Nested quantifiers for $y < x^2$

Question

1. $∀x ∃y P(x, y)$

   For every $x$ there exists a $y…$

   This is true, because for every number $x$ there exists at least one number $y$ for the statement to be true. For example, we can choose $x=100$ and $y =10.$

2. $∀y ∃x P(x, y)$

   ?

3. $∃x ∀y P(x, y)$

   There exists a $x$ (such that) for every $y…,$

   This is true, we can increment $x$ and $y$ to infinite, and every
   time the statement will stay true. If we set an end to the infinite
   and give the largest number possible to $x,$ that number will hold
   true for any possible $y.$

4. $∃y ∀ x P(x, y)$

   There exists a $y$ (such that) for every $x…,$ means that there is a
   $y$ that is less than $x^2$ for every possible $x.$ This is true
   because we can give $y$ a negative value, whatever value we will
   give $x$ will still be more than any negative value because it is
   quadratic.

Is this how you correctly understand quantifiers? Or am I understanding wrongly?

Answer 1

P.S. Forgot to add that x and y are REAL numbers

It isn't necessary to specify this, since complex numbers cannot be ordered anyway.

1. $∀x ∃y P(x, y)$

   For every $x$ there exists a $y…$

   This is true, because...

Yes.

2. $∀y ∃x P(x, y)$

This is true, because no matter what $y$ is, there is always some $x$ whose square is bigger, since $\mathbb R$ has no upper bound.

3. $∃x ∀y P(x, y)$

There exists a $x$ (such that) for every $y…,$

This is true, we can increment $x$ and $y$ to infinite, and every time the statement will stay true. If we set an end to the infinite and give the largest number possible to $x,$ that number will hold true for any possible $y.$

Proposition $3$ is false, because no such $x$ exists, precisely because we cannot "set an end to the infinite".

4. $∃y ∀ x P(x, y)$

There exists a $y$ (such that) for every $x…,$ means that the is a $y$ that is less than $x^2$ for every possible $x.$ This is true because...

Yes. But do be careful though: "there is a $y$ that is less than $x^2$ for every possible $x$" is ambiguous (it contains a hanging quantifier), and can be interpreted either as $∃y ∀x P(x, y)$ or $∀x ∃y P(x, y).$