Quote:

Explain the difference between the first-order prefixes:

(a) $\displaystyle \exists x\forall y$ and $\displaystyle \forall x\exists y$;

(b) $\displaystyle \exists x\forall y\exists z$ and $\displaystyle \forall x\exists y\forall z$;

(c) $\displaystyle \forall x\exists y\forall z\exists w$ and $\displaystyle \exists x\forall y\exists z\forall w$.

First, I notice that each prefix can be obtained by negation from the other one in the pair. Also, I can show a formula such that different prefixes produce different truth values. For example, consider (c):