bijection between two sets of functions

Hi

here's a problem I am trying to solve. Suppose A,B,C,D are sets

where is set of all functions from A to C. and is

set of all functions from B to D. Following is my work. I have already prove that

Suppose

I have also already proved that, for any sets A and B,

that means that there is a bijection from

to .

is the set of all subsets of

. But subsets of are just relations from A to C. So

is set of all relations from A to C. In similar way,

is set of all relations from B to D. Functions are special kind of relations , So

Since , there is some bijection between these two sets. Let it be h

From here on, I am stuck. We have a bijection between one family of relations to another family of relations. What we need is a bijection between the subsets of

these families.

(Emo)

Re: bijection between two sets of functions

Quote:

Originally Posted by

**issacnewton** Suppose A,B,C,D are sets

where

is set of all functions from A to C. and

is

set of all functions from B to D.

I suppose you mean functions or mappings. Then, consider bijections and . Now, define such that . Easily proved, is bijective hence .

Edited: Wrong argument.

Edited: Wrong day for posting.

Re: bijection between two sets of functions

Here is a picture.

https://lh4.googleusercontent.com/-D...0/diagram4.png

Note that since and are bijections, there exist and such that , , and , where is the identity function on . Therefore, you can multiply various equalities by these functions. E.g., implies .

Re: bijection between two sets of functions

Right. A quick, mental (and wrong) reasoning led me to . Better appproach: denoting we have , etc.

Re: bijection between two sets of functions

Re: bijection between two sets of functions

Quote:

Originally Posted by

**emakarov** Um, why is it wrong?

Perhaps too much whisky and champagne last night. :)

Re: bijection between two sets of functions

thanks. I was almost there as I have proved the bijection between the set of relations from A to C to the set of relations from B to D.