Binary relation

An example of a binary relation R between two finite sets of natural numbers, A and B. Note that R is a subset of the Cartesian product, A × B. In this example, R = {(a, b) ∈ A × B: a < b}.

In mathematics, a binary relation associates some elements of one set called the domain with some elements of another set (possibly the same) called the codomain.^[1] Precisely, a binary relation over sets $X$ and $Y$ is a set of ordered pairs $(x,y)$ , where $x$ is an element of $X$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y} is an element of $Y$ .^[2] It encodes the common concept of relation: an element $x$ is related to an element Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y} , if and only if the pair $(x,y)$ belongs to the set of ordered pairs that defines the binary relation.

An example of a binary relation is the "divides" relation over the set of prime numbers Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {P} } and the set of integers $\mathbb {Z}$ , in which each prime $p$ is related to each integer $z$ that is a multiple of $p$ , but not to an integer that is not a multiple of $p$ . In this relation, for instance, the prime number $2$ is related to numbers such as Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -4} , $0$ , $6$ , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 10} , but not to $1$ or Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 9} , just as the prime number $3$ is related to $0$ , $6$ , and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 9} , but not to $4$ or Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 13} .

A binary relation is called a homogeneous relation when $X=Y$ . A binary relation is also called a heterogeneous relation when it is not necessary that $X=Y$ .

Binary relations, and especially homogeneous relations, are used in many branches of mathematics to model a wide variety of concepts. These include, among others:

the "is greater than", "is equal to", and "divides" relations in arithmetic;
the "is congruent to" relation in geometry;
the "is adjacent to" relation in graph theory;
the "is orthogonal to" relation in linear algebra.

A function may be defined as a binary relation that meets additional constraints.^[3] Binary relations are also heavily used in computer science.

A binary relation over sets $X$ and $Y$ can be identified with an element of the power set of the Cartesian product $X\times Y.$ Since a powerset is a lattice for set inclusion ( $\subseteq$ ), relations can be manipulated using set operations (union, intersection, and complementation) and algebra of sets.

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.

A binary relation is the most studied special case Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n=2} of an $n$ -ary relation over sets Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle X_{1},\dots ,X_{n}} , which is a subset of the Cartesian product Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle X_{1}\times \cdots \times X_{n}.} ^[2]

Definition

Given sets $X$ and $Y$ , the Cartesian product $X\times Y$ is defined as Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{(x,y)\mid x\in X{\text{ and }}y\in Y\},} and its elements are called ordered pairs.

A binary relation $R$ over sets $X$ and $Y$ is a subset of $X\times Y.$ ^[2]^[4] The set $X$ is called the domain^[2] or set of departure of $R$ , and the set $Y$ the codomain or set of destination of $R$ . In order to specify the choices of the sets $X$ and $Y$ , some authors define a binary relation or correspondence as an ordered triple Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (X,Y,G)} , where $G$ is a subset of $X\times Y$ called the graph of the binary relation. The statement Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (x,y)\in R} reads " $x$ is $R$ -related to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} " and is denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRy} .^[5]^[6]^[7]^{[lower-alpha 1]} The domain of definition or active domain^[2] of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is the set of all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRy} for at least one Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} . The codomain of definition, active codomain,^[2] image or range of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is the set of all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRy} for at least one Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} . The field of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is the union of its domain of definition and its codomain of definition.^[9]^[10]^[11]

When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = Y,} a binary relation is called a homogeneous relation (or endorelation). To emphasize the fact that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} are allowed to be different, a binary relation is also called a heterogeneous relation.^[12]^[13]^[14] The prefix hetero is from the Greek ἕτερος (heteros, "other, another, different").

A heterogeneous relation has been called a rectangular relation,^[14] suggesting that it does not have the square-like symmetry of a homogeneous relation on a set where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = B.} Commenting on the development of binary relations beyond homogeneous relations, researchers wrote, "... a variant of the theory has evolved that treats relations from the very beginning as heterogeneous or rectangular, i.e. as relations where the normal case is that they are relations between different sets."^[15]

The terms correspondence,^[16] dyadic relation and two-place relation are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a Cartesian product Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \times Y} without reference to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} , and reserve the term "correspondence" for a binary relation with reference to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} .^{[citation needed]}

In a binary relation, the order of the elements is important; if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \neq y} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle yRx} can be true or false independently of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRy} . For example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3} divides Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 9} , but Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 9} does not divide Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3} .

Operations

Union

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} are binary relations over sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R \cup S = \{ (x, y) \mid xRy \text{ or } xSy \}} is the union relation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} .

The identity element is the empty relation. For example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leq} is the union of < and =, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \geq} is the union of > and =.

Intersection

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} are binary relations over sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R \cap S = \{ (x, y) \mid xRy \text{ and } xSy \}} is the intersection relation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} .

The identity element is the universal relation. For example, the relation "is divisible by 6" is the intersection of the relations "is divisible by 3" and "is divisible by 2".

Composition

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a binary relation over sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is a binary relation over sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \circ R = \{ (x, z) \mid \text{ there exists } y \in Y \text{ such that } xRy \text{ and } ySz \}} (also denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R; S} ) is the composition relation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} .

The identity element is the identity relation. The order of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} in the notation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \circ R,} used here agrees with the standard notational order for composition of functions. For example, the composition (is parent of)Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \circ} (is mother of) yields (is maternal grandparent of), while the composition (is mother of)Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \circ} (is parent of) yields (is grandmother of). For the former case, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is the parent of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} is the mother of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is the maternal grandparent of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} .

Converse

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a binary relation over sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^\textsf{T} = \{ (y, x) \mid xRy \}} is the converse relation,^[17] also called inverse relation,^[18] of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} .

For example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =} is the converse of itself, as is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neq} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle >} are each other's converse, as are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leq} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \geq.} A binary relation is equal to its converse if and only if it is symmetric.

Complement

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a binary relation over sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} then ${\bar {R}}=\{(x,y)\mid \neg xRy\}$ (also denoted by $\neg R$ ) is the complementary relation of $R$ over $X$ and $Y$ .

