Principle of Duality

Principle of Duality

The principle of duality theorem says that, starting with a Boolean relation, you can derive another Boolean relation by

1. Changing each OR sign to an AND sign

2. Changing each AND sign to an OR sign and

3. Complementing any 0 or 1 appearing in the expression.

For example : Dual of relation A + Ā = 1 is A • Ā = 0

Illustrative Examples

Example: 1

Simplify: x + x'y.

Solution :

Example: 2

Apply DeMorgans theorem to simplify 

Solution :

Example: 3

Prove the following Boolean identities.

Solution :

Example: 4

Solution :

Example: 5

Solution :

Examples with Solution

Example: 6

If A and B are Boolean variables and if A = 1 and   = 0, find B.

Solution :

Given  = 0 

A + B = 1, Now if A = 1 we have,

1 + B = 1 and therefore B can be either 0 or 1.

Example: 7

Simplify A+ AB+ Ā+B.

Solution :

A + AB+ Ā + B = A(1 + B) + Ā + B = A + Ā + B = 1 + B = 1.

Example: 8

Apply DeMorgan's theorem to the following expression. ((A+B+ C) D)'

Solution :

Example: 9

Using Boolean laws and rules simplify the logic expression.

Z = (A' + B) (A + B)

Solution:

Example: 10

Prove the following using DeMorgan's theorem

Solution:

Example: 11

Apply DeMorgan's theorem for the function .

Solution :

Example: 12

Find the complement of A+ BC + AB.

Solution :

Example: 13

Simplify the following Boolean expression to a minimum number of literals

 (BC'+A'D) (AB'+CD').

Solution :

= 0

• A Ā = 0

Example: 14

Simplify the Boolean expression. F= x'y'z' + x'yz + xy'z' + xyz'.

Solution :

F = x'y'z' + x'yz + xy'z' + xyz'        

Example: 15

Simplify the given Boolean expression

F = x' + xy + xz' + xy'z'.

Solution :

Example: 16

Simplify the Boolean expression using laws and rules of Boolean algebra

Z = [AB' (C+BD)+(AB)'] C.

Solution :

Examples for Practice

Example: 17

Simplify the following Boolean expressions to a minimum number of literals: XY+Z+YZ   

 [ Answer: XY + Z]

Example: 18

Simplify the following Boolean expression.

 [ Answer: ]

Example: 19

Simplify the following Boolean expressions to a minimum number of literals : 

 [ Answer: X]

Review Questions

1. What are Boolean variables?

2. Define the following terms: Boolean variable, complement, literal.

3. State the fundamental postulates of Boolean algebra.

4. State various laws of Boolean algebra.

5. State the associative law of Boolean algebra.

6. State and prove DeMorgan's theorem.

7. Explain the principle of duality with the help of example.