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Math Help - Binary Operations!

  1. #1
    Junior Member
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    Binary Operations!

    Let S = {a, b, c, d}.
    How many binary operations can be defined on S?
    I think the answer might be 4^{16}

    How many are commutative?
    I think the answer is 4^{10}

    Assume that x = x * a = a * x for all x in the set S.
    Then how many binary operations are there?
    I think the answer is 4^{9}

    Can any1 tell me if my answers are right or wrong?
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  2. #2
    MHF Contributor

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    The answer really depends upon the definitions is use.
    How are you defining binary operator?
    The most common definition is a function \phi :X \times X \mapsto X.
    If that is you definition then yes 4^{16} is correct for the total number.
    I agree with your other two answers.
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