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#1 2013-11-09 13:30:55

zetafunc.
Guest

Logic Question

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?

#2 2013-11-09 15:20:04

Au101
Member
Registered: 2010-12-01
Posts: 286

Re: Logic Question

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 smile

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#3 2013-11-09 19:57:04

Nehushtan
Member
From: London
Registered: 2013-03-09
Posts: 615
Website

Re: Logic Question

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(x) be a real-valued function of the real variable x. Then f is continuous iff the following holds:

where

denotes the statement
.

You should also be aware that

and
do not commute:
. In the first statement, the y depends on the particular x chosen; different choices of x may require different y. In the second statement, however, there is a unique y for all the x you choose.

For 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:

It’s the same thing.


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#4 2013-11-10 01:00:56

zetafunc.
Guest

Re: Logic Question

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

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