Question

3. Give an example of a predicate P(x,y) such that ∃x∀yP(x,y) and ∀y∃xP (x, y) have different truth values.

Answer 1

Let us give an example of a predicate $P(x,y)$ such that $∃x∀yP(x,y)$ and $∀y∃xP (x, y)$ have different truth values.

Let $P(x,y)$ means $x\ge y,$ and the domains for $x$ and $y$ are the set $\N$ of all positive integers.

Since the set $\N$ does not contain the greatest element, the value of $∃x∀y(x\ge y)$ is false. On the other hand, for any $y\in\N$ let $x=y$. Then $y\ge y=x,$ and hence the value of $∀y∃xP (x, y)$ is true.