$\forall x\ P(x) \rightarrow \exists x\ \neg P(x)$
Answer: Not valid This statement is not valid because it can be false. If the domain is non-empty, the premise $\forall x\ P(x)$ implies that $P(x)$ is true for all $x$. The conclusion $\exists x\ \neg P(x)$ implies that there is at least one $x$ for which $P(x)$ is false. These two statements are contradictory; the premise entails the negation of the conclusion.
$\forall x\ P(x) \rightarrow \exists x\ P(x)$
Answer: Valid This statement is valid under the standard assumption that the domain of discourse is non-empty. If something is true for all elements in a non-empty domain, it must be true for at least one element.
$\exists x\ P(x) \rightarrow \forall x\ P(x)$
Answer: Not valid This statement is not valid because it can be false. For example, if the domain is integers and $P(x)$ is the predicate $x$ is even," then $\exists x\ P(x)$ is true (e.g., 2 is even), but $\forall x\ P(x)$ is false (e.g., 3 is not even). The implication is false when the premise is true and the conclusion is false.
$\exists x\ P(x) \leftrightarrow \forall x\ P(x)$
Answer: Not valid This statement is not valid because the biconditional operator( $\leftrightarrow$) means both directions of the implication must be true. As established in problem C, the implication $\exists x\ P(x) \rightarrow \forall x\ P(x)$ is not always true. Thus the biconditional is not valid.
