# Thread: One and Two Sided Inverses

1. ## One and Two Sided Inverses

I posted a thread not two long ago about one and two sided identities but I'm having a little confusion with inverses. The question is:

"Prove: Let $\displaystyle <A,O>$ be a system with identity $\displaystyle e$ in which $\displaystyle O$ is associative. If $\displaystyle b$ is a left-inverse for $\displaystyle a \in A$ and c a right-inverse for a, then b = c. As corollaries, show that (a) a two inverse is unique, and (b) if $\displaystyle O$ is commutative, then $\displaystyle <A,O>$ has at most one left inverse."

I'm sure there is a super simple solution but it isn't immediately apparent to me, thanks in advance.

2. Originally Posted by jameselmore91
I posted a thread not two long ago about one and two sided identities but I'm having a little confusion with inverses. The question is:

"Prove: Let $\displaystyle <A,O>$
A is a set of objects, O is a binary operation on them?
be a system with identity $\displaystyle e$ in which $\displaystyle O$ is associative. If $\displaystyle b$ is a left-inverse for $\displaystyle a \in A$ and c a right-inverse for a, then b = c. As corollaries, show that (a) a two inverse is unique, and (b) if $\displaystyle O$ is commutative, then $\displaystyle <A,O>$ has at most one left inverse."

I'm sure there is a super simple solution but it isn't immediately apparent to me, thanks in advance.
By the definition of "left inverse", ba= I, the identity. By the definition of "right inverse", ac= I.

Mutiply both sides of ac= I, on the left, by b and see what happens. (Using, of course, the fact that O is associative so b(ac)= (ba)c.)

The last two parts, then, are simple.