Functions

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

Let $S$ and $T$ be sets. A map or function $f:S\to T$ is given by associating to each element $s\in S$ a unique element $f\left(s\right)\in T$. Write $$\begin{array}{rcl}f:S& ⟶& T\\ s& ⟼& f\left(s\right)\text{.}\end{array}$$

Often in mathematics one will try to define a function without being explicitly sure if what has been defined is really a function. In order to check that a function is well defined one must check that

	(a)	If $s\in S$ then $f\left(s\right)\in T$, and
	(b)	If $s_1=s_2$ then $f\left(s_1\right)=f\left(s_2\right)$.

Let $S$ and $T$ be sets. Two functions $f:S\to T$ and $g:S\to T$ are equal if $f$ and $g$ satisfy

if $s\in S$ then $f\left(s\right)=g\left(s\right)$.

Write $f=g$ if $f$ and $g$ are equal functions.

Let $S$ and $T$ be sets and let $f:S\to T$ be a function. Let $R\subseteq S$. The restriction of $f$ to $R$ is the function ${f|}_R$ defined by $$\begin{array}{rcl}{f|}_R:R& ⟶& T\\ r& ⟼& f\left(r\right)\text{.}\end{array}$$

A map $f:S\to T$ is injective if it satisfies the following condition:

If $s_1\text{, }{s}_2\in S$ and $f\left(s_1\right)=f\left(s_2\right)$ then $s_1=s_2$.

A map $f:S\to T$ is surjective or onto if $f$ satisfies

If $t\in T$ then there exists $s\in S$ such that $f\left(s\right)=t$.

A map is bijective if it is both injective and surjective.

Examples

It is useful to visualise a function $f:S\to T$ as a graph with edges $\left(s,f\left(s\right)\right)$ connecting elements $s\in S$ and $f\left(s\right)\in T$. With this idea in mind we have the following examples.

(a)  a function	(b)  not a function	(c)  not a function
(d)  an injective function	(e)  a surjective function	(f)  a bijective function

In these pictures view the elements of the left column as elements of the set $S$ and the elements of the right column as the elements of a set $T$. In order to be a function the graph must have exactly one edge adjacent to each point in $S$. A function is injective if there is at most one edge adjacent to each point in $T$. A function is surjective if there is at least one edge adjacent to each point in $T$.

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)