Given G is a set with associative binary opeartion.
Given for any,$a,b \in G$ , there exists $a'$ in G such that $ be=b, $. And $a'a=e$ First i try to show that $eb=b$.
Given,
$be=b$
$beb=bb$
$b'beb=b'bb$
$b'beb=eb=b$
$b'beb=b$
$eb=b$
Now for any element c of set G i want to show that $ce=c$
$ce=cc'e=ec=c$
To show right inverse :
Given $a'a=e$, $a'aa'=ea'$ $a'eaa'=ea'$
$a'aa'=a'$
$a'aa'=a'e$
$aa'=e$ (i believe cancellation laws can be proved by using left inverse, so using them)
Is this correct ? Thank you