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$$\exists\ x \in \mathbb{N}\ \textrm{such that}\ \forall\ y \in \mathbb{N}, 2x \leq y + 1$$

$$\forall\ y \in \mathbb{N}, \exists\ x \in \mathbb{N}\ \textrm{such that}\ 2x \leq y + 1$$

I'm having trouble understanding the differences between these two statements. To me, they seem to mean the same thing. Can anyone explain in basic terms? Thanks.

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"Every hour a person is injured in a car accident" - "Oh dear, I would stay at home if I were that fellow". – Hagen von Eitzen Sep 20 '12 at 18:23
The difference is that in the first case you have to choose $x$ first, without reference to $y$: the statement must be true for all $y$ for some fixed $x$. In the second case $x$ is chosen after $y$ is introduced, and so $x$ can be a function of $y$. – mjqxxxx Sep 20 '12 at 18:53
It is not too much of an exaggeration to say that the whole rationale of the quantifier-variable analysis of statements of generality which we owe to Frege is to mark this kind of distinction. So asking the question suggests you haven't "got it". So I'd warmly recommend looking at a good intro logic book where it introduces the quantifiers. Paul Teller's Primer is very good and freely available online. – Peter Smith Sep 21 '12 at 6:17
Answers to Confused between Nested Quantifiers question may help you to better understand nested quantifiers. – olek Sep 21 '12 at 11:09
up vote 2 down vote accepted

Let $L(x,y)$ stand for "$x$ loves $y$". Then $\exists x\forall y: L(x,y)$ means "There is someone who loves everyone." and $\forall y\exists x: L(x,y)$ means "Everybody is loved by someone". Clearly, these two are very different.

Now compare the simple mathematical statements.

$\exists x\in\mathbb{N}$ such that $\forall y\in\mathbb{N}$, $x\leq y$.

$\forall y\in\mathbb{N},$ $\exists x$ such that $x\leq y$.

The first one says that there is some natural number $x$ that is smaller or equal than every natural number $y$.

The second statement says that for every natural number $y$, there is a natural number $x$ that is less or equal than $y$.

It turns out that both of these statements are true. But now replace $\mathbb{N}$ by $\mathbb{Z}$. There is no smallest integer, so the first statement becomes wrong then. But the second one would still be true because for every integer $y$, the integer $y-1$ is smaller or equal than y$.

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  1. There is some $x$, such that no matter what $y$ you choose, $x$ will be less than $y+1$. $x=0$ fits the bill.
  2. No matter which $y$ you choose, you can always choose some $x$ s.t. $2x\leq y+1$. E.g. if $y=5$, then $x=1$ satisfies.

Note that if we were dealing with say the integers, the second statement would be true but the first one would be false. It might be instructive to figure out why.

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