NIMCET 2007 — Computer PYQ

We can solve this problem using two standard approaches in Boolean algebra: the Distributive Law method and the standard Algebraic Expansion method.

Method 1: Using the Distributive Law (Quickest)

In Boolean algebra, the Distributive Law of logical OR ($+$) over logical AND ($\cdot$) is state as:

$$A+(B\cdot C)=(A+B)\cdot (A+C)$$

If we look at our given expression carefully, it perfectly matches the right-hand side of this identity:

$$(x+y)\cdot (x+y^{\prime })$$

By mapping our variables to the identity:

• Let $A=x$

• Let $B=y$

• Let $C=y^{\prime }$

We can rewrite the expression as:

$$(x+y)\cdot (x+y^{\prime })=x+(y\cdot y^{\prime })$$

According to the Complementarity Law in Boolean algebra, any variable ANDed with its own complement always results in $0$:

$$y\cdot y^{\prime }=0$$

Substituting this value back into our equation:

$$x+0=x$$