mapping of period n is a bijection

Theorem Suppose X is a set. Then a mapping f:XX of period ( n is a bijection.

Proof. If n=1, the claim is trivial; f is the identity mapping. Suppose n=2,3,. Then for any xX, we have x=f(fn-1(x)), so f is an surjection. To see that f is a injection, suppose f(x)=f(y) for some x,y in X. Since fn is the identityPlanetmathPlanetmath, it follows that x=y.

Title mapping of period n is a bijection
Canonical name MappingOfPeriodNIsABijection
Date of creation 2013-03-22 13:48:57
Last modified on 2013-03-22 13:48:57
Owner Koro (127)
Last modified by Koro (127)
Numerical id 7
Author Koro (127)
Entry type Proof
Classification msc 03E20