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 $∀x ∃y P(x, y).$

Answer 2

Your understanding and examples for your first and last sentence sounds right.

As for the sentence $\forall y \exists x P(x, y)$ in your case, it reads "for any $y \in \mathbb{R}$ there's some $x \in \mathbb{R}$ such that $y<x^2$" which is certainly true as there's no upper bound of the term $x^2$.

As for the sentence $\exists x \forall y P(x, y)$ in your case, it reads "there's some $x \in \mathbb{R}$ for any $y \in \mathbb{R}$ it's the case that $y<x^2$" which is clearly false since if you happens to find such $x$, you can always find a positive $y>x^2$.

In conclusion order matters for mixed quantifiers with different types...