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."Functions from Carnegie Mellon University
Algebraic characterization Total relations can be characterized algebraically by equalities and inequalities involving compositions of relations. To this end, let be two sets, and let For any two sets let be the universal relation between and and let be the identity relation on We use the notation for the converse relation of
is total iff for any set and any implies is total iff If is total, then The converse is true if If then will be not total. If is total, then The converse is true if Observe and apply the previous bullet. If is total, then The converse is true if Definition 5.8, page 57. More generally, if is total, then for any set and any The converse is true if Take and appeal to the previous bullet.
See also Serial relation — a total homogeneous relation
Notes
References
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, Gunther Schmidt & Thomas Strohlein (2012) Gunther Schmidt (2011)