Relation (mathematics)

Given a set {{math|X}}, a relation {{math|R}} over {{math|X}} is a set of ordered pairs of elements from {{math|X}}, formally: {{math|1=R ⊆ {{mset| (x,y) {{!}} x, y ∈ X }}}}.{{sfn|Codd|1970|ps=}}{{sfn|Enderton|1977|loc=Ch 3. p. 40|ps=}}

The statement {{math|(x,y) ∈ R}} reads "{{math|x}} is {{math|R}}-related to {{math|y}}" and is written in infix notation as {{math|xRy}}. The order of the elements is important; if {{math|x ≠ y}} then {{math|yRx}} can be true or false independently of {{math|xRy}}. For example, {{math|3}} divides {{math|9}}, but {{math|9}} does not divide {{math|3}}.

A relation {{math|R}} on a finite set {{math|X}} may be represented as:

• Directed graph: Each member of {{math|X}} corresponds to a vertex; a directed edge from {{math|x}} to {{math|y}} exists if and only if {{math|(x,y) ∈ R}}.
• Boolean matrix: The members of {{math|X}} are arranged in some fixed sequence {{math|x₁}}, ..., {{math|x_n}}; the matrix has dimensions {{math|n × n}}, with the element in line {{math|i}}, column {{math|j}}, being {{text|13px}}, if {{math|(x_i,x_j) ∈ R}}, and {{text|13px}}, otherwise.
• 2D-plot: As a generalization of a Boolean matrix, a relation on the –infinite– set {{math|R}} of real numbers can be represented as a two-dimensional geometric figure: using Cartesian coordinates, draw a point at {{math|(x,y)}} whenever {{math|(x,y) ∈ R}}.

A transitive{{efn|see below}} relation {{math|R}} on a finite set {{math|X}} may be also represented as

• Hasse diagram: Each member of {{math|X}} corresponds to a vertex; directed edges are drawn such that a directed path from {{math|x}} to {{math|y}} exists if and only if {{math|(x,y) ∈ R}}. Compared to a directed-graph representation, a Hasse diagram needs fewer edges, leading to a less tangled image. Since the relation "a directed path exists from {{math|x}} to {{math|y}}" is transitive, only transitive relations can be represented in Hasse diagrams. Usually the diagram is laid out such that all edges point in an upward direction, and the arrows are omitted.

For example, on the set of all divisors of {{math|12}}, define the relation {{math|R_div}} by

: {{math|x R_div y}} if {{math|x}} is a divisor of {{math|y}} and {{math|x ≠ y}}.

Formally, {{math|1=X = {{mset| 1, 2, 3, 4, 6, 12 }}}} and {{math|1=R_div = {{mset| (1,2), (1,3), (1,4), (1,6), (1,12), (2,4), (2,6), (2,12), (3,6), (3,12), (4,12), (6,12) }}}}.

The representation of {{math|R_div}} as a Boolean matrix is shown in the middle table; the representation both as a Hasse diagram and as a directed graph is shown in the left picture.

The following are equivalent:

• {{math|x R_div y}} is true.
• {{math|(x,y) ∈ R_div}}.
• A path from {{math|x}} to {{math|y}} exists in the Hasse diagram representing {{math|R_div}}.
• An edge from {{math|x}} to {{math|y}} exists in the directed graph representing {{math|R_div}}.
• In the Boolean matrix representing {{math|R_div}}, the element in line {{math|x}}, column {{math|y}} is "{{text|13px}}".

As another example, define the relation {{math|R_el}} on {{math|R}} by

: {{math|x R_el y}} if {{math|1=x² + xy + y² = 1}}.

The representation of {{math|R_el}} as a 2D-plot obtains an ellipse, see right picture. Since {{math|R}} is not finite, neither a directed graph, nor a finite Boolean matrix, nor a Hasse diagram can be used to depict {{math|R_el}}.

Some important properties that a relation {{mvar|R}} over a set {{mvar|X}} may have are:

; {{em|Reflexive}}: for all {{math|x ∈ X}}, {{math|xRx}}. For example, {{math|≥}} is a reflexive relation but {{math|>}} is not.

