Translate the following inference to logic notation: For all x, if z plays a game, then z is cooking. For all x, if z cooks, then z is playing a game. There is at least one x for which if z cooks, then z is playing a game. Therefore there is at least one x for which z will play a game if and only if z will cook.
forall x: P(x) -> Q(x). forall x: Q(x) -> P(x). exists x: Q(x) -> P(x). Therefore exists x: P(x) <-> Q(x).