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.
Here is a picture.
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 .