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Math Help - How to find if neutral/idnty element exists for 16 different binary operations of *?

  1. #1
    Senior Member x3bnm's Avatar
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    How to find if neutral/idnty element exists for 16 different binary operations of *?

    Suppose I'm given an operation * on a set A\,\, \text{where}\,\, a,b \in A such that the operation between two elements of set A is also an element of set A


    Now there are 2 \times 2 \times 2 \times 2 = 16 possible binary operations of * on set A.

    If I'm asked to find if there exists a neutral/identity element of a certain table where a * a, a * b, b * a  \text{ and } b * b
    are defined, how do I figure that out?


    What I mean is suppose 1 of 16 table of operations looks like this:


     \begin{array}{|l||r|}     \hline (a * a) & a \\ \hline (a * b) & b \\ \hline     (b * a) & a \\ \hline (b * b) & b \\     \hline \end{array}

    How do I figure out if an identity element exist for this table? Also is it possible to find the identity/neutral element from this table?
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  2. #2
    MHF Contributor

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    Re: How to find if neutral/idnty element exists for 16 different binary operations of

    That binary operation does not have an identity. The definition of identity is "e is an identity for operation "*" if and only if e*x= x*e= x for every x in in the set". Here, a*b= b but b*a= a\ne b so a is NOT an identity. Similarly, b*a= a but a*b= b\ne a so b is not an identity.
    Last edited by HallsofIvy; August 24th 2012 at 06:56 PM.
    Thanks from x3bnm
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  3. #3
    Senior Member x3bnm's Avatar
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    Re: How to find if neutral/idnty element exists for 16 different binary operations of

    Again thanks.
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