# Contents

## Definition

A binary relation from a set $X$ to a set $Y$ is called functional if every element of $X$ is related to at most one element of $Y$. Notice that this is the same thing as a partial function, seen from a different point of view. A (total) function is precisely a relation that is both functional and entire.

## Properties

Like any relation, a functional relation $r$ can be viewed as a span

$\begin{array}{ccc}& & {\Delta }_{r}\\ & {↙}_{{\iota }_{r}}& & {↘}^{{\varphi }_{r}}\\ X& & & & Y\end{array}$\array { & & \Delta_r \\ & \swarrow_{\iota_r} & & \searrow^{\phi_r} \\ X & & & & Y }

Such a span is a relation iff the pairing map from the domain ${\Delta }_{r}$ to $X×Y$ is an injection, and such a relation is functional iff the inclusion map ${\iota }_{r}$ is also an injection. Such a relation is entire iff the inclusion map ${\iota }_{r}$ is a surjection.

(Total) functions can be characterized as the left adjoints in the bicategory of relations, in other words relations $r:X\to Y$ in $\mathrm{Rel}$ for which there exists $s:Y\to X$ satisfying

$r\circ s\le {1}_{Y};\phantom{\rule{2em}{0ex}}{1}_{X}\le s\circ r$r \circ s \leq 1_Y; \qquad 1_X \leq s \circ r

in which case it may be proven that $s={r}^{\mathrm{op}}$. A relation is functional if and only if $r\circ {r}^{\mathrm{op}}\le {1}_{Y}$, and is entire if and only if ${1}_{X}\le {r}^{\mathrm{op}}\circ r$.

Further to this: surjectivity of a function $r:X\to Y$ can be expressed as the condition ${1}_{Y}\le r\circ {r}^{\mathrm{op}}$, and injectivity as ${r}^{\mathrm{op}}\circ r\le {1}_{X}$.

Revised on August 24, 2012 20:06:25 by Urs Schreiber (89.204.138.87)