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#1 2013-11-10 12:30:55

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 2013-11-10 14:20:04

Au101
Power Member

Offline

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

#3 2013-11-10 18:57:04

Nehushtan
Power Member

Offline

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:



Its the same thing.


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#4 2013-11-11 00:00:56

zetafunc.
Guest

Re: Logic Question

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

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