binary operation on a set

Given a set S, a function f: S X S $\displaystyle \longrightarrow$ S is called a binary operation on S. If S is a finite set, then how many different binary operations on S are possible?

I have no clue on this one? Could someone point me in the right direction?

I realize that if S has n elements then SxS would have $\displaystyle n^2$ possible ordered pairs. However, I'm not sure what is meant by how many different binary operations. Would we have addition, subtraction, multiplication, division, exponentiation, etc. times $\displaystyle n^2$ binary operations?

Re: binary operation on a set

Yes I believe that (n)^(n^2) is correct

Re: binary operation on a set

in general, the number of functions f:A→B is:

|B|^{|A|}.

it is easiest to see this when |B| = 2, such as when B = {0,1}, so that the functions f:A→{0,1} can be put in a 1-1 correspondence with the subset of A:

given a subset S of A, we define:

f(a) = 1, if a is in S

f(a) = 0, if a is not in S.

thus the number of functions f:A→{0,1} is the same number as 2^{|A|}, the cardinality of the power set of A.

so, yes, the correct answer is n^{(n2)}.