MHB What is the binary operation in set $S$ and what is the given property?

[Ackbach]

Here is this week's POTW:

-----

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 $a,b\in S$.

-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[Ackbach]

Re: Problem Of The Week # 271 - Jul 10, 2017

This was Problem A-1 in the 2001 William Lowell Putnam Mathematical Competition.

Congratulations to Opalg for his (incredibly concise and, of course, correct) answer, which follows:

Spoiler

In the identity $(a*b)*a = b$, substitute $b*a$ in place of $a$: $$((b*a)*b)*(b*a) = b.$$ But (interchanging $a$ and $b$ in the original identity) $(b*a)*b = a$. Therefore $$a*(b*a) = b.$$