How Do You Prove a Function Is Isomorphic to the Cartesian Product of X?

[Nexttime35]

Homework Statement

See Attachment:
https://www.physicsforums.com/attachment.php?attachmentid=59074&d=1369708771

Homework Equations

As shown in the attachment, I am slightly confused as to where to begin this problem.

I know that I need to prove that a function, f, is 1-1 and onto, in order for the function to be isomorphic to the cartesian product of X, but I am unaware as to where to begin this problem.

Does anyone have any ideas that could help?

Thank you very much,

G.

 

[Dick]

Homework Statement

See Attachment

Homework Equations

As shown in the attachment, I am slightly confused as to where to begin this problem.

I know that I need to prove that a function, f, is 1-1 and onto, in order for the function to be isomorphic to the cartesian product of X, but I am unaware as to where to begin this problem.

Does anyone have any ideas that could help?

Thank you very much,

G.

Shouldn't you attach something? I don't see anything.

 

[Nexttime35]

Here is the attachment:

 

[Dick]

Here is the attachment:

Ok, I see it now. Suppose f(1)=x1 and f(2)=x2 where x1 and x2 are in X. Can't you think of a way to associate that with an ordered pair in X x X?

 

[Nexttime35]

Hmm. I'm still slightly confused about how to link f(1)=x1 and f(2)=x2 with an ordered pair, to prove that f is 1-1 and onto. Any ideas?

 

[Dick]

Hmm. I'm still slightly confused about how to link f(1)=x1 and f(2)=x2 with an ordered pair, to prove that f is 1-1 and onto. Any ideas?

Whatever you are using to display attachements seems to be blocked in the firefox browser or by the website, or they've expired. They've now disappeared. If I open it in Chome I don't see anything either. You should probably just type your problem statement in. I've forgotten how it was exactly stated and it would be best to have it here for reference. But isn't (f(1),f(2)) an ordered pair in X x X? And doesn't an ordered pair also define a function {1,2}->X x X?

 

[Nexttime35]

https://www.physicsforums.com/attachment.php?attachmentid=59112&stc=1&d=1369838320

Here's the link to the problem.

 

[Dick]

https://www.physicsforums.com/attachment.php?attachmentid=59112&stc=1&d=1369838320

Here's the link to the problem.

Thanks. You don't want to prove f is 1-1 and onto. Define a function G:X^{1,2}->X x X, where if f is in X^{1,2} then G(f)=(f(1),f(2)). G is what you want to prove is 1-1 and onto.

 

[Nexttime35]

OK so, here's my first attempt at proving G is 1-1.

G is 1-1: Assume G(f(1)) = G(f(2)). Then since g is 1-1, (f(1)) = (f(2)) which lies in X x X.
Does that follow? Thanks for your help.

 

[Dick]

OK so, here's my first attempt at proving G is 1-1.

G is 1-1: Assume G(f(1)) = G(f(2)). Then since g is 1-1, (f(1)) = (f(2)) which lies in X x X.
Does that follow? Thanks for your help.

You don't seem to be quite getting this. G is supposed to map a function from {1,2} into X into an ordered pair in X x X. Suppose X=N, the natural numbers. If f(1)=9 and f(2)=16 then G(f) is an ordered pair. What is it? Conversely if you are given an ordered pair (1,3), what is the corresponding f? You want to prove there is a 1-1 correspondence between functions and ordered pairs.

 

[Nexttime35]

If f(1) = 9 and f(2) = 16, then G(f) is an ordered pair, (9,16). Given an ordered pair (1,3), the corresponding f would be (f(1),f(3)), correct?
So I want to find a bijection G: X^{1,2} to X x X, and then prove the bijection. Now, my professor said there are two options for the function to prove is 1-1 and onto. What are those two functions?

 

[Dick]

If f(1) = 9 and f(2) = 16, then G(f) is an ordered pair, (9,16). Given an ordered pair (1,3), the corresponding f would be (f(1),f(3)), correct?
So I want to find a bijection G: X^{1,2} to X x X, and then prove the bijection. Now, my professor said there are two options for the function to prove is 1-1 and onto. What are those two functions?

Well, no. The function defined by the ordered pair has to be a function from {1,2} into X. If the ordered pair is (1,3) then define the function f(1)=1 and f(2)=3. So G will map that function to the ordered pair (1,3). Another obvious choice would be to define H(f)=(f(2),f(1)). It should be kind of obvious that G and H are bijections from functions to ordered pairs, if you think about it. See if you can explain why.