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I'm working with the universal and the existential quantifiers and I was wondering is there an order of operations that applies to them? What is the difference between

$(\forall x)P(x)(\exists x)Q(x)$

And

$(\exists x)Q(x)(\forall x)P(x)$

Lets say we have this relationship

(a) $[(\forall x)P(x) \vee (\exists x)Q(x)] \implies (\forall x)[P(x) \vee Q(x) ]$

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    @lampShade: The last line should be the following: $[(\forall x)P(x) \vee (\forall x)Q(x)] \Longleftrightarrow (\forall x)[P(x) \vee Q(x)]$. You can only distribute the same quantifiers.2011-02-03

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Here is the question you probably meant to ask, and if it isn't, then here is the question you should have asked. Is there a difference between the statement $(\forall x)(\exists y) P(x, y)$ and the statement $(\exists y)(\forall x) P(x, y)$? The answer is yes. Consider the following simple example:

For all men, there exists a woman who loves them.

is not the same as

There exists a woman who loves all men.

The point is that when we say $(\forall x)(\exists y)$, the $y$ that is meant is a function of $x$, and this is not true in the other order. This is unfortunately hidden by the notation.

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    @Mariano: ah, I didn't see your comment up top.2011-02-03