Let \(P(x,y)\) be the predicate \(x \lt y\text{.}\) It is true, in the natural numbers, that for all \(x\) there is some \(y\) greater than that \(x\) (since there are infinitely many numbers). However, there is not a natural number \(y\) which is greater than every number \(x\text{.}\) Thus it is possible for \(\forall x \exists y P(x,y)\) to be true while \(\exists y \forall x P(x,y)\) is false.
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