For example, $=$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \neq } are each other's complement, as are $\subseteq$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \not \subseteq } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \supseteq} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \not \supseteq} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \in} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \not \in} , and for total orders also Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \geq} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle >} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leq} .

The complement of the converse relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^\textsf{T}} is the converse of the complement: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{R^\mathsf{T}} = \bar{R}^\mathsf{T}.}

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = Y,} the complement has the following properties:

If a relation is symmetric, then so is the complement.
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.

Restriction

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a binary homogeneous relation over a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is a subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{\vert S} = \{ (x, y) \mid xRy \text{ and } x \in S \text{ and } y \in S \}} is the restriction relation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} .

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a binary relation over sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} and if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is a subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{\vert S} = \{ (x, y) \mid xRy \text{ and } x \in S \}} is the left-restriction relation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} .^{[clarification needed]}

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 "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is parent of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} " to females yields the relation "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is mother of the woman Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 females does relate a woman with her paternal grandmother.

Also, the various concepts of completeness (not to be confused with being "total") do not carry over to restrictions. For example, over the real numbers a property of the relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leq} is that every non-empty subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq \R} with an upper bound in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R} has a least upper bound (also called supremum) in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R.} However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leq} to the rational numbers.

A binary relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} over sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} is said to be contained in a relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} , written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R \subseteq S,} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} , that is, for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in Y,} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRy} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xSy} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is contained in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is contained in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} are called equal written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R = S} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is contained in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} but Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is not contained in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is said to be smaller than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} , written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R \subsetneq S.} For example, on the rational numbers, the relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle >} is smaller than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \geq} , and equal to the composition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle > \circ >} .

Matrix representation

Binary relations over sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} can be represented algebraically by logical matrices indexed by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} with entries in the Boolean semiring (addition corresponds to OR and multiplication to AND) where matrix addition corresponds to union of relations, matrix multiplication corresponds to composition of relations (of a relation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} and a relation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} ),^[19] the Hadamard product corresponds to intersection of relations, the zero matrix corresponds to the empty relation, and the matrix of ones corresponds to the universal relation. Homogeneous relations (when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = Y} ) form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring) where the identity matrix corresponds to the identity relation.^[20]

Examples

2nd example relation
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}	ball	car	doll	cup
John	+	−	−	−
Mary	−	−	+	−
Venus	−	+	−	−

1st example relation
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}	ball	car	doll	cup
John	+	−	−	−
Mary	−	−	+	−
Ian	−	−	−	−
Venus	−	+	−	−

The following example shows that the choice of codomain is important. Suppose there are four objects Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \{ \text{ball, car, doll, cup} \}} and four people Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = \{ \text{John, Mary, Ian, Venus} \}.} A possible relation on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} is the relation "is owned by", given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R = \{ (\text{ball, John}), (\text{doll, Mary}), (\text{car, Venus}) \}.} That is, John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the cup and Ian owns nothing; see the 1st example. As a set, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} does not involve Ian, and therefore Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} could have been viewed as a subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \times \{ \text{John, Mary, Venus} \},} i.e. a relation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and $\{{\text{John, Mary, Venus}}\};$ see the 2nd example. But in that second example, $R$ contains no information about the ownership by Ian.
While the 2nd example relation is surjective (see below), the 1st is not.

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Oceans and continents (islands omitted)

Ocean borders continent
NA SA AF EU AS AU AA

Indian 0 0 1 0 1 1 1

Arctic 1 0 0 1 1 0 0

Atlantic 1 1 1 1 0 0 1

Pacific 1 1 0 0 1 1 1
Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A=\{{\text{Indian}},{\text{Arctic}},{\text{Atlantic}},{\text{Pacific}}\}} , the oceans of the globe, and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle B=\{{\text{NA}},{\text{SA}},{\text{AF}},{\text{EU}},{\text{AS}},{\text{AU}},{\text{AA}}\}} , the continents. Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle aRb} represent that ocean $a$ borders continent $b$ . Then the logical matrix for this relation is:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle R={\begin{pmatrix}0&0&1&0&1&1&1\\1&0&0&1&1&0&0\\1&1&1&1&0&0&1\\1&1&0&0&1&1&1\end{pmatrix}}.}
The connectivity of the planet Earth can be viewed through $RR^{\mathsf {T}}$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^\mathsf{T} R} , the former being a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 \times 4} relation on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , which is the universal relation (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \times A} or a logical matrix of all ones). This universal relation reflects the fact that every ocean is separated from the others by at most one continent. On the other hand, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^\mathsf{T} R} is a relation on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B \times B} which fails to be universal because at least two oceans must be traversed to voyage from Europe to Australia.
Visualization of relations leans on graph theory: For relations on a set (homogeneous relations), a directed graph illustrates a relation and a graph a symmetric relation. For heterogeneous relations a hypergraph has edges possibly with more than two nodes, and can be illustrated by a bipartite graph. Just as the clique is integral to relations on a set, so bicliques are used to describe heterogeneous relations; indeed, they are the "concepts" that generate a lattice associated with a relation.
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The various Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} axes represent time for observers in motion, the corresponding Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} axes are their lines of simultaneity.
Hyperbolic orthogonality: Time and space are different categories, and temporal properties are separate from spatial properties. The idea of simultaneous events is simple in absolute time and space since each time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} determines a simultaneous hyperplane in that cosmology. Hermann Minkowski changed that when he articulated the notion of relative simultaneity, which exists when spatial events are "normal" to a time characterized by a velocity. He used an indefinite inner product, and specified that a time vector is normal to a space vector when that product is zero. The indefinite inner product in a composition algebra is given by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x, z\rangle = x \bar{z} + \bar{x}z\;} where the overbar denotes conjugation.
As a relation between some temporal events and some spatial events, hyperbolic orthogonality (as found in split-complex numbers) is a heterogeneous relation.^[21]
A geometric configuration can be considered a relation between its points and its lines. The relation is expressed as incidence. Finite and infinite projective and affine planes are included. Jakob Steiner pioneered the cataloguing of configurations with the Steiner systems Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname S(t, k, n)} which have an n-element set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname S} and a set of k-element subsets called blocks, such that a subset with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} elements lies in just one block. These incidence structures have been generalized with block designs. The incidence matrix used in these geometrical contexts corresponds to the logical matrix used generally with binary relations.
An incidence structure is a triple Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf D = (V, \mathbf B, I)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf B} are any two disjoint sets and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is a binary relation between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf B} , i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I \subseteq V \times \mathbf B.} The elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} will be called points, those of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf B} blocks, and those of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} flags.^[22]

Types of binary relations

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Examples of four types of binary relations over the real numbers: one-to-one (in green), one-to-many (in blue), many-to-one (in red), many-to-many (in black).

