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Math Help - binary operation on a set

  1. #1
    Member oldguynewstudent's Avatar
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    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?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by oldguynewstudent View Post
    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.
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  3. #3
    Member oldguynewstudent's Avatar
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    Quote Originally Posted by oldguynewstudent View Post
    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?
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  4. #4
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    Quote Originally Posted by oldguynewstudent View Post
    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)
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  5. #5
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    Re: binary operation on a set

    Yes I believe that (n)^(n^2) is correct
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  6. #6
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    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).
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