Putnam 2001 - Problem A-1 (On a binary operation)

Question

Let $*$ be a binary operation acting on a set $S$, such that $$(a*b)*a=b,$$ for all $a,b\in S$.

Prove that $$a*(b*a)=b,$$ for all $a,b \in S$.

Answer 1

We have that $$ a*(b*a)=\big((b*a)*b\big)*(b*a)=b $$ Note that the second "=" holds since $$ (c*b)*c=b, $$ where $c=b*a$.

Answer 2

Because $b*a,b\in S$, so by the formula $((b*a)*b)*(b*a)=b$. Then $a*(b*a)=b$.