سوال ۹۳ کنکور ارشد علوم کامپیوتر ۱۴۰۴

Rewrite each implication using $A \to B \equiv \neg A \vee B$ and simplify with double negation $\neg\neg A \equiv A$.

Statement $\alpha$:

$\alpha \equiv \neg p \to (r \vee q) \equiv \neg(\neg p) \vee (r \vee q) \equiv p \vee q \vee r.$

Statement $\beta$:

$\beta \equiv \neg r \to (\neg p \to q) \equiv \neg(\neg r) \vee (\neg(\neg p) \vee q) \equiv r \vee (p \vee q) \equiv p \vee q \vee r.$

Both statements reduce to the same clause $p \vee q \vee r$, so

$\alpha \equiv \beta.$

Logical equivalence means each tautologically entails the other: $\alpha \models \beta$ and $\beta \models \alpha$ both hold. Hence each one is a logical consequence of the other, which is option 3.

Exam trick: Flatten nested implications into a single disjunction (clause) right away — $A \to B \to C$ collapses to $\neg A \vee \neg B \vee C$. Two formulas that flatten to the same clause are equivalent, so each entails the other.