## Friday, September 15, 2006

### Binary Ops. Topic: Algebra/Sets. Level: AMC/AIME.

Problem: (2001 Putnam - A1) Consider a set $S$ and a binary operation $*$, i.e., for each $a,b \in S$, $a*b \in S$. Assume $(a*b)*a = b$ for all $a, b \in S$. Prove that $a*(b*a) = b$ for all $a, b \in S$.

Solution: If we make the substitution $a = b*a$, we get

$((b*a)*b)*(b*a) = b$.

However, since $(b*a)*b = a$ (by switching $a$ and $b$), we thus have

$a*(b*a) = b$,

as desired. QED.

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Comment: Not an extremely hard problem, but making the right substitution took some time to find. Playing around with the given equality is pretty much the only option in this problem.

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Practice Problem: Find a set $S$ (preferably nontrivial) and operation $*$ satisfying the conditions of the problem above.