Is there some understandable rationale why $\forall x\, \exists y\, P(x,y) \not \equiv \exists y\, \forall x\, P(x,y)$?
I'm looking for a sentence I can explain to students, but I am failing every time I try to come up with one.
Example
Let $P(x,y)$ mean that $x$ is greater than $y$.
- $\forall x\, \exists y\, P(x,y)$ means that for all $x$, there is a number $y$, such that $x$ is greater than $y$.
- $\exists y\, \forall x\, P(x,y)$ means that there is some $y$, that every number $x$ is greater than.
These don't seem to mean different things to me. Is this perhaps an example where they do mean the same thing or am I just translating to English incorrectly?
Best Answer
Take $P(x, y)$ to mean $y$ is a parent of $x$.
Then $\forall x \exists y P(x, y)$ means everybody has a parent, while $\exists y \forall x P(x, y)$ means there is someone who is the parent of every son and daughter.