Binary relation is defined as a relation on set ‘J’ that is a set of ordered pair of elements or in other words binary relation is a sub set of Cartesian product that is (J2 = J x J). If there are two sets ‘J’ and ‘K’ then they are subset of J x K. It is also known as dyadic relation and 2 – place relation. For example: It is a relation between a set of prime numbers ‘K’ and set of integers ‘I’ in such a way that every Prime number ‘K’ is related with every integer ‘I’ which is multiple of ‘K’. Now we will discuss some properties of this relations. All the properties of this relation are given below:
Now we will understand these properties with help of an example.
For example: If there are three relations R1, R2, R3 defined on set A = j, k, l as follows?
R1 = (j, j), (j, k), (j, l), (k, k), (k, l), (l, j), (l, k), (l, l);
R2 = (j, k), (k, j), (j, l), (l, j),
R3 = (j, k), (k, l), (l, j)
Then show that the relations R1, R2, R3 are reflexive, symmetric and transitive relation?
Solution: First we will see the case of reflexive relation. Reflexive relation is a relation ‘R’ on set ‘A’ if each element of ‘A’ is related to itself. For example:
Reflexive: - (j, j), (k, k), (l, l) ∊R1 then relation R1 is said to be reflexive.
Symmetric: - In symmetric relation (j, k) ∊ R1 but (k, l) ∉R1. So relation R1is not symmetric on relation ‘S’.
Transitive: - In transitive relation (k, l) ∊ R1 and (l, j) ∊ R1 but (k, j) ∉ R1. So relation ‘R1’ is not transitive on relation ‘A’.
2. For R2 relation we check.
Reflexive: - (j, j), (k, k), (l, l) are not present in R2. So we can say that relation R2 is not reflexive.
Symmetric: In symmetric relation ordered pair is obtained by interchanging the components of ordered pairs in R2 is also in R2. So relation R1 is symmetric on relation on ‘A’.
Transitive: - In transitive relation (j, k) ∊ R2 and (k, j) ∊ R2 but (j, j) ∉ R2. So relation R2 is not transitive on relation ‘A’.
(3) Now we will see the case of relation R3. In this also first we will check the reflexive relation.
Reflexive: - In relation R3, (j, j), (k, k), (r, r) are not present in R3. So we can say that relation R3 is not reflexive. Now will check the condition for symmetric relation,
Symmetric: - In that case symmetric relation (k, l) ∊R3 but (l, j) ∉R3. So relation R3 is symmetric on relation on ‘A’. Now discuss the transitive relation.
Transitive: - In transitive relation (j, k) ∊ R3 and (k, l) ∊ R3 but (j, l) ∉ R3. So relation R3 is not transitive on relation ‘A’.
This is all about basic properties of binary relation.
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