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I don't even know where to start, my professor just kinda went so fast through this and didn't explain this. I know that it says "For all of x and some of y, but after that, I just get lost. What is A(x,y)? If you can please explain, I truly want to understand this.

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    $\forall n.\exists m.n$m,n\in{\bf N}$). – 2017-02-04

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With $A(x,y)$: $x$ is the child of $y$

You get:

$\forall x \exists y A(x,y)$: Everyone is the child of someone (true)

$\exists y \forall x A(x,y)$: Someone is a parent of everyone! (false)

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A slightly simpler example, which works in any set with at least two elements: Let $A(x,y)$ be $x=y$. Then $\forall x\exists y\,x=y$ is true, because, given any value for $x$, you can choose the same value for $y$ and thereby make $x=y$ true. But $\exists y\forall x\,x=y$ is false, because you can't choose a value for $y$ without knowing $x$ and get $x=y$ to hold for all values of $x$.

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Yes, there are plenty such things. Quantifiers like $\forall$ and $\exists$ are usually not interchangeable.

For example, it's clearly true that $\forall x \in \mathbb{N},\exists y \in \mathbb{N}$, such that $y>x$. For all natural numbers $x$, there is a natural number $y$ which is bigger than $x$.

However, it is clearly not true that $\exists y \in \mathbb{N}, \forall x \in \mathbb{N}$ such that $y>x$ because the latter implies there is a number $y$ greater than all natural numbers.

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As already hinted at: let $A(n,m) = n < m$, which is a valid predicate on $\mathbb{N}$.

true: $$\forall n \exists m : n < m$$

because for every $n$ we can pick $m = n+1$ which satisfies $n < m$. This statement just says, every number has numbers above it.

false: $$\exists m \forall n : n < m$$

There is no $m$ that is larger than all $n$, also witnessed by $m = n+1$, supposing such $m$ would exist.

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$A(x,y)$ can be any proposition with two subjects. $A(x,y)$ could be $x < y$ or $x = y$ or "$x$ is a multiple of $y$" or "$y$ is a multiple of $x$" or "$x$ and $y$ are both integers", "$\sqrt{x^2 + y^3} > 57$. etc.

So the $\forall x \exists y: A(x,y)$ is true means that for all $x$ there is a $y$ that makes the statement true and $\forall x \exists y: A(x,y)$ is false means for all x there is a y that makes the statement false.

Any proposition that for all $x$ is sometimes true and sometimes false depending on $y$ will do.

Take $A(x,y) \iff x=y$ or $A(x,y) \iff x < y$ or $A(x,y) \iff y \in \mathbb Q$ or.... there are an infinite of them.

For there not to be both we must either have:

a statement that has some $x$ where the statement is always true. Ex. $x > -|y|$ is always true if $x > 0$.

or a statement that has some $x$ where the statement is always false. Ex. $x > |y|$ is always false if $x < 0$.