Some important types of binary relations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} over sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} are listed below.

Uniqueness properties:

Injective^[23] (also called left-unique^[24]): for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y \in X} and all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z \in Y,} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRz} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle yRz} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = y} . In other words, every element of the codomain has at most one preimage element. For such a relation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} is called a primary key of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} .^[2] For example, the green and blue binary relations in the diagram are injective, but the red one is not (as it relates both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} ), nor the black one (as it relates both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} ).
Functional^[23]^[25]^[26] (also called right-unique^[24] or univalent^[27]): for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} and all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y, z \in Y,} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRy} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRz} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = z} . In other words, every element of the domain has at most one image element. Such a binary relation is called a partial function or partial mapping.^[28] For such a relation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ X \}} is called a primary key of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} .^[2] For example, the red and green binary relations in the diagram are functional, but the blue one is not (as it relates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} to both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1} ), nor the black one (as it relates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} to both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} ).
One-to-one: injective and functional. For example, the green binary relation in the diagram is one-to-one, but the red, blue and black ones are not.
One-to-many: injective and not functional. For example, the blue binary relation in the diagram is one-to-many, but the red, green and black ones are not.
Many-to-one: functional and not injective. For example, the red binary relation in the diagram is many-to-one, but the green, blue and black ones are not.
Many-to-many: not injective nor functional. For example, the black binary relation in the diagram is many-to-many, but the red, green and blue ones are not.

Totality properties (only definable if the domain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and codomain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} are specified):

Total^[23] (also called left-total^[24]): for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} there exists a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in Y} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRy} . In other words, every element of the domain has at least one image element. In other words, the domain of definition of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . This property, is different from the definition of connected (also called total by some authors)^{[citation needed]} in Properties. Such a binary relation is called a multivalued function. For example, the red and green binary relations in the diagram are total, but the blue one is not (as it does not relate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1} to any real number), nor the black one (as it does not relate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} to any real number). As another example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle >} is a total relation over the integers. But it is not a total relation over the positive integers, because there is no Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} in the positive integers such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 > y} .^[29] However, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <} is a total relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is total: for a given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , choose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = x} .
Surjective^[23] (also called right-total^[24]): for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in Y} , there exists an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRy} . In other words, every element of the codomain has at least one preimage element. In other words, the codomain of definition of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} . For example, the green and blue binary relations in the diagram are surjective, but the red one is not (as it does not relate any real number to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1} ), nor the black one (as it does not relate any real number to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} ).

Uniqueness and totality properties (only definable if the domain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and codomain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} are specified):

A function (also called mapping^[24]): a binary relation that is functional and total. In other words, every element of the domain has exactly one image element. For example, the red and green binary relations in the diagram are functions, but the blue and black ones are not.
An injection: a function that is injective. For example, the green relation in the diagram is an injection, but the red one is not; the black and the blue relation is not even a function.
A surjection: a function that is surjective. For example, the green relation in the diagram is a surjection, but the red one is not.
A bijection: a function that is injective and surjective. In other words, every element of the domain has exactly one image element and every element of the codomain has exactly one preimage element. For example, the green binary relation in the diagram is a bijection, but the red one is not.

If relations over proper classes are allowed:

Set-like (also called local): for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} , the class of all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in Y} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle yRx} , i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{y \in Y, yRx\}} , is a set. For example, the relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \in} is set-like, and every relation on two sets is set-like.^[30] The usual ordering < over the class of ordinal numbers is a set-like relation, while its inverse > is not.^{[citation needed]}

Sets versus classes

Certain mathematical "relations", such as "equal to", "subset of", and "member 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, to model the general concept of "equality" as a binary relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =} , take the domain and codomain to be the "class of all sets", which is not a set in the usual set theory.

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 usual work-around to this problem is to select a "large enough" set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , that contains all the objects of interest, and work with the restriction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =_A} instead of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =} . Similarly, the "subset of" relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \subseteq} needs to be restricted to have domain and codomain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A)} (the power set of a specific set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} ): the resulting set relation can be denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \subseteq_A.} Also, the "member of" relation needs to be restricted to have domain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and codomain Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P(A)} to obtain a binary relation $\in _{A}$ that is a set. Bertrand Russell has shown that assuming Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \in } to be defined over all sets leads to a contradiction in naive set theory, see Russell's paradox.

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 binary relation with its graph in this context.)^[31] With this definition one can for instance define a binary relation over every set and its power set.

Homogeneous relation

A homogeneous relation over a set $X$ is a binary relation over $X$ and itself, i.e. it is a subset of the Cartesian product Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle X\times X.} ^[14]^[32]^[33] It is also simply called a (binary) relation over $X$ .

A homogeneous relation $R$ over a set $X$ may be identified with a directed simple graph permitting loops, where $X$ is the vertex set and $R$ is the edge set (there is an edge from a vertex $x$ to a vertex Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y} if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRy} ). The set of all homogeneous relations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{B}(X)} over a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is the power set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{X \times X}} which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation. Considering composition of relations as a binary operation on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{B}(X)} , it forms a semigroup with involution.

