A. For each of the following formulas, find an interpretation structure in which
it is true and one in which it is false. (Answers to 1, 4, 7, 9, 12, and 15.)
(1) $(\forall x)(Px \land Qx \to Rx)$
(2) $(\exists x)(Px \land \neg Qx \land \neg Rx)$
(3) $(\forall x)(Px \to Qx) \lor (\exists x)(Px \land \neg Rx)$
(4) $Pa \land (\exists x)(Px \land \neg Qxb)$
(5) $(\exists x)(Px \land Qxa) \to (\forall y)(Py \land Ry \to Qby)$
(6) $(\forall x)[Px \land Qx \to (\exists y)(Ry \land \neg Sxy)]$
(7) $(\exists x)[Px \land \neg Qx \land (\forall y)(Rxy \to \neg Sxy)]$
(8) $(\exists x)Px \to (\forall y)(Qy \to Rxy)$
(9) $(\forall x)(Px \to Qxa) \to (\exists y)(Py \land Ry \land \neg Qxy)$
