Question

The negation of the Boolean expression $x\leftrightarrow y$ is equivalent to

• A. $(x\wedge y)\wedge (~x\vee ~y)$
• B. $\left(x\wedge ~y\right)\vee \left(y\wedge ~x\right)$
• C. $(x\wedge ~y)\vee (~x\wedge ~y)$
• D. $(~x\wedge y)\vee (~x\wedge ~y)$

Answer 1

The correct option is B

$\left(x\wedge ~y\right)\vee \left(y\wedge ~x\right)$

Explanation for the correct answer:

The statement $x\leftrightarrow y$ can be wraitten as

$x\leftrightarrow y\equiv \left(x\to y\right)\wedge \left(y\to x\right)$$...\left(i\right)$

The statement $x$implies $y$ can be written as

$\left(x\to y\right)\equiv ~x\vee y$ $...\left(ii\right)$

The statement $y$implies $x$ can be written as

$\left(y\to x\right)\equiv ~y\vee x$ $...\left(iii\right)$

Hence the negation of $\left(i\right)$ can be given as

$~\left(x\leftrightarrow y\right)\equiv ~\left(\left(~x\vee y\right)\wedge \left(~y\vee x\right)\right)$ …[From (ii) and (iii)]

$\Rightarrow$ $~\left(x\leftrightarrow y\right)\equiv ~\left(~x\vee y\right)\vee ~\left(~y\vee x\right)$

$\Rightarrow$ $~\left(x\leftrightarrow y\right)\equiv \left(x\wedge ~y\right)\vee \left(y\wedge ~x\right)$

Hence, option $\left(B\right)$ is the correct answer.