Some important properties that a homogeneous relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} over a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} may have are:

Reflexive: for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRx} . For example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \geq} is a reflexive relation but > is not.
Irreflexive: for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X,} not Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRx} . For example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle >} is an irreflexive relation, but Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \geq} is not.
Symmetric: for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y \in X,} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRy} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle yRx} . For example, "is a blood relative of" is a symmetric relation.
Antisymmetric: for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y \in X,} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRy} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle yRx} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = y.} For example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \geq} is an antisymmetric relation.^[34]
Asymmetric: for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y \in X,} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRy} then not Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle yRx} . A relation is asymmetric if and only if it is both antisymmetric and irreflexive.^[35] For example, > is an asymmetric relation, but Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \geq} is not.
Transitive: for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y, z \in X,} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRy} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle yRz} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRz} . A transitive relation is irreflexive if and only if it is asymmetric.^[36] For example, "is ancestor of" is a transitive relation, while "is parent of" is not.
Connected: for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y \in X,} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \neq y} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRy} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle yRx} .
Strongly connected: for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y \in X,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRy} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle yRx} .
Dense: for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y \in X,} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRy ,} then some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z \in X} exists such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRz} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle zRy} .

A partial order is a relation that is reflexive, antisymmetric, and transitive. A strict partial order is a relation that is irreflexive, asymmetric, and transitive. A total order is a relation that is reflexive, antisymmetric, transitive and connected.^[37] A strict total order is a relation that is irreflexive, asymmetric, transitive and connected. An equivalence relation is a relation that is reflexive, symmetric, and transitive. For example, "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} divides Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} " is a partial, but not a total order on natural numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \N,} "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x < y} " is a strict total order on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \N,} and "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is parallel to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} " is an equivalence relation on the set of all lines in the Euclidean plane.

All operations defined in section § Operations also apply to homogeneous relations. Beyond that, a homogeneous relation over a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} may be subjected to closure operations like:

Reflexive closure: the smallest reflexive relation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} containing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} ,
Transitive closure: the smallest transitive relation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} containing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} ,
Equivalence closure: the smallest equivalence relation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} containing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} .

Calculus of relations

Developments in algebraic logic have facilitated usage of binary relations. The calculus of relations includes the algebra of sets, extended by composition of relations and the use of converse relations. The inclusion Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R \subseteq S,} meaning that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle aRb} implies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle aSb} , sets the scene in a lattice of relations. But since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \subseteq Q \equiv (P \cap \bar{Q} = \varnothing ) \equiv (P \cap Q = P),} the inclusion symbol is superfluous. Nevertheless, composition of relations and manipulation of the operators according to Schröder rules, provides a calculus to work in the power set of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \times B.}

In contrast to homogeneous relations, the composition of relations operation is only a partial function. The necessity of matching target to source of composed relations has led to the suggestion that the study of heterogeneous relations is a chapter of category theory as in the category of sets, except that the morphisms of this category are relations. The objects of the category Rel are sets, and the relation-morphisms compose as required in a category.^{[citation needed]}

Induced concept lattice

Binary relations have been described through their induced concept lattices: A concept Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C \subset R} satisfies two properties:

The logical matrix of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is the outer product of logical vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{i j} = u_i v_j , \quad u, v} logical vectors.^{[clarification needed]}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is maximal, not contained in any other outer product. Thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is described as a non-enlargeable rectangle.

For a given relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R \subseteq X \times Y,} the set of concepts, enlarged by their joins and meets, forms an "induced lattice of concepts", with inclusion Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqsubseteq} forming a preorder.

The MacNeille completion theorem (1937) (that any partial order may be embedded in a complete lattice) is cited in a 2013 survey article "Decomposition of relations on concept lattices".^[38] The decomposition is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R = f E g^\textsf{T}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} are functions, called mappings or left-total, functional relations in this context. The "induced concept lattice is isomorphic to the cut completion of the partial order Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} that belongs to the minimal decomposition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f, g, E)} of the relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} ."

Particular cases are considered below: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} total order corresponds to Ferrers type, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} identity corresponds to difunctional, a generalization of equivalence relation on a set.

Relations may be ranked by the Schein rank which counts the number of concepts necessary to cover a relation.^[39] Structural analysis of relations with concepts provides an approach for data mining.^[40]

Particular relations

Proposition: If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a surjective relation and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^\mathsf{T}} is its transpose, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I \subseteq R^\textsf{T} R} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times m} identity relation.
Proposition: If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a serial relation, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I \subseteq R R^\textsf{T}} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} identity relation.

Difunctional

The idea of a difunctional relation is to partition objects by distinguishing attributes, as a generalization of the concept of an equivalence relation. One way this can be done is with an intervening set $Z=\{x,y,z,\ldots \}$ of indicators. The partitioning relation $R=FG^{\textsf {T}}$ is a composition of relations using functional relations Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F\subseteq A\times Z{\text{ and }}G\subseteq B\times Z.} Jacques Riguet named these relations difunctional since the composition Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle FG^{\mathsf {T}}} involves functional relations, commonly called partial functions.

In 1950 Riguet showed that such relations satisfy the inclusion:^[41]

RR^{\textsf {T}}R\subseteq R

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This terminology recalls the fact that, when represented as a logical matrix, the columns and rows of a difunctional relation can be arranged as a block matrix with rectangular blocks of ones on the (asymmetric) main diagonal.^[42] More formally, a relation $R$ on $X\times Y$ is difunctional if and only if it can be written as the union of Cartesian products Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A_{i}\times B_{i}} , where the Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A_{i}} are a partition of a subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_i} likewise a partition of a subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} .^[43]

Using the notation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{y \mid xRy\} = xR} , a difunctional relation can also be characterized as a relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} such that wherever Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1 R} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2 R} have a non-empty intersection, then these two sets coincide; formally Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1 \cap x_2 \neq \varnothing} implies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1 R = x_2 R.} ^[44]

In 1997 researchers found "utility of binary decomposition based on difunctional dependencies in database management."^[45] Furthermore, difunctional relations are fundamental in the study of bisimulations.^[46]

In the context of homogeneous relations, a partial equivalence relation is difunctional.

