Translate the following inference to logic notation: For all x, if z works, then z is happy. For all x, if z were happy, then z is taking a plane. There is at least one x for which z will work or z will not take a plane. 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). forall x: Q(x) -> R(x). exists x: P(x) or ~R(x). Therefore exists x: R(x) <-> Q(x).