coloring Coloring 2002-08-10 14:27:13 2005-03-03 17:12:22 Definition % 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 %\PMlinkescapeword{theory} A \emph{coloring} of a set $X$ by $Y$ is just a function $f:X\rightarrow Y$. The term coloring is used because the function can be thought of as assigning a ``color'' from $Y$ to each element of $X$. Any coloring provides a partition of $X$: for each $y\in Y$, $f^{-1}(y)$, the set of elements $x$ such that $f(x)=y$, is one element of the partition. Since $f$ is a function, the sets in the partition are disjoint, and since it is a total function, their union is $X$.