Given a set S, a function f: S X S 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 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 binary operations?
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)}.