Translate the following inference to logic notation: For all x, if x_10 sleeps, then x_10 is not an astronaut. There is at least one x for which x_10 will sleep if and only if x_10 is rich. For all x, x_10 is rich if and only if x_10 will cook. For all x, x_10 is an astronaut. Therefore there is at least one x for which x_10 will not cook.
forall x: P(x) -> ~Q(x). exists x: P(x) <-> R(x). forall x: R(x) <-> S(x). forall x: Q(x). Therefore exists x: ~S(x).