i've been confuse on how to show the proof of this abstract problem. please help me.?

Question

suppose that * is an associative and commulative binary operations on a set S, show that H = {a element s such that a * a = a} is closed under *

* means asteris as the binary operation...

Answer 1

To show that H is closed under *, we just have to show that for any a, b in H, a*b is also in H. To do this we have to show that (a*b)*(a*b) = a*b. since * is commutative and associative, (a*b)*(a*b) = (a*a)*(b*b) = a*b. Thus H is closed under *.