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Math Help - proving * is commutative

  1. #1
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    proving * is commutative

    This is the request of the exercise:

    "Knowing (M,*) is a grupoid and the following statements are true:
    1.x*x=x
    2.(x*y)*z=(y*z)*x

    Prove that * is commutative.

    Thanks!
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  2. #2
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    Quote Originally Posted by cristidrincu View Post
    This is the request of the exercise:

    "Knowing (M,*) is a grupoid and the following statements are true:
    1.x*x=x
    2.(x*y)*z=(y*z)*x

    Prove that * is commutative.

    Thanks!
    What do you get if, in (2), you take z= x? Now "multiply", on the right, on both sides of that, by x.
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  3. #3
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    proving * is commutative

    Thanks. I'm new to abstract algebra. Did not know that we can substitute z with x. Thanks again!
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post

    What do you get if, in (2), you take z= x? Now "multiply", on the right, on both sides of that, by x.
    i don't see how this will solve the problem? here is my solution: xy=(xy)(xy)=((xy)x)y=((yx)x)y=(x^2y)y=(xy)y=y^2x=y  x.
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