Total relation

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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."[1]

Algebraic characterization[edit]

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 [2]: 54 
  • is total iff [2]: 54 
  • If is total, then The converse is true if [note 1]
  • If is total, then The converse is true if [note 2][2]: 63 
  • If is total, then The converse is true if [2]: 54 [3]
  • More generally, if is total, then for any set and any The converse is true if [note 3][2]: 57 

Notes[edit]

  1. ^ If then will be not total.
  2. ^ Observe and apply the previous bullet.
  3. ^ Take and appeal to the previous bullet.

References[edit]

  1. ^ Functions from Carnegie Mellon University
  2. ^ a b c d e Schmidt, Gunther; Ströhlein, Thomas (6 December 2012). Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer Science & Business Media. ISBN 978-3-642-77968-8.
  3. ^ Gunther Schmidt (2011). Relational Mathematics. Cambridge University Press. doi:10.1017/CBO9780511778810. ISBN 9780511778810. Definition 5.8, page 57.