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Binary Relations

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A binary relation is a collection of Sets between two Sets ‘M’ and ‘N’ which is the subset of M * N, or we can say that it is a Set of Ordered Pair m, n Є M*N. Here set of ‘M’ and ‘N’ are known as Domain.
Binary Relations can be used in various streams of Math like more than and equals to, division in arithmetic, in Geometry congruency and in many other fields such as in graph theory, in linear Algebra, etc. in binary relation the function comes under its special branch, the requirement of binary relation is also felt in computer science.
There are also kinds of properties which the Binary relations pursue. They are:
Reflexive: such as 'greater than' and 'equal to' is a reflexive relation but only greater than is does not hold this relation.
Co-reflexive: It explains that 'odd' and 'equal to' will be the case of co-reflexive relation.
Symmetric: We can easily define this property by quoting an example like taking the set say of (r, s) which must relate to (s, r), this defines the property of the symmetric relation.
Transitive: To explain this property we will consider the example if n = m and m = o then in brief we can also write it as n = o this is format the transitive property.
There are few of the common examples of binary relation: Lets consider an order relation which has strict orders in themselves they are “greater than”, “greater than and equal to”, “less than or equal to”, “less than”, “is a subset of”. Now we will take the example of equivalence binary relation few of its examples are “equality”, “parallel to”. There are also few relations whose examples can be taken into consideration like dependency relation (a finite, symmetric), independency relation (asymmetric, irreflexive relation).

Binary relation can be defined as a relation on Set P which is a set of Ordered Pair of elements or we can say that binary relation is a sub set of Cartesian product that is (P2 = P x P). Binary relation among two Sets P and Q is a subset of P x Q. Binary relation is also known as dyadic relation and 2 – place relation. For example: It is a relation between set of prime Numbers Q and set of integers R in such a way that every Prime number Q is related with every Integer R which is multiple of Q. Now we will see some Binary Relations properties. Properties of binary relations are mentioned below:

Reflexive relation
Irreflexive relation
Symmetric relation
Transitive relation


Let’s have small introduction about above mentioned properties of binary relation.
Reflexive relation: Reflexive relation can be defined as a relation R on set U which is said to be reflexive relation if and only if < u, u > є R for each element of u of set U. In other words we can say if element of set U are related to itself. For example: ‘b’ is related to itself.

Irreflexive relation: it is defined as a relation R on a set U which is said to be irreflexive relation if and only if < u, u > does not belong to relation R for each element of u of set U.
Symmetric relation: It is defined as a relation R on set U which is said to be symmetric relation if and only if for any u and v element in U, whenever < u, v > є R, <v, u> є R.
Transitive relation: It is defined as relation R on a set U which is said to be transitive relation if and only if for any u, v and w in set U, whenever < u, v > є R, and < v, w > є R, < u, w > є R relation. This is all about the properties of binary relations.
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