IDENTITY RELATION
Identity relation is the one in which every elements maps to itself only.
The rule for identity relation is given below.
"Every element is related to itself only"
Let R be a relation defined on the set A
If R is identity relation, then
R = {(a, a) / for all a ∈ A}
That is, every element of A has to be related to itself only.
In case, there is an ordered pair (a, b) in R, then R is not identity. Because the ordered pair (a, b) does not satisfy the above rule of identity relation.
Examples
Example 1 :
Let A = {1, 2, 3} and R be a relation defined on set A as
R = {(1, 1), (2, 2), (3, 3)}
Verify R is identity.
Solution :
In the set A, we find three elements. They are 1, 2 and 3.
When we look at the ordered pairs of R, we find the following associations.
(1, 1) -----> 1 is related to 1
(2, 2) -----> 2 is related to 2
(3, 3) -----> 3 is related to 3
In R, every element of A is related to itself and not to any other different element.
That is, every element of A is related to itself only.
So, R is identity.
Example 2 :
Let A = {1, 2, 3} and R be a relation defined on set A as
R = {(1, 1), (2, 2), (3, 3), (2, 3)}
Verify R is identity.
Solution :
In the set A, we find three elements. They are 1, 2 and 3.
When we look at the ordered pairs of R, we find the following associations.
(1, 1) -----> 1 is related to 1
(2, 2) -----> 2 is related to 2
(3, 3) -----> 3 is related to 3
(2, 3) -----> 2 is related to 3
In R, every element of A is related to itself and also the element "2" is related to a different element "3".
Here, we can not say that every element of A is related to itself only.
So, R is not identity.
Difference between Reflexive and Identity Relation
The two relations reflexive and identity appear, as if they were same.
But, there is a huge difference between them.
The difference between reflexive and identity relation can be described in simple words as given below.
Reflexive : Every element is related to itself
Identity : Every element is related to itself only
Let us consider an example to understand the difference between the two relations reflexive and 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.
That is,
Reflexive : Every element is related to itself
Identity : Every element is related to itself only
Related Topics
Difference between reflexive and identity relation
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