سوال ۹۳ کنکور ارشد علوم کامپیوتر ۱۴۰۲

Key rule: "$A$ only if $B$" translates to $A \to B$ — the part after only if is the necessary condition, hence the consequent. (Contrast with "$A$ if $B$", which is $B \to A$.)

Here:

• $A = p$ (you succeed in the konkur),
• $B = q \wedge r$ (you solve all exercises and have focus).

So the sentence is

$p \to (q \wedge r).$

Now compare with the options. None of them is literally $p \to (q \wedge r)$, so we look for a logically equivalent form. The contrapositive of $p \to (q \wedge r)$ is

$\neg(q \wedge r) \to \neg p,$

which is exactly option 1.

Why the others fail:

Exam trick: "only if" points the arrow forward ($A \to B$); "if" points it backward ($B \to A$). When the literal $p \to (q\wedge r)$ isn't listed, check its contrapositive $\neg(q\wedge r)\to\neg p$ — and remember $\neg(q\wedge r) = \neg q \vee \neg r$, not $\neg q \wedge \neg r$, which kills option 3.