Translate the following inference to logic notation: For all x, z will play a game or z will not listen to a song. There is at least one x for which z will listen to a song if and only if z will play a game. Therefore there is at least one x for which z will listen to a song if and only if z will play a game.
forall x: P(x) or ~Q(x). exists x: Q(x) <-> P(x). Therefore exists x: Q(x) <-> P(x).