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 elemen ...
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
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 o ...
. 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 ''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, 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 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.
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 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
can be defined using
composition of relations ''AB'' and
inclusion
Inclusion or Include may refer to:
Sociology
* Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society.
** Inclusion (disability rights), promotion of people with disabiliti ...
''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
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
is equivalent to
and also equivalent to
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 Kl ...
& Thomas Ströhlein (1993) ''Relations and Graphs'', pages 15–19, Springer books
References
Further reading
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{{DEFAULTSORT:Ternary Relation
Mathematical relations
ru:Тернарное отношение