Relations

Arun Ram
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

and

Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 22 September 2009

Definitions

A relation on a set $S$ is a subset of $S\times S$. We write $s_1\sim s_2$ if the pair $\left(s_1,s_2\right)$ is in the subset.

A relation is reflexive if, for each $s\in S$, $$s\sim s\text{.}$$

A relation is symmetric if, for all $s_1,s_2\in S,$ $$s_1\sim s_2⟺s_2\sim s_1\text{.}$$

A relation is transitive if, for all $s_1,s_2,s_3\in S$, $$s_1\sim s_2\text{ and }{s}_2\sim s_3⟹s_1\sim s_3\text{.}$$

An equivalence relation on a set $S$ is a relation on $S$ that is reflexive, symmetric and transitive.

Examples

Let $S$ be the set $\left\{1,2,6\right\}$. Then,

	(a)	$R_1=\left\{\left(1,1\right),\left(2,6\right),\left(6,1\right)\right\}$ is a relation on $S$,
	(b)	$R_1$ is not reflexive, not symmetric and not transitive,
	(c)	$R_2=\left\{\left(1,1\right),\left(2,6\right),\left(6,1\right),\left(2,1\right)\right\}$ is a relation on $S$, and
	(d)	$R_2$ is transitive but not reflexive and not symmetric.

More definitions

Let $S$ be a set and let $\sim$ be an equivalence relation on $S$. The equivalence class of an element $s\in S$ is the set $$\left[s\right]=\left\{t\in S|t\sim s\right\}\text{.}$$

A partition of a set $S$ is a collection of subsets $S_{\alpha }$ such that,

	(a)	if $s\in S$ then $s\in S_{\alpha }$ for some $S_{\alpha }$, and
	(b)	if $S_{\alpha }\cap S_{\beta }\ne \varnothing$ then $S_{\alpha }=S_{\beta }$.

Equivalence classes and partitions

This proposition shows that the concepts of an equivalence relation on $S$ and of a partition on $S$ are essentially the same. Each equivalence relation on $S$ determines a partition and visa-versa.

Example

Let $S=\left\{1,2,3,\dots ,10\right\}$. Let $\sim$ be the equivalence relation determined by

$1\sim 5$,	$2\sim 3$,	$9\sim 10$,
$1\sim 7$,	$5\sim 8$,	$4\sim 10$.

Since we are requiring that $\sim$ is an equivalence relation, we are assuming that we have all the other relations we need such that $\sim$ is reflexive, symmetric and transitive;

$1\sim 1$,	$2\sim 2$,	…	$9\sim 9$,	$10\sim 10$,
$5\sim 7$,	$7\sim 8$,	$7\sim 5$,	$5\sim 1$,	…

The equivalence classes are given by $$\begin{array}{rcl}\left[1\right]=\left[5\right]=\left[7\right]=\left[8\right]& =& \left\{1,5,7,8\right\}\text{,}\\ \left[2\right]=\left[3\right]& =& \left\{2,3\right\}\text{,}\\ \left[6\right]& =& \left\{6\right\}\text{, and}\\ \left[4\right]=\left[9\right]=\left[10\right]& =& \left\{4,9,10\right\}\text{,}\end{array}$$ and the sets $$S_1=\left\{1,5,7,8\right\}\text{, }{S}_2=\left\{2,3\right\}\text{, }{S}_3=\left\{6\right\}\text{, and }{S}_4=\left\{4,9,10\right\}$$ form a partition of $S$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)