Ferrers type

A strict order on a set is a homogeneous relation arising in order theory. In 1951 Jacques Riguet adopted the ordering of an integer partition, called a Ferrers diagram, to extend ordering to binary relations in general.^[47]

The corresponding logical matrix of a general binary relation has rows which finish with a sequence of ones. Thus the dots of a Ferrer's diagram are changed to ones and aligned on the right in the matrix.

An algebraic statement required for a Ferrers type relation R is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R \bar{R}^\textsf{T} R \subseteq R.}

If any one of the relations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R, \bar{R}, R^\textsf{T}} is of Ferrers type, then all of them are. ^[48]

Contact

Suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} is the power set of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , the set of all subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} . Then a relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} is a contact relation if it satisfies three properties:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{for all } x \in A, Y = \{ x \} \text{ implies } xgY.}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y \subseteq Z \text{ and } xgY \text{ implies } xgZ.}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{for all } y \in Y, ygZ \text{ and } xgY \text{ implies } xgZ.}

The set membership relation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon = } "is an element of", satisfies these properties so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} is a contact relation. The notion of a general contact relation was introduced by Georg Aumann in 1970.^[49]^[50]

In terms of the calculus of relations, sufficient conditions for a contact relation include Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^\textsf{T} \bar{C} \subseteq \ni \bar{C} \equiv C \overline{\ni \bar{C}} \subseteq C,} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ni} is the converse of set membership (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \in} ).^[51]^: 280

Preorder R\R

Every relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} generates a preorder Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R \backslash R} which is the left residual.^[52] In terms of converse and complements, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R \backslash R \equiv \overline{R^\textsf{T} \bar{R}}.} Forming the diagonal of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^\textsf{T} \bar{R}} , the corresponding row of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^{\textsf{T}}} and column of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{R}} will be of opposite logical values, so the diagonal is all zeros. Then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^\textsf{T} \bar{R} \subseteq \bar{I} \implies I \subseteq \overline{R^\textsf{T} \bar{R}} = R \backslash R} , so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R \backslash R} is a reflexive relation.

To show transitivity, one requires that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (R\backslash R)(R\backslash R) \subseteq R \backslash R.} Recall that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = R \backslash R} is the largest relation such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R X \subseteq R.} Then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(R\backslash R) \subseteq R}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(R\backslash R) (R\backslash R )\subseteq R} (repeat)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \equiv R^\textsf{T} \bar{R} \subseteq \overline{(R \backslash R)(R \backslash R)}} (Schröder's rule)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \equiv (R \backslash R)(R \backslash R) \subseteq \overline{R^\textsf{T} \bar{R}}} (complementation)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \equiv (R \backslash R)(R \backslash R) \subseteq R \backslash R.} (definition)

The inclusion relation Ω on the power set of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} can be obtained in this way from the membership relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \in} on subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega = \overline{\ni \bar{\in}} = \in \backslash \in .} ^[51]^: 283

Fringe of a relation

Given a relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} , its fringe is the sub-relation defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{fringe}(R) = R \cap \overline{R \bar{R}^\textsf{T} R}.}

When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a partial identity relation, difunctional, or a block diagonal relation, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{fringe}(R) = R} . Otherwise the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{fringe}} operator selects a boundary sub-relation described in terms of its logical matrix: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{fringe}(R)} is the side diagonal if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is an upper right triangular linear order or strict order. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{fringe}(R)} is the block fringe if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is irreflexive (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R \subseteq \bar{I}} ) or upper right block triangular. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{fringe}(R)} is a sequence of boundary rectangles when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is of Ferrers type.

On the other hand, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{fringe}(R) = \emptyset} when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a dense, linear, strict order.^[51]

Mathematical heaps

Given two sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} , the set of binary relations between them Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{B}(A,B)} can be equipped with a ternary operation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a, b, c] = a b^\textsf{T} c} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^\mathsf{T}} denotes the converse relation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} . In 1953 Viktor Wagner used properties of this ternary operation to define semiheaps, heaps, and generalized heaps.^[53]^[54] The contrast of heterogeneous and homogeneous relations is highlighted by these definitions:

There is a pleasant symmetry in Wagner's work between heaps, semiheaps, and generalised heaps on the one hand, and groups, semigroups, and generalised groups on the other. Essentially, the various types of semiheaps appear whenever we consider binary relations (and partial one-one mappings) between different sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} , while the various types of semigroups appear in the case where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = B} .
— Christopher Hollings, "Mathematics across the Iron Curtain: a history of the algebraic theory of semigroups"^[55]

Notes

↑ Authors who deal with binary relations only as a special case of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -ary relations for arbitrary Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} usually write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Rxy} as a special case of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Rx_1\dots x_n} (prefix notation).^[8]

