# CSCI 255, ENGR 274, ECE 212 -- 29 August, 2000

## The Boolean Algebra

• Two elements: 0 and 1
• Two binary operators: AND and OR
• One unary operator: NOT

## History of Boolean Algebra

• George Boole, 1815-1864
• Edward V. Huntington
• Developed the textbook (pp. 41-42) axioms in 1904
• Developed a three axiom set in 1933
• Commutativity
• Associativity
• Huntington equation: (x' + y)' + (x' + y')' = x
• Marshall Stone
• In 1936, represented Boolean algebras as "compact zero-dimensional Hausdorff space"
• In 1937, represented Boolean algebras as a field of sets
• Claude Shannon
• In 1938, showed that Boolean algebra can describe logic circuits
• Later developed information theory
• William McCune
• Works with automatic theorem proving at Argonne National Lab
• Used AI program EQP to prove the Robbin's conjecture in 1996

## Laws of Boolean Algebra

 Identity α + 0 = α α 1 = α Annihilation α + 1 = 1 α 0 = 0 Idempotency α + α = α α α = α Involution (α')' = α Complementarity α + α' = 1 α α' = 0 Commutativity α + β = β + α α β = β α Associativity α + (β + γ) = (α + β) + γ α (β γ) = (α β) γ Distributivity α (β + γ) = α β + α γ α + β γ = (α + β) (α + γ) de Morgan's law (α + β)' = α' β' (α β)' = α' + β' absorption α + α β = α α (α + β) = α α + α' β = α + β α (α' + β) = α β

## Duality

Take a Boolean equation. Change AND's to OR's, OR's to AND's, 0's to 1's, and 1's to 0's. The equation still holds.

## An example

Prove the uniting theorem, the equivalence of the following

• x y + x y'
• x

Show the equivalence of the following two Boolean equations

• x' y' z' + x' y' z + x' y z' + x' y z + x y' z' + x y z
• x' + y' z' + y z