Boolean Algebra Discussion D6.1 Sections 13-3 – 13-6

Boolean Algebra and Logic Equations George Boole Switching Algebra Theorems Venn Diagrams

George Boole English logician and mathematician Publishes Investigation of the Laws of Thought in 1854

One-variable Theorems OR Version AND Version X | 0 = X X | 1 = 1 X & 1 = X X & 0 = 0 Note:Principle of Duality You can change # to & and 0 to 1 and vice versa

One-variable Theorems OR Version AND Version X | !X = 1 X | X = X X & !X = 0 X & X = X Note:Principle of Duality You can change | to & and 0 to 1 and vice versa

Two-variable Theorems Commutative Laws Unity Absorption-1 Absorption-2

Commutative Laws X | Y = Y | X X & Y = Y & X

Venn Diagrams X !X

Venn Diagrams XY X & Y

Venn Diagrams X | Y XY

Venn Diagrams ~X & Y X Y

Unity ~X & Y X Y X & Y (X & Y) | (~X & Y) = Y Dual: (X | Y) & (~X | Y) = Y

Absorption-1 X Y X & Y Y | (X & Y) = Y Dual: Y & (X | Y) = Y

Absorption-2 ~X & Y X Y X | (~X & Y) = X | Y Dual: X & (~X | Y) = X & Y

Three-variable Theorems Associative Laws Distributive Laws

Associative Laws X | (Y | Z) = (X | Y) | Z Dual: X & (Y & Z) = (X & Y) & Z

Associative Law X Y Z Y | Z X | (Y | Z) X | Y (X | Y) | Z X | (Y | Z) = (X | Y) | Z

Distributive Laws X & (Y | Z) = (X & Y) | (X & Z) Dual: X | (Y & Z) = (X | Y) & (X | Z)

Distributive Law - a

Distributive Law - b X & (Y | Z) = (X & Y) | (X & Z)

Question The following is a Boolean identity: (true or false) Y | (X & ~Y) = X | Y

Absorption-2 X & ~Y Y X Y | (X & ~Y) = X | Y