Boolean Algebra

Discussion D2.2

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

UnityX' * 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-2X' * 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