# Showing that a subset is closed under a binary operation

1. Feb 19, 2017

### Mr Davis 97

1. The problem statement, all variables and given/known data
Suppose that * is an associative binary operation on a set S. Let $H=\{a \in S ~| ~a*x=x*a, ~ \forall x \in S\}$. Show that * is closed under H.

2. Relevant equations

3. The attempt at a solution
Let b and c be two different elements in H. We need to show that b*c is also in H.

We know that bx = xb, and that cx = xc. Putting these two equations, and using associativity and commutativity the farthest I can get is (bc)xx = xx(bc). I'm not sure how to get (bc)x = x(bc)

2. Feb 19, 2017

### Staff: Mentor

You start with $(bc)x$. And it's " H is closed under * ".