Predicate formula to propositional formula

Question

I have: $$\begin{align} \exists x \forall y P(x,y) \\ \end{align}$$ where $$\begin{align} M=\{a,b\} \\ \end{align}$$ I need to convert this formula to propositional logic. I know that if M is finite then you can eliminate quantifier, but what can I do when there is two quantifiers? Any hints, help would be appreciated

Answer 1

Try to eliminate quantifiers step by step: $$ \begin{aligned} \exists x\forall yP(x,y) &\iff \exists x (P(x,a)\land P(x,b))\\ &\iff \left(P(a,a)\land P(a,b)\right)\lor \left(P(b,a)\land P(b,b)\right). \end{aligned}$$