Sets

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

and

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

Last updates: 24 September 2009

Definitions

A set is a collection of objects which are called elements. Write $s\in S$ if $s$ is an element of a set $S$.

The empty set $\varnothing$ is the set with no elements.

A subset $T$ of a set $S$ is a set $T$ such that if $t\in T$ then $t\in S$. Write $T\subseteq S$ if $T$ is a subset of $S$.

Two sets $S$ and $T$ are equal if $S\subseteq T$ and $T\subseteq S$. Write $T=S$ if $T$ and $S$ are equal sets.

Let $S$ and $T$ be sets. The union of $S$ and $T$ is the set $S\cup T$ of all $u$ such that $u\in S$ or $u\in T$, $$S\cup T=\left\{u|u\in S\text{ or }u\in T\right\}\text{.}$$

Let $S$ and $T$ be sets. The intersection of $S$ and $T$ is the set $S\cap T$ of all $u$ such that $u\in S$ and $u\in T$, $$S\cap T=\left\{u|u\in S\text{ and }u\in T\right\}\text{.}$$

Let $S$ and $T$ be sets. The sets $S$ and $T$ are disjoint if $S\cup T=\varnothing$.

Let $S$ and $T$ be sets. The set $S$ is a proper subset of $T$ if $S\subseteq T$ and $S\ne T$. Write $S⫋T$ if $S$ is a proper subset of $T$.

The product of two sets $S$ and $T$ is the set of all ordered pairs $\left(s,t\right)$ where $s\in S$ and $t\in T$, $$S\times T=\left\{\left(s,t\right)|s\in S\text{, }t\in T\right\}\text{.}$$ More generally, given sets $S_1,\dots ,S_n$, the product $\prod _i{S}_i$ is the set of all tuples $\left(s_1,\dots ,s_n\right)$ such that $s_i\in S_i$.

The elements of a set $S$ are indexed by the elements of a set $I$ if each element of $S$ is labeled by a unique element of $I$. Write $S=\left\{s_i|i\in I\right\}$.

Notation

We will use the following notations: $$\begin{array}{rcl}\mathbb{Z}& =& \left\{\dots ,-2,-1,0,1,2,\dots \right\}\text{is the set of all integers,}\\ {\mathbb{Z}}_{\ge 0}& =& \left\{0,1,2,\dots \right\}\text{is the set of all nonnegative integers,}\\ {\mathbb{Z}}_{>0}& =& \left\{1,2,3,\dots \right\}\text{is the set of all positive integers,}\\ \left[1,n\right]& =& \left\{1,2,\dots ,n\right\},\text{for each }n\in {\mathbb{Z}}_{>0}\text{,}\\ ℝ& & \text{is the set of all real numbers, and}\\ \mathbb{Q}& & \text{is the set of all complex numbers.}\end{array}$$.

Example

Let $S$, $T$, $U$ and $V$ be the sets $S=\left\{1,2\right\}$, $U=\left\{1,2\right\}$, $T=\left\{1,2,3\right\}$ and $V=\left\{2,3\right\}$. Then

	(a)	$S\subseteq U\subseteq T,$
	(b)	$U⊈V$,
	(c)	$U\cup V=T$,
	(d)	$U\cap V=\left\{2\right\}$, and
	(e)	$S\times T=\left\{\left(1,1\right),\left(1,2\right),\left(1,2\right),\left(2,1\right),\left(2,2\right),\left(2,3\right)\right\}$.

Homework

Show that the emptyset is a subset of every set.

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)