# Thread: binary operation on a set

1. ## binary operation on a set

Given a set S, a function f: S X S $\longrightarrow$ 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 $n^2$ 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 $n^2$ binary operations?

2. Originally Posted by oldguynewstudent
Given a set S, a function f: S X S $\longrightarrow$ 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 $n^2$ 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 $n^2$ binary operations?
It's asking how many functions are there from a set with $n^2$ elements to a set with $n$ elements.

3. Originally Posted by oldguynewstudent
Given a set S, a function f: S X S $\longrightarrow$ S is called a binary operation on S. If S is a finite set, then how many different binary operations on S are possible?
With the help of Drexel28, we have a set S with n elements which gives $n^2$ mapped to n. This should give us $n^3$ binary operations, correct?

4. Originally Posted by oldguynewstudent
With the help of Drexel28, we have a set S with n elements which gives $n^2$ mapped to n. This should give us $n^3$ binary operations, correct?

Shouldn't it be (n)^(n^2)

5. ## Re: binary operation on a set

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

6. ## 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).