Give algebraic proof of absorption law of boolean algebra.

Q. Give algebraic proof of absorption law of boolean algebra.

Answer :-

(i)

X+XY=X
It can be proved algebraically as

L.H.S. = X+XY
           = X(1 + Y)

Putting 1+ Y = 1   (ref. properties of 0, 1 Theorem 1)
X.1=X = R.H.S. (ref. properties of 0, 1)

Hence proved.

(ii)

X(X + Y) = X

The algebraic proof of this law.

L.H.S. = X(X+Y)= X.X + XY
= X.X + XY
= X + XY  (X.X=X: Indempotence Law)
= X(1 + Y)
= X.1  (using 1+ Y =1, properties of 0, 1)
= X (X.1=X, using property of 0, 1)
= R.H.S

Hence proved.