Chapter 1.5, Problem 49E

a) Show that $\forall xP\left(x\right)\wedge \exists xQ\left(x\right)$is logically equivalent to $\forall x\exists y\left(P\left(x\right)\wedge Q\left(y\right)\right)$, where all quantifiers have the same nonempty domain.

b) Show that $\forall xP\left(x\right)\wedge \exists xQ\left(x\right)$is equivalent to $\forall x\exists y\left(P\left(x\right)\vee Q\left(y\right)\right)$, where all quantifiers have the same nonempty domain.