سوال ۹۶ کنکور ارشد علوم کامپیوتر ۱۳۹۷

The proposition has three variables $p,q,r$, so its truth table has $2^{3}=8$ rows. The chained biconditional $r \leftrightarrow (p \leftrightarrow q)$ is associative and equals $p \leftrightarrow q \leftrightarrow r$, which is true exactly when the number of false variables among $p,q,r$ is even.

First, $p \leftrightarrow q$ is true iff $p$ and $q$ have the same truth value. Then $r \leftrightarrow (p \leftrightarrow q)$ is true iff $r$ matches that inner value.

Counting the $T$ values in the final column gives $4$.

More generally, a biconditional of $n$ variables is true on exactly half of its $2^{n}$ rows, i.e. $2^{n-1}$ rows. For $n=3$ this is $2^{2}=4$.

Exam trick: A biconditional chain $x_1 \leftrightarrow x_2 \leftrightarrow \cdots \leftrightarrow x_n$ is a parity function; it is true on exactly $2^{n-1}$ of the $2^{n}$ assignments, so you never need to draw the full table.