Thread: 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!

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: