Translate the following inference to logic notation: For all x, z is happy if and only if z will take a plane. There is at least one x for which z will take a plane if and only if z is happy. Therefore there is at least one x for which z will take a plane if and only if z is happy.
forall x: P(x) <-> Q(x). exists x: Q(x) <-> P(x). Therefore exists x: Q(x) <-> P(x).