سوال ۹۳ کنکور ارشد علوم کامپیوتر ۱۴۰۵

Formalize the inner statement. "All humans are intelligent" means: for every $x$, if $x$ is a human then $x$ is intelligent:

$\forall x\,(A(x) \rightarrow B(x)).$

Negate it. The sentence says "It is not the case that ...", so we negate the whole thing:

$\neg\,\forall x\,(A(x) \rightarrow B(x)).$

Push the negation inward. By the quantifier-negation rule $\neg\forall x\,P(x) \equiv \exists x\,\neg P(x)$:

$\exists x\,\neg\big(A(x) \rightarrow B(x)\big).$

Now use the negation of an implication, $\neg(p \rightarrow q) \equiv p \wedge \neg q$:

$\exists x\,\big(A(x) \wedge \neg B(x)\big).$

This reads "there exists a human who is not intelligent," which is exactly the meaning of "not all humans are intelligent." That is option 2.

Pitfall: When negating a universally-quantified implication, the implication becomes a conjunction ($A \wedge \neg B$), not another implication. Options with $A(x) \rightarrow \neg B(x)$ are the classic trap.