9 February, 2000

Everyone's favorite example -- the seven segment display

 _         _    _         _    _    _    _    _    _         _         _    _
| |    |   _|   _|  |_|  |_   |_     |  |_|  |_|  |_|  |_   |     _|  |_   |_ 
|_|    |  |_    _|    |   _|  |_|    |  |_|   _|  | |  |_|  |_   |_|  |_   |

Laws of Boolean Algebra

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)
Identity α + 0 = α α 1 = α
Idempotency α + α = α α α = α
Annihilation α + 1 = 1 α 0 = 0
Complement α + α' = 1 α α' = 0
Involution (α')' = α
de Morgan's law (α + β)' = α' β' (α β)' = α' + β'
absorption α + α β = α α (α + β) = α
absorption 2 α + α' β = α + β α (α' + β) = α β

The example -- continued

Simplifiction

x' y' z' + x' y' z + x' y z' + x' y z + x y' z' + x y z Sum-of-products
x' y' (z' + z) + x' y z' + x' y z + x y' z' + x y z Distributivity
x' y' 1 + x' y z' + x' y z + x y' z' + x y z Complement
x' y' + x' y z' + x' y z + x y' z' + x y z Identity
x' y' + x' y (z' + z) + x y' z' + x y z Distributivity
x' y' + x' y 1 + x y' z' + x y z Complement
x' y' + x' y + x y' z' + x y z Identity
x' (y' + y) + x y' z' + x y z Distributivity
x' 1 + x y' z' + x y z Complement
x' + x y' z' + x y z Identity
x' + x (y' z') + x (y z) Associativity
x' + y' z' + x (y z) Absorption 2
x' + y' z' + y z Absorption 2

A much harder example

Simplifiction

x' y' z' w' + x' y' z' w + x' y' z w' + x' y' z w + x' y z' w' + x' y z w + x y' z' w' + x y' z' w + x y' z w' + x y z' w Sum-of-products
x' y' z' w' + x' y' z w' + x' y' z' w + x' y' z w + x' y z' w' + x' y z w + x y' z' w' + x y' z w' + x y' z' w + x y z' w Associativity
x' y' (z' + z) w' + x' y' (z' + z) w + x' y z' w' + x' y z w + x y' (z' + z) w' + x (y' + y) z' w Distributivity
x' y' w' + x' y' w + x' y z' w' + x' y z w + x y' w' + x z' w Complement and Identity
x' y' w' + x' y z' w' + x' y' w + x' y z w + x y' w' + x z' w Associativity
x' (y' + y z') w' + x' (y' + y z) w + x y' w' + x z' w Distributivity
x' (y' + z') w' + x' (y' + z) w + x y' w' + x z' w Absorption 2
x' y' w' + x' z' w' + x' y' w + x' z w + x y' w' + x z' w Distributivity
x' y' w' + x' y' w + x' z' w' + x' z w + x y' w' + x z' w Communitivity
x' y' (w' + w) + x' z' w' + x' z w + x y' w' + x z' w Distributivity
x' y' + x' z' w' + x' z w + x y' w' + x z' w Complement and Identity
x' y' + x y' w' + x' z' w' + x' z w + x z' w Communitivity and Associativity
(x' + x w') y' + x' z' w' + x' z w + x z' w Communitivity and Associativity
(x' + w') y' + x' z' w' + x' z w + x z' w Absorption 2
x' y' + y' w' + x' z' w' + x' z w + x z' w Distributivity and Communitivity