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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Originally Posted by cristidrincu 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.
Thanks. I'm new to abstract algebra. Did not know that we can substitute z with x. Thanks again!
Originally Posted by HallsofIvy 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: $\displaystyle xy=(xy)(xy)=((xy)x)y=((yx)x)y=(x^2y)y=(xy)y=y^2x=y x.$
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