References

↑ Meyer, Albert (17 November 2021). "MIT 6.042J Math for Computer Science, Lecture 3T, Slide 2" (PDF). Archived (PDF) from the original on 2021-11-17.
↑ ^2.0 ^2.1 ^2.2 ^2.3 ^2.4 ^2.5 ^2.6 ^2.7 Codd, Edgar Frank (June 1970). "A Relational Model of Data for Large Shared Data Banks" (PDF). Communications of the ACM. 13 (6): 377–387. doi:10.1145/362384.362685. S2CID 207549016. Archived (PDF) from the original on 2004-09-08. Retrieved 2020-04-29.
↑ "Relation definition – Math Insight". mathinsight.org. Retrieved 2019-12-11.
↑ Enderton 1977, Ch 3. pg. 40
↑ Ernst Schröder (1895) Algebra und Logic der Relative, via Internet Archive
↑ C. I. Lewis (1918) A Survey of Symbolic Logic, pages 269–279, via internet Archive
↑ Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7, Chapt. 5
↑ Hans Hermes (1973). Introduction to Mathematical Logic. Hochschultext (Springer-Verlag). London: Springer. ISBN 3540058192. ISSN 1431-4657. Sect.II.§1.1.4
↑ Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960]. Axiomatic Set Theory. Dover. ISBN 0-486-61630-4.
↑ Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in 1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7.
↑ Levy, Azriel (2002) [republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979]. Basic Set Theory. Dover. ISBN 0-486-42079-5.
↑ Schmidt, Gunther; Ströhlein, Thomas (2012). [[[:Template:Google books]] Relations and Graphs: Discrete Mathematics for Computer Scientists] Check |url= value (help). Springer Science & Business Media. Definition 4.1.1. ISBN 978-3-642-77968-8.
↑ Christodoulos A. Floudas; Panos M. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science & Business Media. pp. 299–300. ISBN 978-0-387-74758-3.
↑ ^14.0 ^14.1 ^14.2 Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi. ISBN 978-1-4020-6164-6.
↑ G. Schmidt, Claudia Haltensperger, and Michael Winter (1997) "Heterogeneous relation algebra", chapter 3 (pages 37 to 53) in Relational Methods in Computer Science, Advances in Computer Science, Springer books ISBN 3-211-82971-7
↑ Jacobson, Nathan (2009), Basic Algebra II (2nd ed.) § 2.1.
↑ Garrett Birkhoff & Thomas Bartee (1970) Modern Applied Algebra, page 35, McGraw-Hill
↑ Mary P. Dolciani (1962) Modern Algebra: Structure and Method, Book 2, page 339, Houghton Mifflin
↑ John C. Baez (6 Nov 2001). "quantum mechanics over a commutative rig". Newsgroup: sci.physics.research. Usenet: [email protected]. Retrieved November 25, 2018.
↑ Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28. doi:10.1007/978-3-642-01492-5_1, pp. 7-10
↑ Template:Wikibooks inline
↑ Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried (1986). Design Theory. Cambridge University Press. p. 15.. 2nd ed. (1999) ISBN 978-0-521-44432-3
↑ ^23.0 ^23.1 ^23.2 ^23.3 Van Gasteren 1990, p. 45.
↑ ^24.0 ^24.1 ^24.2 ^24.3 ^24.4 Kilp, Knauer, Mikhalev 2000, p. 3.
↑ "Functional relation - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-06-13.
↑ "functional relation in nLab". ncatlab.org. Retrieved 2024-06-13.
↑ Schmidt 2010, p. 49.
↑ Kilp, Knauer, Mikhalev 2000, p. 4.
↑ Yao, Y.Y.; Wong, S.K.M. (1995). "Generalization of rough sets using relationships between attribute values" (PDF). Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33..
↑ Kunen, Kenneth (1980). Set theory: an introduction to independence proofs. North-Holland. p. 102. ISBN 0-444-85401-0. Zbl 0443.03021.
↑ Tarski, Alfred; Givant, Steven (1987). A formalization of set theory without variables. American Mathematical Society. p. 3. ISBN 0-8218-1041-3.
↑ M. E. Müller (2012). Relational Knowledge Discovery. Cambridge University Press. p. 22. ISBN 978-0-521-19021-3.
↑ Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. Springer Science & Business Media. p. 496. ISBN 978-3-540-67995-0.
↑ Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006), A Transition to Advanced Mathematics (6th ed.), Brooks/Cole, p. 160, ISBN 0-534-39900-2
↑ Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography, Springer-Verlag, p. 158.
↑ Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: School of Mathematics – Physics Charles University. p. 1. Archived from the original (PDF) on 2013-11-02. Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".
↑ Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, ISBN 0-12-597680-1, p. 4
↑ R. Berghammer & M. Winter (2013) "Decomposition of relations on concept lattices", Fundamenta Informaticae 126(1): 37–82 doi:10.3233/FI-2013-871
↑ Ki-Hang Kim (1982) Boolean Matrix Theory and Applications, page 37, Marcel Dekker ISBN 0-8247-1788-0
↑ Ali Jaoua, Rehab Duwairi, Samir Elloumi, and Sadok Ben Yahia (2009) "Data mining, reasoning and incremental information retrieval through non enlargeable rectangular relation coverage", pages 199 to 210 in Relations and Kleene algebras in computer science, Lecture Notes in Computer Science 5827, Springer Template:Mr
↑ Riguet, Jacques (January 1950). "Quelques proprietes des relations difonctionelles". Comptes rendus (in French). 230: 1999–2000.
↑ Julius Richard Büchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions. Springer Science & Business Media. pp. 35–37. ISBN 978-1-4613-8853-1.
↑ East, James; Vernitski, Alexei (February 2018). "Ranks of ideals in inverse semigroups of difunctional binary relations". Semigroup Forum. 96 (1): 21–30. arXiv:1612.04935. doi:10.1007/s00233-017-9846-9. S2CID 54527913.
↑ Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer Science & Business Media. p. 200. ISBN 978-3-211-82971-4.
↑ Ali Jaoua, Nadin Belkhiter, Habib Ounalli, and Theodore Moukam (1997) "Databases", pages 197–210 in Relational Methods in Computer Science, edited by Chris Brink, Wolfram Kahl, and Gunther Schmidt, Springer Science & Business Media ISBN 978-3-211-82971-4
↑ Gumm, H. P.; Zarrad, M. (2014). "Coalgebraic Simulations and Congruences". Coalgebraic Methods in Computer Science. Lecture Notes in Computer Science. 8446. p. 118. doi:10.1007/978-3-662-44124-4_7. ISBN 978-3-662-44123-7.
↑ J. Riguet (1951) "Les relations de Ferrers", Comptes Rendus 232: 1729,30
↑ Schmidt, Gunther; Ströhlein, Thomas (2012). [[[:Template:Google books]] Relations and Graphs: Discrete Mathematics for Computer Scientists] Check |url= value (help). Springer Science & Business Media. p. 77. ISBN 978-3-642-77968-8.
↑ Georg Aumann (1971). "Kontakt-Relationen". Sitzungsberichte der mathematisch-physikalischen Klasse der Bayerischen Akademie der Wissenschaften München. 1970 (II): 67–77.
↑ Anne K. Steiner (1970) Review:Kontakt-Relationen from Mathematical Reviews
↑ ^51.0 ^51.1 ^51.2 Gunther Schmidt (2011) Relational Mathematics, pages 211−15, Cambridge University Press ISBN 978-0-521-76268-7
↑ In this context, the symbol Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \backslash} does not mean "set difference".
↑ Viktor Wagner (1953) "The theory of generalised heaps and generalised groups", Matematicheskii Sbornik 32(74): 545 to 632 Template:Mr
↑ C.D. Hollings & M.V. Lawson (2017) Wagner's Theory of Generalised Heaps, Springer books ISBN 978-3-319-63620-7 Template:Mr
↑ Christopher Hollings (2014) Mathematics across the Iron Curtain: a history of the algebraic theory of semigroups, page 265, History of Mathematics 41, American Mathematical Society ISBN 978-1-4704-1493-1

