REFLEXIVE RELATION

Reflexive relation is the one in which every element maps to itself. 

The rule for reflexive relation is given below. 

"Every element is related to itself"

Let R be a relation defined on the set A

If R is reflexive relation, 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 relation defined on  set A as

R  = {(1, 1), (2, 2), (3, 3)}

Verify R is reflexive. 

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, it is clear that every element of A is related to itself. 

So, R is reflexive relation. 

Solved Problems

Problem 1 : 

Let A  =  {2, 3, 7}, R be a relation defined on set as

R  =  {(2, 2), (3, 3), (2, 3) (3, 7)}

Determine whether R is reflexive relation.

Solution :

If the relation on set A is reflexive, every element of A has to be related to itself. 

In the set A, we find three elements. They are 1, 2 and 3.

So, we must have

2 has to be related to itself ------> (2, 2)

3 has to be related to itself ------> (3, 3)

7 has to be related to itself ------> (7, 7)

But, the ordered pair (7,7) is not in R. 

So, R is not reflexive. 

Problem 2 :

Let A  =  {1, 2, 3} and R be a relation defined on set A as

R  = {(1, 1), (2, 2), (3, 3), (1, 2)}

Determine whether R is reflexive relation.

Solution :

If the relation on set A is reflexive, every element of A has to be related to itself. 

In the set A, we find three elements. They are 1, 2 and 3.

So, we must have

1 has to be related to itself ------> (1, 1)

2 has to be related to itself ------> (2, 2)

3 has to be related to itself ------> (3, 3)

We have all the three ordered pairs (1, 1), (2, 2) and (3, 3) in R.  

So, R is reflexive. 

Problem 3 :

Let A  =  {1, 2, 3} and R be a relation defined on  set A as

R  = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}

Determine whether R is reflexive relation.

Solution :

If the relation on set A is reflexive, every element of A has to be related to itself. 

In the set A, we find three elements. They are 1, 2 and 3.

So, we must have

1 has to be related to itself ------> (1, 1)

2 has to be related to itself ------> (2, 2)

3 has to be related to itself ------> (3, 3)

We have all the three ordered pairs (1, 1), (2, 2) and (3, 3) in R.  

So, R is reflexive. 

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

Symmetric relation

Transitive relation

Equivalence relation

Identity relation

Inverse relation

Difference between reflexive and identity relation

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