Boolean Algebra

Discussion D6.1

Sections 13-3 – 13-6

Boolean Algebra andLogic Equations

• George Boole - 1854

• Switching Algebra Theorems

• Venn Diagrams

George BooleEnglish logician and mathematician

Publishes Investigation of theLaws 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

X Y

X & Y

Venn Diagrams

X | Y

X Y

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

0 0 0 0 0 0 00 0 1 1 1 0 10 1 0 1 1 1 10 1 1 1 1 1 11 0 0 0 1 1 11 0 1 1 1 1 11 1 0 1 1 1 11 1 1 1 1 1 1

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)

X Y

Z

X | (Y & Z) = (X | Y) & (X | Z)

Distributive Law - a

Distributive Law - b

X & (Y | Z) = (X & Y) | (X & Z)

X Y

Z

Question

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

Absorption-2X & ~Y

Y X

Y | (X & ~Y) = X | Y