Bibliography

Schmidt, Gunther (2010). Relational Mathematics. Berlin: Cambridge University Press. ISBN 9780511778810.
Schmidt, Gunther; Ströhlein, Thomas (2012). [[[:Template:Google books]] "Chapter 3: Heterogeneous relations"] Check |chapter-url= value (help). Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer Science & Business Media. ISBN 978-3-642-77968-8.
Ernst Schröder (1895) Algebra der Logik, Band III, via Internet Archive
Codd, Edgar Frank (1990). The Relational Model for Database Management: Version 2 (PDF). Boston: Addison-Wesley. ISBN 978-0201141924. Archived (PDF) from the original on 2022-10-09.
Enderton, Herbert (1977). Elements of Set Theory. Boston: Academic Press. ISBN 978-0-12-238440-0.
Kilp, Mati; Knauer, Ulrich; Mikhalev, Alexander (2000). Monoids, Acts and Categories: with Applications to Wreath Products and Graphs. Berlin: De Gruyter. ISBN 978-3-11-015248-7.
Van Gasteren, Antonetta (1990). On the Shape of Mathematical Arguments. Berlin: Springer. ISBN 9783540528494.
Peirce, Charles Sanders (1873). "Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic". Memoirs of the American Academy of Arts and Sciences. 9 (2): 317–178. Bibcode:1873MAAAS...9..317P. doi:10.2307/25058006. hdl:2027/hvd.32044019561034. JSTOR 25058006. Retrieved 2020-05-05.
Schmidt, Gunther (2010). Relational Mathematics. Cambridge: Cambridge University Press. ISBN 978-0-521-76268-7.

External links

Template:Springer

Template:Order theory Template:Set theory Template:Functions navbox

[9] Authors who deal with binary relations only as a special case of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -ary relations for arbitrary Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} usually write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Rxy} as a special case of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Rx_1\dots x_n} (prefix notation).^[8]

[1] Meyer, Albert (17 November 2021). "MIT 6.042J Math for Computer Science, Lecture 3T, Slide 2" (PDF). Archived (PDF) from the original on 2021-11-17.

[Codd1970-2] 2.0 ^2.1 ^2.2 ^2.3 ^2.4 ^2.5 ^2.6 ^2.7 Codd, Edgar Frank (June 1970). "A Relational Model of Data for Large Shared Data Banks" (PDF). Communications of the ACM. 13 (6): 377–387. doi:10.1145/362384.362685. S2CID 207549016. Archived (PDF) from the original on 2004-09-08. Retrieved 2020-04-29.

[3] "Relation definition – Math Insight". mathinsight.org. Retrieved 2019-12-11.

[4] Enderton 1977, Ch 3. pg. 40

[Schroder.1895-5] Ernst Schröder (1895) Algebra und Logic der Relative, via Internet Archive

[Lewis.1918-6] C. I. Lewis (1918) A Survey of Symbolic Logic, pages 269–279, via internet Archive

[gs-7] Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7, Chapt. 5

[8] Hans Hermes (1973). Introduction to Mathematical Logic. Hochschultext (Springer-Verlag). London: Springer. ISBN 3540058192. ISSN 1431-4657. Sect.II.§1.1.4

[suppes-10] Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960]. Axiomatic Set Theory. Dover. ISBN 0-486-61630-4.

[smullyan-11] Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in 1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7.

[levy-12] Levy, Azriel (2002) [republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979]. Basic Set Theory. Dover. ISBN 0-486-42079-5.

[Schmidt-13] Schmidt, Gunther; Ströhlein, Thomas (2012). [[[:Template:Google books]] Relations and Graphs: Discrete Mathematics for Computer Scientists] Check |url= value (help). Springer Science & Business Media. Definition 4.1.1. ISBN 978-3-642-77968-8.

[FloudasPardalos2008-14] Christodoulos A. Floudas; Panos M. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science & Business Media. pp. 299–300. ISBN 978-0-387-74758-3.

[Winter2007-15] 14.0 ^14.1 ^14.2 Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi. ISBN 978-1-4020-6164-6.

[16] G. Schmidt, Claudia Haltensperger, and Michael Winter (1997) "Heterogeneous relation algebra", chapter 3 (pages 37 to 53) in Relational Methods in Computer Science, Advances in Computer Science, Springer books ISBN 3-211-82971-7

[17] Jacobson, Nathan (2009), Basic Algebra II (2nd ed.) § 2.1.

[18] Garrett Birkhoff & Thomas Bartee (1970) Modern Applied Algebra, page 35, McGraw-Hill

[19] Mary P. Dolciani (1962) Modern Algebra: Structure and Method, Book 2, page 339, Houghton Mifflin

[20] John C. Baez (6 Nov 2001). "quantum mechanics over a commutative rig". Newsgroup: sci.physics.research. Usenet: [email protected]. Retrieved November 25, 2018.

[droste-21] Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28. doi:10.1007/978-3-642-01492-5_1, pp. 7-10

[22] Template:Wikibooks inline

[23] Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried (1986). Design Theory. Cambridge University Press. p. 15.. 2nd ed. (1999) ISBN 978-0-521-44432-3

[vangasteren1990-24] 23.0 ^23.1 ^23.2 ^23.3 Van Gasteren 1990, p. 45.

