Given a set X with a binary operation *, not necessarily associative or commutative, but such that (x * y) * x = y for all x, y in X. Show that x * (y * x) = y for all x, y in X.
Solution
If the relation given is to yield an expression ending in (y * x), we must substitute (y * x) for x. So we consider ( (y * x) * y) * (y * x). Indeed that suffices without more. [If we regard (y * x) as z, then it evaluates to y. On the other hand, we can evaluate ( (y * x) * y) to x, so that the expression becomes x * (y * x).]
© John Scholes
jscholes@kalva.demon.co.uk
16 Dec 2001