Triadic relation
Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...

 is formally defined as a set of pairs, i.e. a subset of the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...

  of some sets A and B, so a ternary relation is a set of triples, forming a subset of the Cartesian product of three sets A, B and C.

An example of a ternary relation in elementary geometry is the collinearity of points.

Binary functions

A function in two variables, taking values in two sets A and B, respectively, is formally a function that associates to every pair (a,b) in an element ƒ(ab) in C. Therefore its graph consists of pairs of the form . 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 , for all a in A and b in B.

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 = , by stipulating that 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 = { } represents the hours on a clock face
Clock face
A clock face is the part of an analog clock that displays the time through the use of a fixed numbered dial or dials and moving hands. In its most basic form, recognized universally throughout the world, the dial is numbered 1–12 indicating the hours in a 12-hour cycle, and a short hour hand...

, then holds and does not hold.
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