IDENTITY RELATION
About "Identity relation"
The rule for identity relation is given below.
" Every element is related to itself only "
Let R be an identit-relation defined on the set A and
a ∈ A, then R = { (a, a) / for all a ∈ A }
That is, every element of A has to be related to itself only.
Example :
Let A = {1, 2, 3 } and R be an identity-relation defined on set A.
Then, R = { (1, 1), (2, 2), (3, 3) }
Here, every element of A is related to it self and not to any other different element.
Apart from the identity relation, we have some other relations on sets.
Let us discuss the other different types of relations here:
The other different types of relations on sets are
(i) Reflexive relation
(ii) Symmetric relation
(iii) Transitive relation
(iv) Equivalence relation
(v) Inverse relation
Let us discuss the above different types of relations in detail.
Reflexive relation
The rule for reflexive relation is given below.
" Every element is related to itself "
Let R be a reflexive relation defined on the set A and
a ∈ A, then R = { (a, a) / for all a ∈ A }
That is, every element of A has to be related to itself.
Example :
Let A = {1, 2, 3 } and R be a reflexive relation defined on set A.
Then, R = { (1, 1), (2, 2), (3, 3) }
Symmetric relation
Let R be a symmetric relation defined on the set A and
a, b ∈ A, then R = { (a, b), (b, a) / for all a, b ∈ A }
That is, if "a" is related to "b", then "b" has to be related to "a" for all "a" and "b" belonging to A.
Example :
Let A be the set of two male children in a family, R be a relation defined on set A and R = "is brother of". Verify whether R is symmetric.
Let a, b ∈ A.
If "a" is brother of "b", then "b" has to be brother of "a".
Clearly, R = { (a, b), (b, a) }
Hence, R is symmetric.
Transitive relation
Let R be a transitive relation defined on the set A and a, b, c ∈ A,
then R = { (a, b), (b, c), (a, c) / for all a,b,c ∈ A }
That is,
If "a" is related to "b" and "b" is related to "c", then "a" has to be related to "c".
Example :
Let A = { 1, 2, 3 }, R be a relation defined on set A as "is less than" and R = { (1, 2), (2, 3), (1,3) } Verify R is transitive.
From the given set A, let a = 1, b = 2 and c = 3.
Then, we have
(a, b ) = (1, 2) --> 1 is less than 2
(b, c ) = (2, 3) --> 2 is less than 3
(a, c ) = (1, 3) --> 1 is less than 3
That is, if 1 is less than 2 and 2 is less than 3, then 1 is less than 3
Clearly, the above points prove that R is transitive.
Important note :
For a particular ordered pair in R, if we have (a, b) and we don't have (b, c), then we don't have to check transitive for that ordered pair.
So, we have to check transitive, only if we find both (a, b) and (b, c) in R.
Equivalence relation
As we have seen rules for reflexive, symmetric and transitive relations, we don't have any specific rule for equivalence relation.
Let R be a relation defined on a set A.
If the three relations reflexive, symmetric and transitive hold in R, then it is an equivalence relation.
To verify equivalence relation, we have to check whether the three relations reflexive, symmetric and transitive hold.
Inverse relation
Let R be a relation defined on the set A, a, b ∈ A and R = { (a, b) }, then the inverse relation of R is
R⁻¹ = { (b, a) }
In the given relation, if "a" is related to "b", then in the inverse relation "b" will be related to "a".
Example :
Let R = { (1, 2), (2, 3), (7, 5) }
Then R⁻¹ = { (2, 1), (3, 2), (5, 7) }
Difference between reflexive and identity relation
Reflexive = " Every element is related to itself "
Identity = " Every element is related to itself only "
Let us consider an example to have better understanding of the difference between the two relations. (Reflexive vs Identity)
Let A = {1, 2, 3}
Let R₁ and R₂ be two relations defined on set A such that
R₁ = { (1,1) , (2,2) , (3,3) , (1,2) }
R₂ = { (1,1) , (2,2) , (3,3) }
When we look at R₁ , every element of A is related to itself and also, the element "1" is related to a different element "2".
More details about R₁
(i) "1" is related to "1", "2" is related to "2" and "3" is related to "3"
(ii) Apart from "1" is related to "1", "1" is also related to "2"
Here we can not say that "1" is related to "1" only. Because "1" is related to "2" also.
This is the point which makes the reflexive relation to be different from identity relation.
Hence R₁ is reflexive relation
When we look at R₂, every element of A is related to it self and no element of A is related to any different element other than the same element.
More details about R₂
(i) "1" is related to "1", "2" is related to "2" and "3" is related to "3"
(ii) "1" is related to "1" and it is not related to any different element.
The same thing happened to "2" and "3".
(iii) From the second point, it is very clear that every element of R is related to itself only. No element is related to any different element
This is the point which makes identity relation to be different from reflexive relation.
Hence R₂ is identity relation
Important terms related to relations
If a relation does mapping from the set A to set B, then we can define the following terms.
Domain : Set A
Co domain : Set B
Range : Elements of B involved in mapping.
Let R and R⁻¹ be the two relations which are inverse to each other.
Then we have,
Domain (R) = Range (R⁻¹)
Range (R) = Domain (R⁻¹)
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