good at discrete math? can you answer this???

Question

Let A be a set containing 23 elements and B a set containing 13 elements.
(a) How many functions are there from A to B?
(b) How many injective functions from A to B?
(c) How many surjective functions from A to B?

Answer 1

(a)
element a1 could map to any element in B (b1 to b13)
element a2 could map to any element in B (b1 to b13)
...
Total number of ways: 13^23

(b) injective: no two elements of a can map to the same element of b.

If you can't limit the domain to a subset of A, then none (because when you want to map the 14th element of A, you will have run out of element in B).

Otherwise, you can restrict the domain to any 13 elements of A (there are 1,144,066 ways to pick 13 elements out of 23); In each subset, the first element can map to any of the 13, the second to any of the remaining 12...
1,144,066 times 13! (where ! means factorial)

(c) surjective: each element of B is used (at least once).

I'll have to think about this one...

You could rank all the elements of B from the most used to the least used. Find out how many ways there are to get 23 by adding 13 numbers together, the least number being 1.

e.g., 23 = 8+3+1+1+1+1+1+1+1+1+1+1+1+1

For each of these ways, there are 13! ways to arrange the elements of B in that order.

Then (using that particular one as an example), There are 23C8 ways to select the 8 elements of A that will map to the most used element of B, 15C2 ways to select the next two elements of A that will map to the second-most used element of B, then the remaining elements of A must be assigned to each element of B (no freedom, as all possible arrangements have been taken care of in the 13! ways this particular distribution could be arranged).

You have to do that for all possible arrangements that add up to 23.