GATE Overflow Test Series | Mixed Subjects | Test 1 | Question: 6

Question

The following propositional statement simplifies to
$\ (\neg P\leftrightarrow \neg Q)\vee (\neg P\leftrightarrow Q)$

• A. $P\wedge Q$
• B. True
• C. $P\rightarrow Q$
• D. False

Answer 1

We know that $$\neg P\leftrightarrow \neg Q\equiv P\leftrightarrow Q\equiv (P\rightarrow Q)\wedge (Q\rightarrow P)$$
 $$\equiv(\neg P\wedge \neg Q)\vee(P\wedge Q)\equiv P\odot Q$$
 and $$\neg P\leftrightarrow Q\equiv  P\leftrightarrow \neg Q\equiv(\neg P\rightarrow Q)\wedge (Q\rightarrow \neg P)$$ $$\equiv( P\wedge \neg Q)\vee(\neg P\wedge Q)\equiv P\oplus Q$$

$(P_1\oplus P_2\oplus \ldots P_n \equiv\overline{(P_1\odot P_2\odot \ldots P_n)}$, is valid for only even $n)$

Now,
$(P\odot Q) + (P\oplus Q)\equiv(P\odot Q)+\overline{(P\odot Q)}\equiv T $

$\because [X+\overline{X}\equiv T]$

So,the correct answer is $(B)$.