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 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 can be given on triples of points, where a triple is in the relation if the three points are collinear. Another geometric example can be obtained by considering triples consisting of two points and a line, where a triple is in the ternary relation if the two points determine (are incident with) the line.
A function in two variables, mapping two values from sets A and B, respectively, to a value in C associates to every pair (a,b) in an element f(a, b) 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 f a ternary relation between A, B and C, consisting of all triples, satisfying,, and
See main article: article and Cyclic order.
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, 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 represents the hours on a clock face, then holds and does not hold.
See main article: article and Betweenness relation.
See main article: article and Ternary equivalence relation.
See main article: article.
The ordinary congruence of arithmetics
a\equivb\pmod{m}
See main article: article.
A typing relation indicates that e is a term of type σ in context Γ, and is thus a ternary relation between contexts, terms and types.
Given homogeneous relations A, B, and C on a set, a ternary relation can be defined using composition of relations AB and inclusion . Within the calculus of relations each relation A has a converse relation AT and a complement relation . Using these involutions, Augustus De Morgan and Ernst Schröder showed that is equivalent to and also equivalent to . The mutual equivalences of these forms, constructed from the ternary relation are called the Schröder rules.