Translate the following inference to logic notation: For all x, if x_6 were a janitor, then x_6 is running. There is at least one x for which if x_6 listens to a song, then x_6 is a janitor. For all x, if x_6 goes running, then x_6 is cooking. For all x, if x_6 cooks, then x_6 is listening to a song. Therefore there is at least one x for which x_6 will go running if and only if x_6 will cook.
forall x: P(x) -> Q(x). exists x: R(x) -> P(x). forall x: Q(x) -> S(x). forall x: S(x) -> R(x). Therefore exists x: Q(x) <-> S(x).