; {{em|Irreflexive}} (or {{em|strict}}): for all {{math|x ∈ X}}, not {{math|xRx}}. For example, {{math|>}} is an irreflexive relation, but {{math|≥}} is not.

The previous 2 alternatives are not exhaustive; e.g., the red relation {{math|1=y = x²}} given in the diagram below is neither irreflexive, nor reflexive, since it contains the pair {{math|(0,0)}}, but not {{math|(2,2)}}, respectively.

; {{em|Symmetric}}: for all {{math|x, y ∈ X}}, if {{math|xRy}} then {{math|yRx}}. For example, "is a blood relative of" is a symmetric relation, because {{mvar|x}} is a blood relative of {{mvar|y}} if and only if {{mvar|y}} is a blood relative of {{mvar|x}}.

; {{em|Antisymmetric}}: for all {{math|x, y ∈ X}}, if {{math|xRy}} and {{math|yRx}} then {{math|1=x = y}}. For example, {{math|≥}} is an antisymmetric relation; so is {{math|>}}, but vacuously (the condition in the definition is always false).{{sfn|Smith|Eggen|St. Andre|2006|p=160|ps=}}

; {{em|Asymmetric}}: for all {{math|x, y ∈ X}}, if {{math|xRy}} then not {{math|yRx}}. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.{{sfn|Nievergelt|2002|p=[https://books.google.com/books?id=_H_nJdagqL8C&pg=PA158 158]|ps=}} For example, {{math|>}} is an asymmetric relation, but {{math|≥}} is not.

Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation {{math|xRy}} defined by {{math|x > 2}} is neither symmetric (e.g. {{math|5R1}}, but not {{math|1R5}}) nor antisymmetric (e.g. {{math|6R4}}, but also {{math|4R6}}), let alone asymmetric.

; {{em|Transitive}}: for all {{math|x, y, z ∈ X}}, if {{math|xRy}} and {{math|yRz}} then {{math|xRz}}. A transitive relation is irreflexive if and only if it is asymmetric.{{sfn|Flaška|Ježek|Kepka|Kortelainen|2007|loc=p.1 Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".|ps=}} For example, "is ancestor of" is a transitive relation, while "is parent of" is not.

; {{em|Connected}}: for all {{math|x, y ∈ X}}, if {{math|1=x ≠ y}} then {{math|xRy}} or {{math|yRx}}. For example, on the natural numbers, {{math|<}} is connected, while "is a divisor of{{-"}} is not (e.g. neither {{math|5R7}} nor {{math|7R5}}).

; {{em|Strongly connected}}: for all {{math|x, y ∈ X}}, {{math|xRy}} or {{math|yRx}}. For example, on the natural numbers, {{math|≤}} is strongly connected, but {{math|<}} is not. A relation is strongly connected if, and only if, it is connected and reflexive.

File:The four types of binary relations.pngs: one-to-one (in green), one-to-many (in blue), many-to-one (in red), many-to-many (in black). 2D-plot representation is used.]]

Uniqueness properties:

; Injective{{efn|name="heterogeneous"|These properties also generalize to heterogeneous relations.}} (also called left-unique) : For all {{math|x, y, z ∈ X}}, if {{math|xRy}} and {{math|zRy}} then {{math|1=x = z}}. For example, the green and blue relations in the diagram are injective, but the red one is not (as it relates both {{math|−1}} and {{math|1}} to {{math|1}}), nor is the black one (as it relates both {{math|−1}} and {{math|1}} to {{math|0}}).

; FunctionalVan Gasteren 1990, p. 45.{{Cite web|title=Functional relation - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Functional_relation|access-date=2024-06-13|website=encyclopediaofmath.org}}{{Cite web|title=functional relation in nLab|url=https://ncatlab.org/nlab/show/functional+relation|access-date=2024-06-13|website=ncatlab.org}}{{efn|name="heterogeneous"}} (also called right-unique,{{harvnb|Kilp|Knauer|Mikhalev|2000|p=3}}. The same four definitions appear in the following: {{harvnb|Pahl|Damrath|2001|p=506}}, {{harvnb|Best|1996|pp=19–21}}, {{harvnb|Riemann|1999|pp=21–22}} right-definite{{sfn|Mäs|2007|ps=}} or univalent{{sfn|Schmidt|2010|loc=Chapt. 5|ps=}}) : For all {{math|x, y, z ∈ X}}, if {{math|xRy}} and {{math|xRz}} then {{math|1=y = z}}. Such a relation is called a {{em|partial function}}. For example, the red and green relations in the diagram are functional, but the blue one is not (as it relates {{math|1}} to both {{math|−1}} and {{math|1}}), nor is the black one (as it relates 0 to both −1 and 1).

