MappingOfDegreeNIsASurjection 2003-08-01 14:41:30 2004-03-12 01:57:16 Proof period of mapping 0 % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\sR}[0]{\mathbb{R}} \newcommand{\sC}[0]{\mathbb{C}} \newcommand{\sN}[0]{\mathbb{N}} \newcommand{\sZ}[0]{\mathbb{Z}} {\bf Theorem} Suppose $X$ is a set. Then a mapping $f:X\to X$ \PMlinkname{of period}{PeriodOfMapping} $n$ is a bijection. {\bf Proof.} If $n=1$, the claim is trivial; $f$ is the identity mapping. Suppose $n=2,3,\ldots$. Then for any $x\in X$, we have $x=f\big(f^{n-1}(x)\big)$, so $f$ is an surjection. To see that $f$ is a injection, suppose $f(x)=f(y)$ for some $x,y$ in $X$. Since $f^n$ is the identity, it follows that $x=y$. $\Box$