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In mathematics, a ternary relation or triadic relation is a
finitary relation In mathematics, a finitary relation over sets is a subset of the Cartesian product ; that is, it is a set of ''n''-tuples consisting of elements ''x'i'' in ''X'i''. Typically, the relation describes a possible connection between the eleme ...
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 elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...
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 ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
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 points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
. 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 Incident may refer to: * A property of a graph in graph theory * ''Incident'' (film), a 1948 film noir * Incident (festival), a cultural festival of The National Institute of Technology in Surathkal, Karnataka, India * Incident (Scientology), a ...
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 ''A''3 = , 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 A clock face is the part of an analog clock (or watch) that displays time through the use of a flat dial with reference marks, and revolving pointers turning on concentric shafts at the center, called hands. In its most basic, globally recogni ...
, 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 ''a'' − ''b'', formally may be considered as a ternary relation. However, usually, this instead is considered as a family of
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...
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. Each equivalence relatio ...
; while the combined ternary relation in general is not studied as one relation.


Typing relation

A ''typing relation'' \Gamma\vdash e\!:\!\sigma indicates that e is a term of type \sigma in context \Gamma, and is thus a ternary relation between contexts, terms and types.


Schröder rules

Given
homogeneous relation In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''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 (A,\ B,\ C) 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 ''AB'' ⊆ ''C''. Within the
calculus of relations In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for ...
each relation ''A'' has a
converse relation In mathematics, the converse relation, or transpose, 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&n ...
''A''T and a complement relation \bar . Using these
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
s, Augustus De Morgan and Ernst Schröder showed that (A,\ B,\ C)is equivalent to (\bar, B^T, \bar) and also equivalent to (A^T,\ \bar,\ \bar). The mutual equivalences of these forms, constructed from the ternary are called the Schröder rules.
Gunther Schmidt Gunther Schmidt (born 1939, Rüdersdorf) is a German mathematician who works also in informatics. Life Schmidt began studying Mathematics in 1957 at Göttingen University. His academic teachers were in particular Kurt Reidemeister, Wilhelm K ...
& Thomas Ströhlein (1993) ''Relations and Graphs'', pages 15–19,
Springer books Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...


References


Further reading

* * * * * * {{DEFAULTSORT:Ternary Relation Mathematical relations ru:Тернарное отношение