Translate the following inference to logic notation: For all x, if x_1 were a cashier, then x_1 is listening to a song. For all x, if x_1 sleeps, then x_1 is listening to a song. For all x, if x_1 listens to a song, then x_1 is sleeping. There is at least one x for which x_1 is not a cashier. For all x, if x_1 listens to a song, then x_1 is a cashier. Therefore there is at least one x for which x_1 will not sleep.
forall x: P(x) -> Q(x). forall x: R(x) -> Q(x). forall x: Q(x) -> R(x). exists x: ~P(x). forall x: Q(x) -> P(x). Therefore exists x: ~R(x).