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**zetafunc.****Guest**

I'm having a little trouble understanding how to interpret some logic statements.

I'm aware that

reads as "For all x, there is at least one y such that P(x,y) is true."Is it correct to say that

reads as "There is at least one x such that P(x,y) is true for all y"?Furthermore, how do I interpret these statements?

Can we simply interpret them as "there exists an x AND there exists a y" and similarly for the subsequent statement? Or is this wrong?

**Au101****Member**- Registered: 2010-12-01
- Posts: 351

It's been a while and you should wait for someone to confirm, but as far as I know yes, everything you've written is fine

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zetafunc. wrote:

I'm aware that

reads as "For all x, there is at least one y such that P(x,y) is true."Is it correct to say that

reads as "There is at least one x such that P(x,y) is true for all y"?

That is correct. As an illustration, let us apply this to the definition of continuity: let f(

where

denotes the statement .You should also be aware that

and do not commute: . In the first statement, theFor example, in the definition of continuity above, let us reverse the order of

and :This is no longer the definition of continuity, but of uniform continuity, which is a stronger condition than continuity.

zetafunc. wrote:

Furthermore, how do I interpret these statements?

Can we simply interpret them as "there exists an x AND there exists a y" and similarly for the subsequent statement? Or is this wrong?

Right again. And this time commutes with , as does with . For example, the definition of continuity above can be restated as follows:

Its the same thing.

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**zetafunc.****Guest**

Thanks a lot -- beautifully explained, makes perfect sense now.

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