This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2009)

In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place.

Just as a binary relation is formally defined as a set of pairs, i.e. a subset of the Cartesian product A × B of some sets A and B, so a ternary relation is a set of triples, forming a subset of the Cartesian product A × B × C of three sets A, B and C.

[edit] Example – graph of a binary function

A function ƒ: A × B → C in two variables, taking values in two sets A and B, respectively, is formally a function that associates to every pair (a,b) in A × B an element ƒ(a, b) in C. Therefore its graph consists of pairs of the form ((a, b), ƒ(a, b)). Such pairs in which the first element is itself a pair are often identified with triples. This makes the graph of ƒ a ternary relation between A, B and C, consisting of all triples (a, b, ƒ(a, b)), for all a in A and b in B.

[edit] Example – cyclic orders

Given any set A whose elements are arranged on a circle, one can define a ternary relation R on A, i.e. a subset of A³ = A × A × A, by stipulating that R(a, b, c) holds if and only if the elements a, b and c are pairwise different and when going from a to c in a clockwise direction one passes through b. For example if A = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 } represents the hours on a clock face, then R(8, 12, 4) holds and R(12, 8, 4) does not hold.