Total relation
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In mathematics, a binary relation R ⊆ A×B is total (or left total) if the source set A equals the domain {x : there is a y with xRy }. Conversely, R is called right total if B equals the range {y : there is an x with xRy }.
When f: A → B is a function, the domain of f is all of A, 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 A, 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]
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, 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