Suppose the logical expression $(((\neg$ $P$) $\leftrightarrow$ $Q$) $\rightarrow$ $R$) $\vee$ ($P$ $\leftrightarrow$ $R$) is FALSE.

Question

Let $P$, $Q$, and $R$ be statement variables. Suppose the logical expression

$(((\neg$$P$) $\leftrightarrow$ $Q$) $\rightarrow$ $R$) $\vee$ ($P$ $\leftrightarrow$ $R$)

is FALSE.

What are the possible truth values for $P$, $Q$, and $R$?

Answer 1

The easiest (and also most tedious) route is to just make a truth table for that compound statement and see which values of $P$, $Q$ and $R$ make it false.

Answer 2

Both $(\neg P\leftrightarrow Q)\to R$ and $P\leftrightarrow R$ must be false. The latter is false only when $P=1,R=0$ or $P=0,R=1$.

For the former to be false, $R=0,\neg P\leftrightarrow Q=1$. This leaves you with $P=1,R=0$. Now try out $Q=0,1$ to see which one makes $\neg P\leftrightarrow Q=1$.

You should get $P=1,R=0,Q=0$.