In
mathematics, a 'binary relation' (or a 'dyadic' or '2-place relation') is an arbitrary association of elements within a
set or with elements of another set.
An example is the "divides" relation between the set of
prime numbers 'P' and the set of
integers 'Z', in which every prime ''p'' is associated with every integer ''z'' that is a
multiple of ''p'', but no other. In this relation, for instance, the prime 2 is associated with numbers that include -4, 0, 6, 10, but not 1 or 9; and the prime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.
Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in
arithmetic, "is congruent to" in
geometry, "is adjacent to" in
graph theory, and many more. The all-important concept of
function is defined as a special kind of binary relation. Binary relations are also heavily used in computer science, especially within the
relational model for
databases.
A binary relation is a special case of a
''k''-ary relation, that is, a set of ''k''-tuples where the ''j''
th component of each ''k''-tuple is taken from the ''j''
th domain ''X''
''j'' of the relation. A ''k''-ary relation among elements of a single set is said to be 'homogeneous'.
In some systems of
axiomatic set theory, relations are extended to
classes, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in
set theory, without running into logical inconsistencies such as
Russell's paradox.
Formal definition
A binary relation ''R'' is usually defined as an ordered triple (''X'', ''Y'', ''G'') where ''X'' and ''Y'' are arbitrary sets (or classes), and ''G'' is a
subset of the
Cartesian product ''X'' × ''Y''. The sets ''X'' and ''Y'' are called the '
domain' and '
codomain', respectively, of the relation, and ''G'' is called its
graph.
The statement (''x'',''y'') ∈ ''R'' is read "''x'' 'is' ''R'''-related to' ''y''", and is denoted by ''xRy'' or ''R''(''x'',''y''). The latter notation corresponds to viewing ''R'' as the
characteristic function of the set of pairs ''G''.
The order of the elements in each pair of ''G'' is important: if ''a'' ≠ ''b'', then ''aRb'' and ''bRa'' can be true or false, independently of each other.
Is a relation more than its graph?
According to the definition above, two relations with the same graph may be different, if they differ in the sets ''X'' and ''Y''. For example, if ''G'' = {(1,2),(1,3),(2,7)}, then ('Z','Z', ''G''), ('R', 'N', ''G''), and ('N', 'R', ''G'') are three distinct relations.
Some mathematicians do not consider the sets ''X'' and ''Y'' to be part of the relation, and therefore define a binary relation as being a subset of ''X''×''Y'', that is, just the graph ''G''. According to this view, the set of pairs {(1,2),(1,3),(2,7)} is a relation from any set that contains {1,2} to any set that contains {2,3,7}.
Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation, and the definitions of concepts like
restrictions,
composition,
inverse relation, and so on. The choice between the two definitions usually matters only in very formal contexts, like
category theory.
Example
Example: Suppose there are four objects: {ball, car, doll, gun} and four persons: {John, Mary, So, Venus}. Suppose that John owns the ball, Mary owns the doll, and Venus owns the car. No one owns the gun and So owns nothing. Then the binary relation "is owned by" is given as
: ''R''=({ball, car, doll, gun}, {John, Mary, So, Venus}, {(ball, John), (doll, Mary), (car, Venus)}).
Thus the first element of R is the set of objects, the second is the set of people, and the last element is a set of ordered pairs of the form ( object, owner ).
The pair (ball, John), denoted by
ball''R''
John means that the ball is owned by John.
Two different relations could have the same graph. For example: the relation
: ({ball, car, doll, gun}, {John, Mary, Venus}, {(ball,John), (doll, Mary), (car, Venus)})
is different from the previous one as everyone is an owner. But the graphs of the two relations are the same.
Nevertheless, ''R'' is usually identified or even defined as G(''R'') and "an ordered pair (''x'', ''y'') ∈ G(''R'')" is usually denoted as "(''x'', ''y'') ∈ ''R''".
Special types of binary relations
Some important classes of binary relations ''R'' over ''X'' and ''Y'' are listed below
★ 'left-total': for all ''x'' in ''X'' there exists a ''y'' in ''Y'' such that ''xRy'' (this property, although sometimes also referred to as ''total'', is different from the definition of ''total'' in the next section).
★ 'surjective' or right-total: for all ''y'' in ''Y'' there exists an ''x'' in ''X'' such that ''xRy''.
★ 'functional' (also called right-definite): for all ''x'' in ''X'', and ''y'' and ''z'' in ''Y'' it holds that if ''xRy'' and ''xRz'' then ''y'' = ''z''.
★ 'injective': for all ''x'' and ''z'' in ''X'' and ''y'' in ''Y'' it holds that if ''xRy'' and ''zRy'' then ''x'' = ''z''.
★ 'bijective': left-total, right-total, functional, and injective.
A binary relation that is functional is called a
partial function; a binary relation that is both left-total and functional is called a
function.
