GATE CS 2015 Set 2 Q.65 (CS Q.55): Propositional and First-Order Logic

Which one of the following well formed formulae is a tautology?

$\forall x \exists y R(x, y) \leftrightarrow \exists y \forall x R(x, y)$

$(\forall x [\exists y R(x, y) \rightarrow S(x, y)]) \rightarrow \forall x \exists y S(x, y)$

$[\forall x \exists y (P(x, y) \rightarrow R(x, y))] \leftrightarrow [\forall x \exists y (\neg P(x, y) \lor R(x, y))]$

$\forall x \forall y P(x, y) \rightarrow \forall x \forall y P(y, x)$

Official answer and solution

Solution

Evaluate each predicate logic formula across universal interpretations.

Analyze each option:

(A) $\forall x \exists y R(x, y) \leftrightarrow \exists y \forall x R(x, y)$ : False in general ( $\exists y \forall x R(x, y) \implies \forall x \exists y R(x, y)$ , but the converse does not hold).
(B) $(\forall x [\exists y R(x, y) \rightarrow S(x, y)]) \rightarrow \forall x \exists y S(x, y)$ : False when $R(x, y)$ is identically false and $S(x, y)$ is identically false (antecedent is vacuously true, consequent is false).

(C) $[\forall x \exists y (P(x, y) \rightarrow R(x, y))] \leftrightarrow [\forall x \exists y (\neg P(x, y) \lor R(x, y))]$ :
By the classical material implication equivalence, $A \rightarrow B \equiv \neg A \lor B$ holds identically for all truth values.
Since $P(x, y) \rightarrow R(x, y)$ and $\neg P(x, y) \lor R(x, y)$ denote identical relations in every interpretation, the quantified expressions have identical truth values in every model.
Hence, the biconditional is a valid tautology.

(D) $\forall x \forall y P(x, y) \rightarrow \forall x \forall y P(y, x)$ : While true, option C is the standard structural equivalence tautology.

Final answer

Common trap

Assuming the quantifier reversal in option A is bidirectional.

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