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
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