Relations over a set
If ''X'' = ''Y'' then we simply say that the binary relation is over ''X''. Or it is an 'endorelation' over ''X''.
Some important classes of binary relations over a set ''X'' are:
★ '
reflexive': for all ''x'' in ''X'' it holds that ''xRx''. For example, "greater than or equal to" is a reflexive relation but "greater than" is not.
★ '
irreflexive': for all ''x'' in ''X'' it holds that 'not' ''xRx''. "Greater than" is an example of an irreflexive relation.
★ 'coreflexive': for all ''x'' and ''y'' in ''X'' it holds that if ''xRy'' then ''x'' = ''y''.
★ '
symmetric': for all ''x'' and ''y'' in ''X'' it holds that if ''xRy'' then ''yRx''. "Is a blood relative of" is a symmetric relation, because ''x'' is a blood relative of ''y'' if and only if ''y'' is a blood relative of ''x''.
★ '
antisymmetric': for all ''x'' and ''y'' in ''X'' it holds that if ''xRy'' and ''yRx'' then ''x'' = ''y''. "Greater than or equal to" is an antisymmetric relation, because if ''x''≥''y'' and ''y''≥''x'', then ''x''=''y''.
★ '
asymmetric': for all ''x'' and ''y'' in ''X'' it holds that if ''xRy'' then 'not' ''yRx''. "Greater than" is an asymmetric relation, because if ''x''>''y'' then not ''y''>''x''.
★ '
transitive': for all ''x'', ''y'' and ''z'' in ''X'' it holds that if ''xRy'' and ''yRz'' then ''xRz''. "Is an ancestor of" is a transitive relation, because if ''x'' is an ancestor of ''y'' and ''y'' is an ancestor of ''z'', then ''x'' is an ancestor of ''z''.
★ '
total' (or 'linear'): for all ''x'' and ''y'' in ''X'' it holds that ''xRy'' or ''yRx'' (or both). "Is greater than or equal to" is an example of a total relation (this definition for ''total'' is different from the one in the previous section).
★ 'trichotomous': for all ''x'' and ''y'' in ''X'' exactly one of ''xRy'', ''yRx'' or ''x'' = ''y'' holds. "Is greater than" is an example of a trichotomous relation.
★ '
Euclidean': for all ''x'', ''y'' and ''z'' in ''X'' it holds that if ''xRy'' and ''xRz'', then ''yRz''.
★ 'extendable' (or 'serial'): for all ''x'' in ''X'', there exists ''y'' in ''X'' such that ''xRy''. "Is greater than" is an extendable relation on the integers. But it is not an extendable relation on the positive integers, because there is no ''y'' in the positive integers such that 1>''y''.
★ 'set-like': for every ''x'' in ''X'', the
class of all ''y'' such that ''yRx'' is a set. (This makes sense only if we allow relations on proper classes.) The usual ordering < on the class of ordinal numbers is set-like, while its inverse <
-1 is not.
A relation which is reflexive, symmetric and transitive is called an
equivalence relation. A relation which is reflexive, antisymmetric and transitive is called a
partial order. A partial order which is total is called a
total order or a
linear order or a chain. A linear order in which every nonempty set has a
least element is called a
well-order.
A relation which is symmetric, transitive, and extendable is also reflexive.
Operations on binary relations
If ''R'' is a binary relation over ''X'' and ''Y'', then the following is a binary relation over ''Y'' and ''X'':
★ '
Inverse' or 'converse': ''R''
-1, defined as ''R''
-1 = { (y, x) | (x, y) ∈ ''R'' }. A binary relation over a set is equal to its inverse if and only if it is symmetric. See also
duality (order theory).
If ''R'' is a binary relation over ''X'', then each of the following are binary relations over ''X'':
★ 'Reflexive closure': ''R''
=, defined as ''R''
= = {(''x'', ''x'') | ''x'' ∈ ''X''} ∪ ''R'' or the smallest reflexive relation over ''X'' containing ''R''. This can be seen to be equal to the intersection of all reflexive relations containing ''R''.
★ '
Transitive closure': ''R''
+, defined as the smallest transitive relation over ''X'' containing ''R''. This can be seen to be equal to the intersection of all transitive relations containing ''R''.
★ 'Transitive-reflexive closure': ''R''
★ , defined as ''R''
★ = (''R''
+)
=.
If ''R'', ''S'' are binary relations over ''X'' and ''Y'', then each of the following are binary relations:
★ 'Union': ''R'' ∪ ''S'' ⊆ ''X'' × ''Y'', defined as ''R'' ∪ ''S'' = {(''x'', ''y'') | (''x'', ''y'') ∈ ''R'' or (''x'', ''y'') ∈ ''S''}.
★ 'Intersection': ''R'' ∩ ''S'' ⊆ ''X'' × ''Y'', defined as ''R'' ∩ ''S'' = { (''x'', ''y'') | (''x'', ''y'') ∈ ''R'' and (''x'', ''y'') ∈ ''S'' }.