Totality properties:

; {{em|Serial}}{{efn|name="heterogeneous"}} (also called {{em|total}} or {{em|left-total}}): For all {{math|x ∈ X}}, there exists some {{math|y ∈ X}} such that {{math|xRy}}. Such a relation is called a multivalued function. For example, the red and green relations in the diagram are total, but the blue one is not (as it does not relate {{math|−1}} to any real number), nor is the black one (as it does not relate {{math|2}} to any real number). As another example, {{math|>}} is a serial relation over the integers. But it is not a serial relation over the positive integers, because there is no {{mvar|y}} in the positive integers such that {{math|1 > y}}.{{sfn|Yao|Wong|1995|ps=}} However, {{math|<}} is a serial relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is serial: for a given {{mvar|x}}, choose {{math|1=y = x}}.

; Surjective{{efn|name="heterogeneous"}} (also called right-total or onto): For all {{math|y ∈ Y}}, there exists an {{math|x ∈ X}} such that {{math|xRy}}. For example, the green and blue relations in the diagram are surjective, but the red one is not (as it does not relate any real number to {{math|−1}}), nor is the black one (as it does not relate any real number to {{math|2}}).

:

Relations that satisfy certain combinations of the above properties are particularly useful, and thus have received names by their own.

; {{em|Equivalence relation}}: A relation that is reflexive, symmetric, and transitive. It is also a relation that is symmetric, transitive, and serial, since these properties imply reflexivity.

Orderings:

; {{em|Partial order}}: A relation that is reflexive, antisymmetric, and transitive.

; {{em|Strict partial order}}: A relation that is irreflexive, asymmetric, and transitive.

; {{em|Total order}}: A relation that is reflexive, antisymmetric, transitive and connected.{{sfn|Rosenstein|1982|p=4|ps=}}

; {{em|Strict total order}}: A relation that is irreflexive, asymmetric, transitive and connected.

Uniqueness properties:

; One-to-one{{efn|name="heterogeneous"}}: Injective and functional. For example, the green relation in the diagram is one-to-one, but the red, blue and black ones are not.

; One-to-many{{efn|name="heterogeneous"}}: Injective and not functional. For example, the blue relation in the diagram is one-to-many, but the red, green and black ones are not.

; Many-to-one{{efn|name="heterogeneous"}}: Functional and not injective. For example, the red relation in the diagram is many-to-one, but the green, blue and black ones are not.

; Many-to-many{{efn|name="heterogeneous"}}: Not injective nor functional. For example, the black relation in the diagram is many-to-many, but the red, green and blue ones are not.

Uniqueness and totality properties:

; A {{em|function}}{{efn|name="heterogeneous"}}: A relation that is functional and total. For example, the red and green relations in the diagram are functions, but the blue and black ones are not.

; An {{em|injection}}{{efn|name="heterogeneous"}}: A function that is injective. For example, the green relation in the diagram is an injection, but the red, blue and black ones are not.

; A {{em|surjection}}{{efn|name="heterogeneous"}}: A function that is surjective. For example, the green relation in the diagram is a surjection, but the red, blue and black ones are not.

; A {{em|bijection}}{{efn|name="heterogeneous"}}: A function that is injective and surjective. For example, the green relation in the diagram is a bijection, but the red, blue and black ones are not.

