Boolean Algebra Expressions

Boolean Algebra is based on two values that is 0 and 1 or we can say two values. So we can say Functions in boolean algebra are binary valued.
Following laws are used in simplify the boolean algebra expressions. These laws are explained one by one.

1) Commutative Law:
(i) X + Y = Y + X,
(ii) X . Y = Y. X,

2) Associative Law:
(i) (X + Y) + Z = X + (Y + Z),
(ii) (X. Y). Z = X. (Y. Z),

3) Distributive Law
(i) X. (Y +Z) = X. Y + X. Z
(ii) X + (Y. Z) = (X + Y) (X + Z)

4) Identity Law
(i) X + X = X
(ii) X. X = X

5) Negation Law
(i) (X’) = X’
(ii) (X’’) = X

6) Redundant Law
(i) X + X. Y = X
(ii) X. (X + Y) = X

7)
(i) 0 + X = X
(ii) 1. X = X
(iii) 1 + X = 1
(iv) 0 A = 0

8)
(i) X’ + X = 1
(ii)X’. X = 1

9)
(i) X + X’. Y = X + Y
(ii) X. (X’ + Y) = X. Y

A theorem is named as De Morgan's Theorem is also used in solving Boolean expression. This is given as:
(i)   =X’ + Y’
(ii)   = X’+ Y’
 

Expression “X’ (X + Y) + (Y + XX) (X +Y’)”is representing Boolean expression. Prove that
X’ (X + Y) + (Y + XX) (X + Y’) = (X + Y). It can be simplified using the various laws of Boolean algebra as shown below:
X’ (X + Y) + (Y + XX) (X + Y’) = X’X+ X’Y + (Y + X) X + (Y + X) Y’,
 
= X’Y+ (Y + X) X + (Y + X) Y’
= X’Y + YX + XX + YY’ + XY’
= X’Y + YX + X + XY’
= X’Y + XY + XT + XY’
= X’Y + X (Y + T +Y’)
= X’Y + X
= X + X’Y
= (X +X’) (X + Y)
= (X + Y).
Here X’ denotes    and Y’ denotes