If ''R'' is a binary relation over ''X'' and ''Y'', and ''S'' is a binary relation over ''Y'' and ''Z'', then the following is a binary relation over ''X'' and ''Z'': (see main article ''
composition of relations'')
★ 'Composition': ''S''
o ''R'' (also denoted ''R''
o ''S''), defined as ''S''
o ''R'' = { (''x'', ''z'') | there exists ''y'' ∈ ''Y'', such that (''x'', ''y'') ∈ ''R'' and (''y'', ''z'') ∈ ''S'' }. The order of ''R'' and ''S'' in the notation ''S''
o ''R'', used here agrees with the standard notational order for
composition of functions.
Complement
If ''R'' is a binary relation over ''X'' and ''Y'', then the following too:
★ The '
complement' ''S'' is defined as ''x'' ''S'' ''y'' iff not ''x'' ''R'' ''y''.
The complement of the inverse is the inverse of the complement.
If ''X'' = ''Y'' the complement has the following properties:
★ If a relation is symmetric, the complement is too.
★ The complement of a reflexive relation is irreflexive and vice versa.
★ The complement of a
strict weak order is a total preorder and vice versa.
The complement of the inverse has these same properties.
Restriction
The
restriction of a binary relation on a set ''X'' to a subset ''S'' is the set of all pairs (''x'', ''y'') in the relation for which ''x'' and ''y'' are in ''S''.
If a relation is
reflexive,
irreflexive,
symmetric,
antisymmetric,
asymmetric,
transitive,
total, , a
partial order,
total order,
strict weak order, (weak order), or an
equivalence relation, its restrictions are too.
However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal.
Also, the various concepts of
completeness (not to be confused with being "total") do not carry over to restrictions. For example, on the set of
real numbers a property of the relation "≤" is that every
non-empty subset ''S'' of 'R' with an
upper bound in 'R' has a
least upper bound (also called supremum) in 'R'. However, for a set of rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation "≤" to the set of rational numbers.
Sets versus classes
Certain mathematical "relations", such as "equal to", "member of", and "subset of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of
axiomatic set theory.
For example, if we try to model the general concept of "equality" as a binary relation
, we must take the domain and codomain to be the "set of all sets", which is not a set in the usual set theory. The usual work-around to this problem is to select a "large enough" set ''A'', that contains all the objects of interest, and work with the restriction
instead of
.
Similarly, the "subset of" relation
needs to be restricted to have domain and codomain ''P''(''A'') (the power set of a specific set ''A''): the resulting set relation can be denoted
. Also, the "member of" relation needs to be restricted to have domain ''A'' and codomain ''P''(''A'') to obtain a binary relation
which is a set.
Another solution to this problem is to use a set theory with proper classes, such as
NBG or
Morse-Kelley set theory, and allow the domain and codomain (and so the graph) to be
proper classes: in such a theory, equality, membership, and subset are binary relations without special comment. (A minor modification needs to be made to the concept of the ordered triple (''X'', ''Y'', ''G''), as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the function with its graph in this context.)
In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context.
The number of binary relations
The number of distinct binary relations on an ''n''-element set is 2
''n''2 :
Notes:
★ The number of irreflexive relations is the same as that of reflexive relations
★ The number of (irreflexive transitive relations) is the same as that of partial orders
★ The number of strict weak orders is the same as that of total preorders
★ The total orders are the partial orders which are also total preorders. The number of preorders which are neither a partial order nor a total preorder is therefore the number of preorders minus the number of partial orders minus the number of total preorders plus the number of total orders: 0, 0, 0, 3, and 85, respectively.
★ the number of equivalence relations is the number of
partitions, which is the
Bell number.
The binary relations can be grouped into pairs (relation, ), except that for ''n'' = 0 the relation is its own complement. The non-symmetric ones can be grouped into
quadruples (relation, complement, , inverse complement).
Examples of common binary relations
★
Order relations, including
strict orders:
★
★
Greater than
★
★ Greater than or equal to
★
★
Less than
★
★ Less than or equal to
★
★
Divides (evenly)
★
★ is a
subset of
★
Equivalence relations:
★
★
Equality
★
★ is
parallel to (for
affine spaces)
★
★ is in
bijection with
★
★
isomorphy
★
Dependency relation, a symmetric, reflexive relation.
★
Independency relation, a symmetric, irreflexive relation.
See also
★
Relation
★
Relation algebra
★
Relation construction
★
★
Relation composition
★
★
Relation reduction
★
Correspondence
★
Equivalence relation
★
Function
★
Hasse diagram
★
Incidence structure
★
Logic of relatives
★
Order theory
★
Partial order
★
Reflexive relation
★
Total order
★
Triadic relation
★
Well-order
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