; {{em|Union}}{{efn|name="heterogeneous operation"|This operation also generalizes to heterogeneous relations.}}: If {{math|R}} and {{math|S}} are relations over {{math|X}} then {{math|1 = R ∪ S = {{mset| (x, y) {{!}} xRy or xSy }}}} is the {{em|union relation}} of {{math|R}} and {{math|S}}. The identity element of this operation is the empty relation. For example, {{math|≤}} is the union of {{math|<}} and {{math|1==}}, and {{math|≥}} is the union of {{math|>}} and {{math|1==}}.

; {{em| Intersection}}{{efn|name="heterogeneous operation"}}: If {{math|R}} and {{math|S}} are relations over {{math|X}} then {{math|1 = R ∩ S = {{mset| (x, y) {{!}} xRy and xSy }}}} is the {{em|intersection relation}} of {{math|R}} and {{math|S}}. The identity element of this operation is the universal relation. For example, "is a lower card of the same suit as" is the intersection of "is a lower card than" and "belongs to the same suit as".

; {{em| Composition}}{{efn|name="heterogeneous operation"}}: If {{math|R}} and {{math|S}} are relations over {{math|X}} then {{math|1 = S ∘ R = {{mset| (x, z) {{!}} there exists y ∈ X such that xRy and ySz }}}} (also denoted by {{math|R; S}}) is the relative product of {{math|R}} and {{math|S}}. The identity element is the identity relation. The order of {{math|R}} and {{math|S}} in the notation {{math|S ∘ R}}, used here agrees with the standard notational order for composition of functions. For example, the composition "is mother of" {{math|∘}} "is parent of" yields "is maternal grandparent of", while the composition "is parent of" {{math|∘}} "is mother of" yields "is grandmother of". For the former case, if {{math|x}} is the parent of {{math|y}} and {{math|y}} is the mother of {{math|z}}, then {{math|x}} is the maternal grandparent of {{math|z}}.

; {{em| Converse}}{{efn|name="heterogeneous operation"}}: If {{math|R}} is a relation over sets {{math|X}} and {{math|Y}} then {{math|1=R^T = {{mset| (y, x) {{!}} xRy }}}} is the converse relation of {{math|R}} over {{math|Y}} and {{math|X}}. For example, {{math|1==}} is the converse of itself, as is {{math|≠}}, and {{math|<}} and {{math|>}} are each other's converse, as are {{math|≤}} and {{math|≥}}.

; {{em| Complement}}{{efn|name="heterogeneous operation"}}: If {{math|R}} is a relation over {{math|X}} then {{math|1 = {{overline|R}} = {{mset| (x, y) {{!}} x, y ∈ X and not xRy }}}} (also denoted by {{math|{{strikethrough|R}}}} or {{math|¬R}}) is the complementary relation of {{math|R}}. For example, {{math|1==}} and {{math|≠}} are each other's complement, as are {{math|⊆}} and {{math|⊈}}, {{math|⊇}} and {{math|⊉}}, and {{math|∈}} and {{math|∉}}, and, for total orders, also {{math|<}} and {{math|≥}}, and {{math|>}} and {{math|≤}}. The complement of the converse relation {{math|R^T}} is the converse of the complement: $\backslash overline\{R^\backslash mathsf\{T\}\}\; =\; \backslash bar\{R\}^\backslash mathsf\{T\}.$

; {{em| Restriction}}{{efn|name="heterogeneous operation"}}: If {{math|R}} is a relation over {{math|X}} and {{math|S}} is a subset of {{math|X}} then {{math|1=R_{{!}}S = {{mset| (x, y) {{!}} xRy and x, y ∈ S }}}} is the {{em|{{visible anchor|restriction relation|Restriction relation|Restriction of a homogeneous relation}}}} of {{math|R}} to {{math|S}}. The expression {{math|1=R_{{!}}S = {{mset| (x, y) {{!}} xRy and x ∈ S }}}} is the {{em|{{visible anchor|left-restriction relation|Left-restriction relation}}}} of {{math|R}} to {{math|S}}; the expression {{math|1=R^{{!}}S = {{mset| (x, y) {{!}} xRy and y ∈ S }}}} is called the {{em|{{visible anchor|right-restriction relation|Right-restriction relation}}}} of {{math|R}} to {{math|S}}. If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, then so too are its restrictions. However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal. For example, restricting the relation "{{math|x}} is parent of {{math|y}}" to women yields the relation "{{math|x}} is mother of the woman {{math|y}}"; its transitive closure does not relate a woman with her paternal grandmother. On the other hand, the transitive closure of "is parent of" is "is ancestor of"; its restriction to women does relate a woman with her paternal grandmother.

