I've been given the following question:
Let P and Q be predicates on the set S, where S has two elements, say S={a, b}. Then the statement $\forall xP(x)$ can also be written in full detail as $P(a) \wedge P(b)$. Rewrite the statements below in a similar fashion, using P, Q and logical operators.
I'm having trouble with these 2: $$\forall x\exists y(P(x) \vee Q(y))$$ and $$\exists xP(x) \wedge \exists yQ(y)$$
For the first one I have: $$\equiv (P(a) \wedge (Q(a) \vee Q(b)) \wedge (P(b) \wedge (Q(a) \vee Q(b))$$
And the second one I have simply: $$\equiv (P(a) \vee P(b))\wedge (Q(a) \vee Q(b)$$
Haven't done anything like this before so any insight is appreciated!