A useful rule: $$\forall x (\alpha) = \neg \exists (x) (\neg \alpha) $$i.e.; If some property $\alpha$ is true for all $x$, then it is equivalent ot say that no $x$ exists such that property $\alpha$ does not hold for it.
Starting with choices:
- $\forall x (\exists z (\neg \beta) \to \forall y (\alpha)) $
$\implies \forall x (\neg \exists z (\neg \beta) \vee \forall y (\alpha)) $
$\implies \forall x (\forall z ( \beta) \vee \forall y (\alpha)) $
$\implies \neg \exists x \neg (\forall z ( \beta) \vee \forall y (\alpha)) $
$\implies \neg \exists x (\neg \forall z ( \beta) \wedge \neg \forall y (\alpha)) $
So, A is not matching with the logical statement in question.
- $\forall x (\forall z (\beta) \to \exists y (\neg \alpha)) $
$\implies \forall x (\neg \forall z (\beta) \vee \exists y (\neg \alpha)) $
$\implies \neg \exists x \neg(\neg \forall z (\beta) \vee \exists y (\neg \alpha)) $
$\implies \neg \exists x (\forall z (\beta) \wedge \neg \exists y (\neg \alpha)) $
$\implies \neg \exists x (\forall z (\beta) \wedge \forall y (\alpha)) $
Hence, matches with the given statement.
- $\forall x (\forall y (\alpha) \to \exists z (\neg \beta)) $
$\implies \forall x (\neg \forall y (\alpha) \vee \exists z (\neg \beta)) $
$\implies \neg \exists x \neg(\neg \forall y (\alpha) \vee \exists z (\neg \beta)) $
$\implies \neg \exists x (\forall y (\alpha) \wedge \neg \exists z (\neg \beta)) $
$\implies \neg \exists x (\forall y (\alpha) \wedge \forall z (\beta)) $
Hence, matches with the given statement.
- $\forall x (\exists y (\neg \alpha) \to \exists z (\neg \beta)) $
$\implies \forall x (\neg \exists y (\neg \alpha) \vee \exists z (\neg \beta)) $
$\implies \forall x (\forall y ( \alpha) \vee \exists z (\neg \beta)) $
$\implies \neg \exists x \neg (\forall y ( \alpha) \vee \exists z (\neg \beta)) $
$\implies \neg \exists x (\neg \forall y ( \alpha) \wedge \neg \exists z (\neg \beta)) $
$\implies \neg \exists x (\neg \forall y ( \alpha) \wedge \forall z (\beta)) $
So, D is not matching with the logical statement in question.
Thus both (A) and (D) are not logically equivalent to the given statement.
In GATE 2013 marks were given to all for this question