Boolean Algebra

Module M4.1

Section 5.1

Boolean Algebra andLogic Equations

• Switching Algebra Theorems

• Venn Diagrams

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

Generalized De Morgan’s Theorem

• NOT all variables

• Change & to # and # to &

• NOT the result

• --------------------------------------------

• F = X & Y # X & Z # Y & Z

• F = !((!X # !Y) & (!X # !Z) & (!Y # !Z))

• F = !(!(X & Y) & !(X & Z) & !(Y & Z))

X

Y

X

Z

Y

Z

F

F = !(!(X & Y) & !(X & Z) & !(Y & Z))

X

Y

X

Z

Y

Z

F

F = !(!(X & Y) & !(X & Z) & !(Y & Z))

NAND Gate

X Y

X Z

Y Z

F

F = X & Y # X & Z # Y & Z

Question

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