# Total relation

In mathematics, a binary relation RX×Y between two sets X and Y is total (or left total) if the source set X equals the domain {x : there is a y with xRy }. Conversely, R is called right total if Y equals the range {y : there is an x with xRy }.

When f: XY is a function, the domain of f is all of X, hence f is a total relation. On the other hand, if f is a partial function, then the domain may be a proper subset of X, in which case f is not a total relation.

"A binary relation is said to be total with respect to a universe of discourse just in case everything in that universe of discourse stands in that relation to something else."[1]

## Algebraic characterization

Total relations can be characterized algebraically by equalities and inequalities involving compositions of relations. To this end, let ${\displaystyle X,Y}$ be two sets, and let ${\displaystyle R\subseteq X\times Y.}$ For any two sets ${\displaystyle A,B,}$ let ${\displaystyle L_{A,B}=A\times B}$ be the universal relation between ${\displaystyle A}$ and ${\displaystyle B,}$ and let ${\displaystyle I_{A}=\{(a,a):a\in A\}}$ be the identity relation on ${\displaystyle A.}$ We use the notation ${\displaystyle R^{\top }}$ for the converse relation of ${\displaystyle R.}$

• ${\displaystyle R}$ is total iff for any set ${\displaystyle W}$ and any ${\displaystyle S\subseteq W\times X,}$ ${\displaystyle S\neq \emptyset }$ implies ${\displaystyle SR\neq \emptyset .}$[2]: 54
• ${\displaystyle R}$ is total iff ${\displaystyle I_{X}\subseteq RR^{\top }.}$[2]: 54
• If ${\displaystyle R}$ is total, then ${\displaystyle L_{X,Y}=RL_{Y,Y}.}$ The converse is true if ${\displaystyle Y\neq \emptyset .}$[note 1]
• If ${\displaystyle R}$ is total, then ${\displaystyle {\overline {RL_{Y,Y}}}=\emptyset .}$ The converse is true if ${\displaystyle Y\neq \emptyset .}$[note 2][2]: 63
• If ${\displaystyle R}$ is total, then ${\displaystyle {\overline {R}}\subseteq R{\overline {I_{Y}}}.}$ The converse is true if ${\displaystyle Y\neq \emptyset .}$[2]: 54 [3]
• More generally, if ${\displaystyle R}$ is total, then for any set ${\displaystyle Z}$ and any ${\displaystyle S\subseteq Y\times Z,}$ ${\displaystyle {\overline {RS}}\subseteq R{\overline {S}}.}$ The converse is true if ${\displaystyle Y\neq \emptyset .}$[note 3][2]: 57

## Notes

1. ^ If ${\displaystyle Y=\emptyset \neq X,}$ then ${\displaystyle R}$ will be not total.
2. ^ Observe ${\displaystyle {\overline {RL_{Y,Y}}}=\emptyset \Leftrightarrow RL_{Y,Y}=L_{X,Y},}$ and apply the previous bullet.
3. ^ Take ${\displaystyle Z=Y,S=I_{Y}}$ and appeal to the previous bullet.

## References

1. ^
2. Schmidt, Gunther; Ströhlein, Thomas (6 December 2012). Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer Science & Business Media. ISBN 978-3-642-77968-8.
3. ^ Gunther Schmidt (2011). Relational Mathematics. Cambridge University Press. doi:10.1017/CBO9780511778810. ISBN 9780511778810. Definition 5.8, page 57.