A relation {{math|R}} over sets {{math|X}} and {{math|Y}} is said to be {{em|{{visible anchor|contained in|Containment of relations}}}} a relation {{math|S}} over {{math|X}} and {{math|Y}}, written {{math|R ⊆ S}}, if {{math|R}} is a subset of {{math|S}}, that is, for all {{math|x ∈ X}} and {{math|y ∈ Y}}, if {{math|xRy}}, then {{math|xSy}}. If {{math|R}} is contained in {{math|S}} and {{math|S}} is contained in {{math|R}}, then {{math|R}} and {{math|S}} are called equal written {{math|1=R = S}}. If {{math|R}} is contained in {{math|S}} but {{math|S}} is not contained in {{math|R}}, then {{math|R}} is said to be {{em|{{visible anchor|smaller|Smaller relation}}}} than {{math|S}}, written {{math|R ⊊ S}}. For example, on the rational numbers, the relation {{math|>}} is smaller than {{math|≥}}, and equal to the composition {{math|> ∘ >}}.

• A relation is asymmetric if, and only if, it is antisymmetric and irreflexive.
• A transitive relation is irreflexive if, and only if, it is asymmetric.
• A relation is reflexive if, and only if, its complement is irreflexive.
• A relation is strongly connected if, and only if, it is connected and reflexive.
• A relation is equal to its converse if, and only if, it is symmetric.
• A relation is connected if, and only if, its complement is anti-symmetric.
• A relation is strongly connected if, and only if, its complement is asymmetric.{{sfn|Schmidt|Ströhlein|1993|ps=}}
• If {{math|R}} and {{math|S}} are relations over a set {{math|X}}, and {{math|R}} is contained in {{math|S}}, then
• If {{math|R}} is reflexive, connected, strongly connected, left-total, or right-total, then so is {{math|S}}.
• If {{math|S}} is irreflexive, asymmetric, anti-symmetric, left-unique, or right-unique, then so is {{math|R}}.
• A relation is reflexive, irreflexive, symmetric, asymmetric, anti-symmetric, connected, strongly connected, and transitive if its converse is, respectively.

• Order relations, including strict orders:
• Greater than
• Greater than or equal to
• Less than
• Less than or equal to
• Divides (evenly)
• Subset of
• Equivalence relations:
• Equality
• Parallel with (for affine spaces)
• Is in bijection with
• Isomorphic
• Tolerance relation, a reflexive and symmetric relation:
• Dependency relation, a finite tolerance relation
• Independency relation, the complement of some dependency relation
• Kinship relations

The above concept of relation has been generalized to admit relations between members of two different sets.

Given sets {{math|X}} and {{math|Y}}, a heterogeneous relation {{math|R}} over {{math|X}} and {{math|Y}} is a subset of {{math|1= {{mset| (x,y) {{!}} x∈X, y∈Y }}}}.{{sfn|Codd|1970|ps=}}{{harvnb|Enderton|1977|loc=Ch 3. p. 40}}

When {{math|1=X = Y}}, the relation concept described above is obtained; it is often called homogeneous relation (or endorelation){{sfn|Müller|2012|p=22|ps=}}{{sfn|Pahl|Damrath|2001|p=496|ps=}} to distinguish it from its generalization.

The above properties and operations that are marked "{{efn|name="heterogeneous"}}" and "{{efn|name="heterogeneous operation"}}", respectively, generalize to heterogeneous relations.

An example of a heterogeneous relation is "ocean {{math|x}} borders continent {{math|y}}".

The best-known examples are functions{{efn|that is, right-unique and left-total heterogeneous relations}} with distinct domains and ranges, such as {{math|sqrt : N → R{{sub|+}}}}.

• Incidence structure, a heterogeneous relation between set of points and lines
• Order theory, investigates properties of order relations
• Relation algebra

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