Simplifying Boolean Expressions

Boolean Operators (T/F)x y x AND y

F F F

F T F

T F F

T T T

x y x OR y

F F F

F T T

T F T

T T T

x y x XOR y

F F F

F T T

T F T

T T F

x NOT x

F T

T F

Boolean Operators (1/0)x y x AND y

0 0 0

0 1 0

1 0 0

1 1 1

x y x OR y

0 0 0

0 1 1

1 0 1

1 1 1

x y x XOR y

0 0 0

0 1 1

1 0 1

1 1 0

x NOT x

0 1

1 0

Boolean Operators Symbols

Operator Symbol

NOT ā (overbar), a’, ~a

AND · (mult. dot)

OR +

XOR (plus sign with (plus sign with circle around it)circle around it)

Simplifying Boolean Expressions Commutative laws

A + B = B + AA · B = B · A

Associative lawsA + (B + C) = (A + B) + CA · (B · C) = (A · B) · C

Distributive lawsA · (B + C) = A · B + A · CA + (B · C) = (A + B) · (A + C)

Simplifying Boolean Expressions Tautology laws

A · A = AA + A = AA + ~A = 1A · ~A = 0

Absorption LawA + (A · B) = AA · (A + B) = A

Simplifying Boolean Expressions

Identities0 · A = 0

0 + A = A

A + 1 = 1

1 · A = A

A = A

ComplementA + ~A · B = A + B

Examples A + A + A + A = A

Using the Tautology law

A Bigger ExampleSimplify ~A · B + A · ~B + ~A · ~B

~A · B + A · ~B + ~A · ~B ~A · B + (A · ~B + ~A · ~B) Associative~A · B + (~B · (A + ~A)) Distributive~A · B + ~B & Tautology~A + ~B Complement

Verify with a truth table!

Practice Show that A + B · C = (A + B) · (A + C) is

true using a truth table.

Practice Show that A + ~A · B = A + B

Practice Simplification Simplify A + AB + ~B and verify with a truth

table

De Morgan’s Laws~(A · B) = ~A + ~B~A · ~B = ~(A+B)

1. Take a term ~A · ~B

2. NOT the individual members of the term A · B

3. Change the operator i.e. · to +, or + to ·A + B

4. NOT the entire term~(A+B)

De Morgan’s Law Example

f = ~A · ~B + (~A + ~B)

= ~~( ~A · ~B + (~A + ~B) ) NOT NOT

= ~( (A + B) · ~(~A + ~B) ) De Morgan’s

= ~( (A + B) · (A·B) ) De Morgan’s

= ~( A·(A·B) + B·(A·B) ) Distributive

= ~( A·B + A·B ) Tautology

= ~(A·B) Tautology