# 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."

## Algebraic characterization

Total relations can be characterized algebraically by equalities and inequalities involving compositions of relations. To this end, let $X,Y$ be two sets, and let $R\subseteq X\times Y.$ For any two sets $A,B,$ let $L_{A,B}=A\times B$ be the universal relation between $A$ and $B,$ and let $I_{A}=\{(a,a):a\in A\}$ be the identity relation on $A.$ We use the notation $R^{\top }$ for the converse relation of $R.$ • $R$ is total iff for any set $W$ and any $S\subseteq W\times X,$ $S\neq \emptyset$ implies $SR\neq \emptyset .$ : 54
• $R$ is total iff $I_{X}\subseteq RR^{\top }.$ : 54
• If $R$ is total, then $L_{X,Y}=RL_{Y,Y}.$ The converse is true if $Y\neq \emptyset .$ [note 1]
• If $R$ is total, then ${\overline {RL_{Y,Y}}}=\emptyset .$ The converse is true if $Y\neq \emptyset .$ [note 2]: 63
• If $R$ is total, then ${\overline {R}}\subseteq R{\overline {I_{Y}}}.$ The converse is true if $Y\neq \emptyset .$ : 54 
• More generally, if $R$ is total, then for any set $Z$ and any $S\subseteq Y\times Z,$ ${\overline {RS}}\subseteq R{\overline {S}}.$ The converse is true if $Y\neq \emptyset .$ [note 3]: 57