When is $(\forall x \in U, P(x)) \implies (\exists x \in U, P(x))$ false?

Question

Consider the statement $$(\forall x \in U, P(x)) \implies (\exists x \in U, P(x))$$

Write down a domain $U$ and a predicate $P$ for which this statement is false.

What property exists that can be true for all members in a group but false for one member? (if that makes any sense?)

Answer 1

Let $P(x)$ indicate $x\neq x,$ and let $U=\emptyset.$

Answer 2

Let $U$ be empty, $P(x)$ be anything you want.