1 vote

Let $P$ and $Q$ be two propositions $\neg (P \leftrightarrow Q)$ is equivalent to

- $P\leftrightarrow \neg Q$
- $\neg P\leftrightarrow Q$
- $\neg P \leftrightarrow \neg Q$
- $Q\rightarrow P$

2 votes

Best answer

0 votes

$\sim$(P $\rightarrow$ Q)

= $\sim$( (P $\rightarrow$ Q) ^ (Q $\rightarrow$ P) )

= $\sim$ ( ($\sim$P v Q) ^ ( $\sim$Q v P) )

= $\sim$($\sim$P v Q) v $\sim$($\sim$Q v P)

= (P ^ $\sim$Q) v (Q v $\sim$P) = P$\bigoplus$Q

Using distributive law. (a + bc) = (a + b).(b + c)

= ( (P ^ $\sim$Q) v Q) ^ ( (P ^ $\sim$Q) v $\sim$P)

= ( ( P v Q)^($\sim$Q v Q)) ^ ( ( $\sim$P v P) ^ ($\sim$Q v $\sim$P) )

= ( ( P v Q) ^ ( $\sim$Q v $\sim$ P)

Now here we have two choices.

1. ($\sim$P $\rightarrow$ Q) ^ ( Q $\rightarrow$ $\sim$P) = $\sim$P $\leftrightarrow$ Q

2. ($\sim$Q $\rightarrow$ P) ^ (P $\rightarrow$ $\sim$ Q) = P $\leftrightarrow$ $\sim$Q.

**So both 1 and 2 option is correct.**