mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a ternary relation or triadic relation is a

finitary relation In mathematics, a finitary relation over a sequence of sets is a subset of the Cartesian product ; that is, it is a set of ''n''-tuples , each being a sequence of elements ''x'i'' in the corresponding ''X'i''. Typically, the relation descri ...

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 In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...

is formally defined as a set of ''pairs'', i.e. a subset of the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

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 In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...

. 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.

Examples

Binary functions

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

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 , 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 A clock face is the part of an analog clock (or watch) that displays time through the use of a flat dial (measurement), dial with reference marks, and revolving pointers turning on concentric shafts at the center, called hands. In its most basi ...

, then holds and does not hold.

Betweenness relations

Ternary equivalence relation

Congruence relation

The ordinary congruence of arithmetics :

a \equiv b \pmod

which holds for three integers ''a'', ''b'', and ''m'' if and only if ''m'' divides , formally may be considered as a ternary relation. However, usually, this instead is considered as a family of

s 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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...

; while the combined ternary relation in general is not studied as one relation.

Typing relation

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.

Schröder rules

Given

homogeneous relation In mathematics, a homogeneous relation (also called endorelation) on a set ''X'' is a binary relation between ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...

s ''A'', ''B'', and ''C'' on a set, a ternary relation can be defined using

composition of relations In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplica ...

''AB'' and

inclusion Inclusion or Include may refer to: Sociology * Social inclusion, action taken to support people of different backgrounds sharing life together. ** Inclusion (disability rights), promotion of people with disabilities sharing various aspects of lif ...

. Within the calculus of relations each relation ''A'' has a

converse relation In mathematics, the converse of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms ...

''A''^T and a complement relation . Using these

involution Involution may refer to: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiati ...

Augustus De Morgan Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He is best known for De Morgan's laws, relating logical conjunction, disjunction, and negation, and for coining the term "mathematical induction", the ...

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.