(1)$ (\forall a \in S)(\exists e \in S) $
(2)$(\exists e \in S)(\forall a \in S)$
Are $(1)\equiv(2) $ or the order in which I write them has some different meaning ? Basically, I have been told that (1) and (2) are not the same mathematical expressions ... Could someone please explain why aren't they the same ?
On
$\forall$ means "for all" but sometimes more cleanly translates as "for any" or "for every."
$\exists$ means "there exists" but might be interpreted as "we can find."
For any point, $a,$ on the number line, there is a point $e$ on the number line that is greater than $a.$
Clearly true.
There exists a point on the number line $e,$ for every point $a$ on the number line $a$ is greater than $e.$
This implies that $e$ is the smallest point. Or that there is a least element which might be true in $\mathbb N$ but is not true in $\mathbb Z$ or $\mathbb R$
Everyone is someone's child, but nobody is everyone's parent.
In symbols: taking "$C(a, b)$" to mean "$a$ is $b$'s child," we have $\forall x\exists y C(x, y)$ but $\neg \exists y\forall x C(x, y)$.
"$\exists x\forall y\varphi(x, y)$" says that not only does every $y$ have an $x$ such that $\varphi(x, y)$ holds, but there is one single $x$ which works for every $y$. This is in general a much stronger claim.
Another example worth considering is that the statement "There is no largest number" can be written in two natural ways: "For all $x$ there is some $y$ such that $x<y$" and "NOT(there is some $y$ such that for all $x$, $x<y$)." In this example, not only are "$\forall x\exists y$" and "$\exists y\forall x$" not equivalent, they're actually direct opposites!