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
:
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; while the combined ternary relation in general is not studied as one relation.
Typing relation
A ''typing relation''
indicates that
is a term of type
in context
, and is thus a ternary relation between contexts, terms and types.
Schröder rules
Given
homogeneous relations ''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 ''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
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 are called the
Schröder rules.
[ Gunther Schmidt & 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
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{{DEFAULTSORT:Ternary Relation
Mathematical relations
ru:Тернарное отношение