سوال ۹۲ کنکور ارشد علوم کامپیوتر ۱۴۰۵

Introduce two atomic propositions:

• $P$ = “Ali is admitted to the university.”
• $Q$ = “Ali is successful.”

Translate the inner statement. “$P$ is a necessary condition for $Q$” means that $Q$ cannot hold without $P$ — i.e. success requires admission. Symbolically this is the conditional

$Q \to P.$

Careful: a necessary condition sits on the right of the arrow ($Q\to P$), while a sufficient condition sits on the left ($P\to Q$). Do not swap them.

Negate it. The whole sentence is “It is not the case that ($Q \to P$)”:

$\neg(Q \to P).$

Using $Q \to P \equiv \neg Q \lor P$ and De Morgan:

$\neg(Q \to P) \equiv \neg(\neg Q \lor P) \equiv Q \land \neg P.$

Read it back. $Q \land \neg P$ = “Ali is successful and Ali is not admitted,” i.e. Ali is successful but has not been admitted to the university.

This is exactly option 4.

Exam trick: The negation of an implication is never another implication — it is a conjunction: $\neg(A\to B)\equiv A\land\neg B$ (the antecedent holds, the consequent fails). Here the antecedent is $Q$ (success) and the consequent is $P$ (admission), so the negation asserts success together with non-admission.