relation

In set theory, relations are often used to model more complex concepts. A two-place predicate is a relation. A function can be defined as a relation such that no different pairs share the same left element. (The function maps every left element of a pair onto the right element of the pair.)

Mathematical reasoning about relations often uses certain properties that relations can possess. For example, it is often of interest when a relation is

• reflexive (every element relates to itself)
• symmetric (whenever the relation relates x to y, it also relates y to x)
• transitive (whenever the relation relates x to y and y to z, it also relates x to z)
• equivalence relations (i.e. reflexive, symmetric, and transitive)
• antisymmetric (whenever the relation relates x to y, it does not relate y to x)
• antireflexive (no element is related to itself)
• functional (the relation is a function)

There are also some well-known operations that build relations out of relations. For example,

• the converse of a relation is the relation obtained by swapping the left and the right in every pair
• the join of two relations R1 and R2 is the relation that relates all x and z whenever R1 relates x to some y and R2 relates that y to z
• the union of two relations relates all pairs in either
• the intersection of two relation relates all pairs in both
• the difference of two relations relates all pairs in the first that are not on the second
• the closure of a relation w.r.t. some property is the smallest relation including it that possesses the property; for example, the symmetric closure of a relation is its union with its converse

For example, consider relationships by birth:

• is (natural) child of: antireflexive, antisymmetric
• is parent of: the converse of the previous
• is ancestor of: the transitive closure of the previous
• has the same parents as: an equivalence relation
• has the same children as: reflexive, symmetric, but not transitive
• is grandparent of: the join of is parent of with itself

A (binary) relation R from a set X to a set Y is a subset of the cartesian product X × Y. Two elements x, y are said to be related if the pair (x, y) is in R. As a shorthand, we can write x R y. This is the sense the word that is used in the relational database model. A function is a special case of a relation, namely one where every x in X is related to exactly one y in Y.

A particulary important case is when X and Y is the same set. Then the relation is said to be on X. (Note that a finite set with a relation defined on it is an ordered graph, except some definitions of graphs allow more than one edge between a given pair of nodes). A relation R on a set X might be:

• reflexive: x R x for all x in X.
• irreflexive: not x R x for any x in X.
• transitive: if x R y and y R z then x R z.
• symmetric: if x R y then y R x.
• antisymmetric: x R y and y R x only if x = y.
• total: either x R y or y R x for all x,y.

(Note that symmetry and antisymmetry are not mutually exclusive: e.g. the relation {(1,1), (2,2), (3,3)} on the set {1,2,3} has both properties.)

These properties can be used to classify relations: a relation that is reflexive, transitive and symmetric is called an equivalence relation, while one that is reflexive, transitive and antisymmetric is a partial order. For example, "x ≤ y" and "x divides y" are partial orders on the integers, while "x = y" and "x = y (modulo N)" are equivalence relations.

One can also view an relation f from X to Y as a function f:X→℘(Y) from X to the power set of Y: it maps each element x in X to a subset of Y (namely the set of elements that are related to x). This is sometimes written f:X→→Y.

Re*la"tion (r?-l?"sh?n), n. [F. relation, L. relatio. See Relate.]

1.

The act of relating or telling; also, that which is related; recital; account; narration; narrative; as, the relation of historical events.

oet's relation doth well figure them. Bacon.

2.

The state of being related or of referring; what is apprehended as appertaining to a being or quality, by considering it in its bearing upon something else; relative quality or condition; the being such and such with regard or respect to some other thing; connection; as, the relation of experience to knowledge; the relation of master to servant.

Any sort of connection which is perceived or imagined between two or more things, or any comparison which is made by the mind, is a relation. I. Taylor.

3.

Reference; respect; regard.

I have been importuned to make some observations on this art in relation to its agreement with poetry. Dryden.

4.

Connection by consanguinity or affinity; kinship; relationship; as, the relation of parents and children.

Relations dear, and all the charities Of father, son, and brother, first were known. Milton.

5.

A person connected by cosanguinity or affinity; a relative; a kinsman or kinswoman.

For me . . . my relation does not care a rush. Ld. Lytton.

6. Law (a)

The carrying back, and giving effect or operation to, an act or proceeding frrom some previous date or time, by a sort of fiction, as if it had happened or begun at that time. In such case the act is said to take effect by relation.

The act of a relator at whose instance a suit is begun.

Wharton. Burrill.

Syn. -- Recital; rehearsal; narration; account; narrative; tale; detail; description; kindred; kinship; consanguinity; affinity; kinsman; kinswoman.

© Webster 1913.