Boolean Algebra: Understanding the Foundation of Digital Logic, Slides of Computer Science

Download Boolean Algebra: Understanding the Foundation of Digital Logic and more Slides Computer Science in PDF only on Docsity!

Boolean Algebra

Introduction

Boolean Algebra

• We observed in our introduction that early in the development of computer hardware, a decision was made to use binary circuits because it greatly simplified the electronic circuit design.
• In order to work with binary circuits, it is helpful to have a conceptual framework to manipulate the circuits algebraically, building only the final “most simple” result.
• George Boole (1813-1864) developed a mathematical structure to deal with binary operations with just two values. Today, we call these structures Boolean Algebras.

Axioms

Axiom # 4: Associative

If a and b are elements of B (a + b) + c = a + (b + c), and (a * b) * c = a * (b * c)

Axiom # 5: Identity Element

B has identity elements with respect to + and * 0 is the identity element for +, and 1 is the identity element for * a + 0 = a and a * 1 = a

Axiom # 6: Distributive

• is distributive over + and + is distributive over * a * (b + c) = (a * b) + (a * c), and a + (b * c) = (a + b) * (a + c)

Axiom #7: Complement Element For every a in B there is an element a' in B such that a + a' = 1, and a * a' = 0

Terminology

• Element 0 is called “FALSE”.
• Element 1 is called “TRUE”.
• ‘+’ operation “OR”,‘*’ operation “AND” and ’ operation “NOT”.
• Juxtaposition implies * operation: ab = a * b
• Operator order of precedence is: (), ’, *, +. a+bc = a+(bc) ≠ (a+b)c ab’ = a(b’) ≠ (a*b)’
• Single Bit Boolean Algebra( 1’ = 0 and 0’ = 1)

• 0 1 0 0 1 1 1 1

n-bit Boolean Algebra

• Single bit Boolean Algebra can be extended to n-bit Boolean Algebra by define sum(+), product(*) and complement (‘) as bit-wise operations
• Let a = 1101010, b = 1011011
• a + b = 1101010 + 1011011 = 1111011
• a * b = 1101010 * 1011011 = 1001010
• a’ = 1101010’ = 0010101 Principle of Duality The dual of a statement S is obtained by interchanging * and +; 0 and 1. Dual of (a1)(0+a’) = 0 is (a+0)+(1*a’) = 1 Dual of any theorem in a Boolean Algebra is also a theorem. This is called the Principle of Duality.

Named Theorems

All of the following theorems can be proven based on the axioms. They are used so often that they have names.

Idempotent a + a = a a * a = a

Boundedness a + 1 = 1 a * 0 = 0

Absorption a + (ab) = a a(a+b) = a

Associative (a+b)+c=a+(b+c) (ab)c=a(bc)

The theorems can be proven for a two-bit Boolean Algebra using a truth table, but you must use the axioms to prove it in general for all Boolean Algebras.

Proof using Theorems

• Use the properties of Boolean Algebra to reduce

(x + y)(x + x) to x. Warning, make sure you use

the laws precisely.

(x + y)(x + x) Given

(x + y)x Idempotent

x(x + y) Commutative

x Absorption

Unlike truth tables, proofs using Theorems are valid for any boolean algebra, but just bits.

Sources

• Lipschutz, Discrete Mathematics
• Mowle, A Systematic Approach to Digital Logic

Design