Translate the following inference to logic notation: For all x, x will work if and only if x is fast. For all x, if x listens to a song, then x is working. For all x, x will not sleep or x will listen to a song. There is at least one x for which x will sleep or x is fast. Therefore there is at least one x for which x will work.
forall x: P(x) <-> Q(x). forall x: R(x) -> P(x). forall x: ~S(x) or R(x). exists x: S(x) or Q(x). Therefore exists x: P(x).