The statement $~ (p \leftrightarrow\sim q)$ is

Moderate

Solution

$\begin{array}{l} ~ (p \leftrightarrow\sim q) \\ \equiv\sim ((p \to\sim q) \land (~ q \to p) \\ \equiv\sim ((~ p \lor ~ q) \land (q \lor p)) \end{array}$
$\begin{array}{l} \equiv\sim (~ q \lor ~ p) \lor (~ (q \lor p)) \\ \equiv (~ (q) \land ~ (~ p)) \lor (~ q \land ~ p) \\ \equiv (q \land p) \lor (~ q \land ~ p) \\ \equiv [(q \land p) \lor (~ q)] \land [(q \land p) \lor (~ p)] \\ \equiv [(q \lor (~ q)) \land (p \lor (~ q))] \land [(q \lor (~ p)) \\ \land (p \lor (~ p))] \\ \equiv [t \land (~ q \lor p)] \land [((~ p) \lor q)) \land t] \\ \equiv (~ q \lor p) \land (~ p \lor q) \\ \equiv (q \to p) \land (p \to q) \\ \equiv p \leftrightarrow q \end{array}$
Alternative Solution. Use the following table.
p q $~ q$ $p \leftrightarrow\sim q$ $~ (p \leftrightarrow\sim q)$ $p \leftrightarrow q$
$T$ $T$ $F$ $F$ $T$ $T$
$T$ $F$ $T$ $T$ $F$ $F$
$F$ $T$ $F$ $T$ $F$ $F$
$F$ $F$ $T$ $F$ $T$ $T$
From the last two colums, we get $t ~ (p \leftrightarrow (~ q)) \equiv p \leftrightarrow q$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

SCAN ME