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."

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 For any two sets A,B, let be the universal relation between A and B, and let be the identity relation on A. We use the notation R^\top for the converse relation of R.