Problem:
Consider a binary operation '*' on the set $S$. It is given that $(a*b)*a = b, \forall a,b \in S$. PT: $a*(b*a) = b \forall a,b \in S$
My attempt:
Given : $(a*b) *a = b \forall a,b \in S$...........1)
Or $((a*b)*a)*(a*b) = b* (a*b)$
Let $X = (a*b)$. Clearly $X \in S$.
Or $(X*a)*X = b*(a*b)$
But from 1)we get :
$b*(a*b) = a \forall ab \in S$.…..........2)
We rename the variables as $a,b = b,a$ ans rewrite 2)
Thus we have : $a*(b*a) = b , \forall a,b \in S$
Is the last part correct where i rename the variables? I couldn't rigorously prove that such a transformation is valid. If valid/ invalid please explain why.
You can do it that way, but it seems clumsy.
The assumption is that $(a*b)*a=b$, for every $a,b\in S$.
In particular, $(b*a)*b=a$, when $a,b\in S$. Hence $$ a*(b*a)=\bigl((b*a)*b\bigr)*(b*a) $$ On the other hand, for every $x\in S$, $(x*b)*x=b$, so with $x=b*a$ we obtain $$ a*(b*a)=\bigl((b*a)*b\bigr)*(b*a)= (x*b)*x=b $$ as wanted.