Predicate Logic and Quantifiers Introduction

Question

I’m a little bit stuck here, I really don’t know statement which make $\exists x\forall y \; p(x,y)$ true. I really don't understand.

Answer 1

Perhaps this task will become easier if we translate the statement as much as possible.

$\exists x \forall y P(x,y)$

There is at least one $x$ such that, for every $y$, $P(x,y)$.

The $P(x,y)$ is a statement that relates $x$ and $y$. In order for it to be true, it must be the case that for at least one element in the domain of $x$, the statement relating $x$ and $y$ is true for every element in the domain of $y$.

Here is an example of one such statement... Let $x$ and $y$ be arbitrary elements in the set of the real numbers. Let $P(x,y)$ be the statement "$x \cdot y = 0$"

It is true that there is at least one real number $x$ such that, for every real number $y$, $x \cdot y = 0$?

Yes! If $x$ is $0$, then this is the case.