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*.

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

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*.

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 *A*^{3} = *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.

The ordinary congruence of arithmetics

a
≡
b
(
mod
m
)
which holds for three integers *a*, *b*, and *m* if and only if *m* divides *a* − *b*, formally may be considered as a ternary relation. However, usually, this instead is considered as a family of binary relations between the *a* and the *b*, indexed by the modulus *m*. For each fixed *m*, indeed this binary relation has some natural properties, like being an equivalence relation; while the combined ternary relation in general is not studied as one relation.

A *typing relation*
Γ
⊢
e
:
σ
indicates that
e
is a term of type
σ
in context
Γ
, and is thus a ternary relation between contexts, terms and types.