Help to prove this predicate

Question

Please help me to prove the following. Please explain step by step, I'm new to this subject.

For any non-empty set $S$ and for any two predicates $p$ defined on $S^2$, prove or disprove the following.

1. $∃x∃y\colon p(x,y) ⇔ ∃y∃x\colon p(x,y)$
2. $∃y∀x\colon p(x,y) \Rightarrow ∃y \colon p(x,y)$

Thanks in advance...

Answer 1

Assunme $\exists x\exists y\colon p(x,y)$. So let $x_0$ be such that $\exists y\colon p(x_0,y)$. Now let $y_0$ be such that $p(x_0,y_0)$. Hence $\exists x\colon p(x,y_0)$ and finally $\exists y\exists x\colon p(x,y)$. The reverse direction follows likewise.

Assume $\exists y\forall x\colon p(x,y)$. So let $y_0$ be such that $\forall x\colon p(x,y_0)$. Then especially $p(x,y_0)$ and hence $\exists y\colon p(x,y)$.