##### Example on Boolen Expression

Consider the Boolean Operator @ with the following properties:

$\text{x @ 1} = \overline{x}$$\text{x @ 0} = \text{x}$$\text{x @ x} = \text{0}$,  $\text{x @ }\overline{\text{x}} = 1$. Then $\text{x @ y}$ is equivalent to

A) $\text{x}\overline{y} + \bar{\text{x}}y$

B) $\text{x}\overline{y} + \bar{\text{x}}\overline{y}$

C) $\bar{\text{x}}y + \text{x}y$

D) $\text{x}y + \bar{\text{x}}\bar{y}$

Let us approach this question as follows:

We need to find $\text{x @ y}$

Given,

$\text{x @ 1} = \overline{x} ​​$

$\text{x @ 0} = \text{x}$

$\text{x @ x} = \text{0}$

$\text{x @ }\overline{\text{x}} = 1$

Now, to find $\text{x @ y}$, we replace $y$ by $1, 0, \text{x},\ \ and \ \ \ \bar{\text{x}}$, and find that the expression of option (A) holds True.

$\therefore$   Ans (A)