[kilp2000-25] 24.0 ^24.1 ^24.2 ^24.3 ^24.4 Kilp, Knauer, Mikhalev 2000, p. 3.

[26] "Functional relation - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-06-13.

[27] "functional relation in nLab". ncatlab.org. Retrieved 2024-06-13.

[28] Schmidt 2010, p. 49.

[29] Kilp, Knauer, Mikhalev 2000, p. 4.

[30] Yao, Y.Y.; Wong, S.K.M. (1995). "Generalization of rough sets using relationships between attribute values" (PDF). Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33..

[31] Kunen, Kenneth (1980). Set theory: an introduction to independence proofs. North-Holland. p. 102. ISBN 0-444-85401-0. Zbl 0443.03021.

[32] Tarski, Alfred; Givant, Steven (1987). A formalization of set theory without variables. American Mathematical Society. p. 3. ISBN 0-8218-1041-3.

[Müller2012-33] M. E. Müller (2012). Relational Knowledge Discovery. Cambridge University Press. p. 22. ISBN 978-0-521-19021-3.

[PahlDamrath2001-p496-34] Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. Springer Science & Business Media. p. 496. ISBN 978-3-540-67995-0.

[35] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006), A Transition to Advanced Mathematics (6th ed.), Brooks/Cole, p. 160, ISBN 0-534-39900-2

[36] Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography, Springer-Verlag, p. 158.

[37] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: School of Mathematics – Physics Charles University. p. 1. Archived from the original (PDF) on 2013-11-02. Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".

[38] Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, ISBN 0-12-597680-1, p. 4

[39] R. Berghammer & M. Winter (2013) "Decomposition of relations on concept lattices", Fundamenta Informaticae 126(1): 37–82 doi:10.3233/FI-2013-871

[40] Ki-Hang Kim (1982) Boolean Matrix Theory and Applications, page 37, Marcel Dekker ISBN 0-8247-1788-0

[41] Ali Jaoua, Rehab Duwairi, Samir Elloumi, and Sadok Ben Yahia (2009) "Data mining, reasoning and incremental information retrieval through non enlargeable rectangular relation coverage", pages 199 to 210 in Relations and Kleene algebras in computer science, Lecture Notes in Computer Science 5827, Springer Template:Mr

[42] Riguet, Jacques (January 1950). "Quelques proprietes des relations difonctionelles". Comptes rendus (in French). 230: 1999–2000.

[Büchi1989-43] Julius Richard Büchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions. Springer Science & Business Media. pp. 35–37. ISBN 978-1-4613-8853-1.

[44] East, James; Vernitski, Alexei (February 2018). "Ranks of ideals in inverse semigroups of difunctional binary relations". Semigroup Forum. 96 (1): 21–30. arXiv:1612.04935. doi:10.1007/s00233-017-9846-9. S2CID 54527913.

[BrinkKahl1997-45] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer Science & Business Media. p. 200. ISBN 978-3-211-82971-4.

[46] Ali Jaoua, Nadin Belkhiter, Habib Ounalli, and Theodore Moukam (1997) "Databases", pages 197–210 in Relational Methods in Computer Science, edited by Chris Brink, Wolfram Kahl, and Gunther Schmidt, Springer Science & Business Media ISBN 978-3-211-82971-4

[47] Gumm, H. P.; Zarrad, M. (2014). "Coalgebraic Simulations and Congruences". Coalgebraic Methods in Computer Science. Lecture Notes in Computer Science. 8446. p. 118. doi:10.1007/978-3-662-44124-4_7. ISBN 978-3-662-44123-7.

[48] J. Riguet (1951) "Les relations de Ferrers", Comptes Rendus 232: 1729,30

[Schmidt_p.77-49] Schmidt, Gunther; Ströhlein, Thomas (2012). [[[:Template:Google books]] Relations and Graphs: Discrete Mathematics for Computer Scientists] Check |url= value (help). Springer Science & Business Media. p. 77. ISBN 978-3-642-77968-8.

[50] Georg Aumann (1971). "Kontakt-Relationen". Sitzungsberichte der mathematisch-physikalischen Klasse der Bayerischen Akademie der Wissenschaften München. 1970 (II): 67–77.

[51] Anne K. Steiner (1970) Review:Kontakt-Relationen from Mathematical Reviews

[GS11-52] 51.0 ^51.1 ^51.2 Gunther Schmidt (2011) Relational Mathematics, pages 211−15, Cambridge University Press ISBN 978-0-521-76268-7

[53] In this context, the symbol Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \backslash} does not mean "set difference".

[54] Viktor Wagner (1953) "The theory of generalised heaps and generalised groups", Matematicheskii Sbornik 32(74): 545 to 632 Template:Mr

[55] C.D. Hollings & M.V. Lawson (2017) Wagner's Theory of Generalised Heaps, Springer books ISBN 978-3-319-63620-7 Template:Mr

[56] Christopher Hollings (2014) Mathematics across the Iron Curtain: a history of the algebraic theory of semigroups, page 265, History of Mathematics 41, American Mathematical Society ISBN 978-1-4704-1493-1

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Binary relation

Page version status

Contents

Definition

Operations

Union

Intersection

Composition

Converse

Complement

Restriction

Matrix representation

Examples

Types of binary relations

Sets versus classes

Homogeneous relation

Calculus of relations

Induced concept lattice

Particular relations

Difunctional

Ferrers type

Contact

Preorder R\R

Fringe of a relation

Mathematical heaps

See also

Notes

References

Bibliography

External links

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Binary relation

Definition

Operations

Union

Intersection

Composition

Converse

Complement

Restriction

Matrix representation

Examples

Types of binary relations

Sets versus classes

Homogeneous relation

Calculus of relations

Induced concept lattice

Particular relations

Difunctional

Ferrers type

Contact

Preorder R\R

Fringe of a relation

Mathematical heaps

See also

Notes

References

Bibliography

External links

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