Let A be a set of size n, B a set of size m, C a set of size k. n≤m≤k
How many ordered pairs of functions (f,g) are there such that f:A-->B, g:B-->C and gºf is injective?
(The working out is what I really want to know).
There are injections .
Now for to be an injection it is not necessary for to be injective.
However, must be injective on the image of
How many functions are there such that the restriction to is an injection?
We have m!/(m-n)! injective functions f from A to B.
We have k!/(k-n)! injective functions g from Im(f) to C.
Why are we counting the the functions g from B to C that are not in the image of f?
I know we have k^(m-n) such as these, but why are we counting them?