Total relation

In mathematics, a binary relation R ⊆ X×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: X → Y 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]

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}

• Serial relation — a total homogeneous relation

• Gunther Schmidt & Michael Winter (2018) Relational Topology
• C. Brink, W. Kahl, and G. Schmidt (1997) Relational Methods in Computer Science, Advances in Computer Science, page 5, ISBN 3-211-82971-7
• Gunther Schmidt & Thomas Strohlein (2012)[1987] Relations and Graphs, p. 54, at Google Books
• Gunther Schmidt (2011) Relational Mathematics, p. 57, at Google Books