Translate the following inference to logic notation: For all x, x will not cook. There is at least one x for which if x were a doctor, then x is cooking. For all x, if x cooks, then x is a doctor. For all x, x is a doctor if and only if x will play a game. Therefore there is at least one x for which x will not play a game.
forall x: ~P(x). exists x: Q(x) -> P(x). forall x: P(x) -> Q(x). forall x: Q(x) <-> R(x). Therefore exists x: ~R(x).