# Binary Relation

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

**AKA:**R, One Set Binary Relation.**Context:**- It can range from being a Binary Typed Relation to being an Binary Untyped Relation.
- It can range from being a Binary Reflexive Relation to being an Binary Irreflexive Relation.
- It can range from being a Binary Symmetric Relation to being an Binary Antisymmetric Relation.
- It can range from being a Binary Transitive Relation to being an Binary Intransitive Relation.
- It can range from being a One-to-One Relation, One-to-Many Relation, or a Many-to-Many Relation.
- It can range from being a Binary Semantic Relation to being a Binary Syntactic Relation.
- It can range from being a Total Relation or a Non-Total Relation (to some given set).
- It can be a Surjective Relation, Injective Relation.
- It can be in an Inverse Relation with another Binary Relation (with switched parameters).
- It can be:
- a Binary Set Relation, restricted to sets.
- …

**Example:**- The Subsumption Relation (IsA) is a binary relation. E.g. The tree of life relation always compares two classes of animals.
- The Parthood Relation (PartOf) is a binary relation.
- The LessThan relation on integers.

**Counter-Example(s):**- any Unary Relation.
- any N-ary Relation.

**See:**Mapping, Russell's Paradox, Ordered Pair, Divisibility, Inequality (Mathematics).

## 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*AA and^{2}= . More generally, a binary relation between two sets*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*⊆*A*_{1}× … ×*A*_{n}, that is, a set of*n*-tuples where the*j*th component of each*n*-tuple is taken from the*j*th domain*A*_{j}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.

- In mathematics, a

### 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 $\displaystyle x\in A \rbrace.$

- 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,

### 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 R*_{i,j}= (E_{i};E_{j}) represents the relation between E_{i}and E_{j}, where E_{i}is the first argument and E_{j}is the second. In addition, R_{ij}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: R
_{1,2}= (“Dole”, “Elizabeth”), R_{2,1}= (“Elizabeth”, “Dole”), R_{1,3}= (“Dole”, “Salisbury, N.C.”), R_{3,1}= (“Salisbury, N.C.”, “Dole”), R_{2,3}= (“Elizabeth”, “Salisbury, N.C.”), and R_{3,2}= (“Salisbury, N.C.”, “Elizabeth”)