GATE 2026 DA Q.48: solution and answer

Official answer: B, C, DSource: Official answer key, GATE 2026 DA (IIT Guwahati)

1. A formula is valid if it is true in every interpretation. In standard first-order logic every domain is non-empty. We check each option for a counter-model.

2. (A) $\forall xP(x)\Rightarrow \exists xP(x)$: if $P$ holds for every element and the domain has at least one element, then $P$ holds for some element. No counter-model exists, so (A) is valid.

3. (B) $\exists xP(x)\Rightarrow \forall xP(x)$: take domain $\{1,2\}$ with $P(1)$ true and $P(2)$ false. Then $\exists xP(x)$ is true and $\forall xP(x)$ is false, so the implication is false in this interpretation. (B) is NOT valid.

4. (C) $\exists xP(x)\iff \forall xP(x)$: a biconditional needs both directions. The direction $\forall \to \exists$ holds, but $\exists \to \forall$ fails in the same counter-model as in (B). So (C) is false there and NOT valid.

5. (D) $\forall xP(x)\Rightarrow \exists x\neg P(x)$: take any non-empty domain in which $P$ is true everywhere. Then $\forall xP(x)$ is true, while $\exists x\neg P(x)$ is false, so the implication is false. (D) is NOT valid. (It is still satisfiable: it is true in any interpretation where $P$ fails for some element, because the antecedent is then false. Satisfiable is not the same as valid.)

6. Hence the statements that are NOT valid are (B), (C) and (D).