# Binary Relation

A binary relation is a finitary relation $\displaystyle{ R(D,B) }$ with two relation arguments (it pairs each Set Member of Relation Domain Set $\displaystyle{ D }$ with at least one Set Member of relation range set B).

## References

### 2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/binary_relation Retrieved:2016-1-18.
• In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of . The terms correspondence', dyadic relation and 2-place relation are synonyms for binary relation.

An example is the “divides” relation between the set of prime numbers P and the set of integers Z, in which every prime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). In this relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and the prime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.

Binary relations are used in many branches of mathematics to model concepts like “is greater than", “is equal to", and "divides" in arithmetic, “is congruent to” in geometry, "is adjacent to" in graph theory, "is orthogonal to" in linear algebra and many more. The concept of function is defined as a special kind of binary relation. Binary relations are also heavily used in computer science.

A binary relation is the special case of a n-ary relation R ⊆ A1 × … × An, that is, a set of n-tuples where the jth component of each n-tuple is taken from the jth domain Aj of the relation. An example for a ternary relation on Z×Z×Z is "lies between … and ...", containing e.g. the triples , , and .

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.

### 2009

• http://planetmath.org/encyclopedia/NullaryRelation.html
• Basically, a binary relation $R$ involves objects coming from two collections $A,B$, where the objects are paired up so that each pair consists of an object from $A$, and an object from $B$.
• More formally, a binary relation is a subset $R$ of the Cartesian product of two sets $A$ and $B$. One may write
• $\displaystyle a\: R\: b$
• to indicate that the ordered pair $(a, b)$ is an element of $R$. A subset of $A\times A$ is simply called a binary relation on $A$. If $R$ is a binary relation on $A$, then we write
• $\displaystyle a_1 \: R \: a_2 \: R \: a_3 \: … \: a_{n-1} \: R \: a_n$

to mean $a_1 \: R\: a_2, a_2\} R\: a_3, \ldots,$ and $a_{n-1}\: R \: a_n$.

• Given a binary relation $R\subseteq A\times B$, the domain $\operatorname{dom}(R)$ of $R$ is the set of elements in $A$ forming parts of the pairs in $R$. In other words,
• $\displaystyle \operatorname{dom}(R):=\lbrace x\in A\mid (x,y)\in R$ for some $\displaystyle y \in B \rbrace$
• and the range $\operatorname{ran}(R)$ of $R$ is the set of parts of pairs of $R$ coming from $B$:
• $\displaystyle \operatorname{ran}(R):=\lbrace y\in B\mid (x,y)\in R$ for some $\displaystyle x\in A \rbrace.$

### 2002

• (Roth & Yih, 2002) ⇒ Dan Roth, and Wen-tau Yih. (2002). “Probabilistic Reasoning for Entity & Relation Recognition.” In: Proceedings of the 20th International Conference on Computational Linguistics (COLING 2002).
• Definition 2.2 (Relation) A (binary) relation Ri,j = (Ei;Ej) represents the relation between Ei and Ej, where Ei is the first argument and Ej is the second. In addition, Rij can range over a set of entity types CR.
• Figure2 : Dole ’s wife, Elizabeth, is a native of Salisbury, N.C.
• Example 2.2 In the sentence given in Figure 2, there are six relations between the entities: R1,2 = (“Dole”, “Elizabeth”), R2,1 = (“Elizabeth”, “Dole”), R1,3 = (“Dole”, “Salisbury, N.C.”), R3,1 = (“Salisbury, N.C.”, “Dole”), R2,3 = (“Elizabeth”, “Salisbury, N.C.”), and R3,2 = (“Salisbury, N.C.”, “Elizabeth”)