algebra

In 1930s Claude Shannon showed that Boolean algebra provides the theory needed for the analysis and design of digital circuits built from electromechanical relays Why? on/off (circuits) are analogs of true/false (logic) Boolean Algebra has remained relevant up to this day since it addresses the structure and behavior of circuits in an abstract, technology independent form.

Boolean Algebra

Huntington’s Axioms/Postulates

Outline

1

Boolean Algebra

2

Huntington’s Axioms/Postulates

3

Fundamental Theorems

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

What are axioms/postulates?

Axioms/postulates are rules that govern the manner in which algebraic operations can be used.

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

Huntington’s Axioms/Postulates

Axioms for Boolean algebra Published by American mathematician Edward V Huntington in 1904 Employs 3 operations: AND(x), OR(+), NOT(’)

Boolean Algebra

Huntington’s Axioms/Postulates

Huntington’s Axioms 1A & 1B

Closure Axioms 1A. If a and b are in K, then a + b is in K 1B. If a and b are in K, then ab is in K

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

Huntington’s Axioms 2A & 2B

Zero Axiom 2A. There is an element 0 in K such that a + 0 = a Unit Axiom 2B. There is an element 1 in K such that a1 = a

Boolean Algebra

Huntington’s Axioms/Postulates

Huntington’s Axioms 3A & 3B

Commutativity Axioms 3A. For all a and b in K, a + b = b + a 3B. For all a and b in K, ab = ba

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

Huntington’s Axioms 4A & 4B

Distributivity Axioms 4A. For all a, b and c in K, a + bc = (a + b)(a + c) 4B. For all a, b and c in K, a(b + c) = ab + ac

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

Huntington’s Axioms 5A & 5B

Inverse Axioms 5A. For each a in K there is an inverse or complement element a’ in K such that a + a’ = 1 5B. For each a in K there is an inverse or complement element a’ in K such that aa’ = 0

Boolean Algebra

Huntington’s Axioms/Postulates

Huntington’s Axiom 6

6. There are at least two distinct elements in K

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

Boolean Algebra and gate networks/circuits

Boolean algebra expressions are easily implementable in gate network/circuit form variables = values operations = gates

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

Circuit Interpretation of Boolean Algebra Axioms Zero Axiom

Unit Axiom

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

Circuit Interpretation of Boolean Algebra Axioms

Distributivity Axiom

Boolean Algebra

Huntington’s Axioms/Postulates

Outline

1

Boolean Algebra

2

Huntington’s Axioms/Postulates

3

Fundamental Theorems

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

What is a Theorem?

A true statement or proposition about an algebra derived from axioms.

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

Why are boolean algebra theorems important?

Along with axioms, theorems provide the rules needed to simplify logic circuits, change gate types, etc

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

Why would we like to simplify circuits?

Hardware availability constraints(gate types, limit to number of inputs, etc) Better performance

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorem 1

Uniqueness Theorems 1A. The zero element 0 is unique 1B. The unit element 1 is unique

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorem 2

Idempotence 1A. For each a in K, a + a = a 1B. For each a in K, aa = a

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorem 3

Unit Property 3A. For each a in K, a + 1 = 1 Zero Property 3B. For each a in K, a0 = 0 Identity Property 3C. For each a in K, a+0 = a and a1 = a

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorem 4

Involution 4. For each a in K, (a’)’ = a

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorem 5

Inverse 5a. For each a in K, the inverse a’ is unique 5b. For each a in K, aa’ = 0 and a + a’= 1

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

Fundamental Theorem 6

Commutativity 6A. For all a, b, and c in K, a + b = b + a 6B. For all a, b, and c in K, ab = ba Associativity 6C. For all a, b, and c in K, (a + b) + c = a + (b + c) 6D. For all a, b, and c in K, (ab)c = a(bc) Distributivity 6E. For all a, b, and c in K, a(b + c) = ab + ac 6F. For all a, b, and c in K, a + (bc) = (a + b)(a + c)

Boolean Algebra

Huntington’s Axioms/Postulates

Other Simplification Theorem 7

De Morgan’s Theorems 7A. For all a and b in K, (a + b)’ = a’b’ 7B. For all a and b in K, (ab)’ = a’ + b’

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Other Simplification Theorem 8

Absorption 8A. For all a and b in K, a + ab = a 8B. For all a and b in K, a(a + b) = a 8C. For all a and b in K, (ab)+(ab’) = a 8D. For all a and b in K, (a+b)(a+b’) = a

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

Other Simplification Theorem

Duality 9. The dual of any statement in a Boolean algebra is the statement obtained by interchanging AND and OR , and simultaneously inter-changing the elements 0 and 1 in the statement Example, given the distributive law a(b + c) = ab + ac the dual expression is a+bc = (a + b)(a + c) Multiplication and Factoring 10. For all a, b, and c in K, (a + b)(a’ + c) = ac + a’b 11. For all a, b, and c in K, ab + a’c = (a + c)(a’ + b)

Boolean Algebra

Huntington’s Axioms/Postulates

Application

Prove the equivalence of E1 and E2 E1 = a’b’h + c’fgh + d’fgh + e’fgh E2 = ((a + b)(cde + (fg)’))’h Options Use truth table (so many rows!)

Use theorems

Fundamental Theorems

Boolean Algebra

Example

Simplify c + (bc)’

Huntington’s Axioms/Postulates

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Example

Simplify c + (bc)’ 1

c + (bc)’ — original expression

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Example

Simplify c + (bc)’ 1

c + (bc)’ — original expression

2

c + (b’ + c’) — DeMorgan’s

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Example

Simplify c + (bc)’ 1

c + (bc)’ — original expression

2

c + (b’ + c’) — DeMorgan’s

3

(c + c’) + b’ — Comm, Assoc

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Example

Simplify c + (bc)’ 1

c + (bc)’ — original expression

2

c + (b’ + c’) — DeMorgan’s

3

(c + c’) + b’ — Comm, Assoc

4

1 + b’ — Complement

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Example

Simplify c + (bc)’ 1

c + (bc)’ — original expression

2

c + (b’ + c’) — DeMorgan’s

3

(c + c’) + b’ — Comm, Assoc

4

1 + b’ — Complement

5

1 — Identity

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Example

Simplify c + (bc)’ 1

c + (bc)’ — original expression

2

c + (b’ + c’) — DeMorgan’s

3

(c + c’) + b’ — Comm, Assoc

4

1 + b’ — Complement

5

1 — Identity

6

Revelation: It’s always true!

Fundamental Theorems

Boolean Algebra

Huntington’s Axioms/Postulates

Fundamental Theorems

Difference between expressions and functions

Boolean expressions is an implementation of a Boolean function Boolean functions define relationships between inputs and output; tied to a truth table Every Boolean expression defines a unique Boolean function A Boolean function can be defined by many different but equivalent Boolean expressions

Boolean Algebra

Huntington’s Axioms/Postulates

So...

Any questions?

Fundamental Theorems