For a relation R in set A

 

Reflexive

Relation is reflexive

If (a, a) ∈ R for every a ∈ A

 

Symmetric

Relation is symmetric,

If (a, b) ∈ R, then (b, a) ∈ R

 

Transitive

Relation is transitive,

If (a, b) ∈ R & (b, c) ∈ R, then (a, c) ∈ R

 

If relation is reflexive, symmetric and transitive,

it is an equivalence relation .


Let’s take an example.

 

Let us define Relation R on Set A = {1, 2, 3}

We will check reflexive, symmetric and transitive

 

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

Check Reflexive

If the relation is reflexive,

then (a, a) ∈ R for every a ∈ {1,2,3}

 

Since (1, 1) ∈ R ,(2, 2) ∈ R & (3, 3) ∈ R

∴ R is reflexive

 

Check symmetric

To check whether symmetric or not,

If (a, b) ∈ R, then (b, a) ∈ R

 

Here (1, 2) ∈ R , but (2, 1) ∉ R

∴ R is not symmetric

 

Check transitive

To check whether transitive or not,

If (a , b ) ∈ R  & (b , c ) ∈ R ,  then (a , c ) ∈ R

 

Here, (1, 2) ∈ R and (2, 3) ∈ R and (1, 3) ∈ R

∴  R is transitive

 

Hence, R is reflexive and transitive but not symmetric

 


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

 

View Answer

 


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

Check Reflexive

If the relation is reflexive,

then (a, a) ∈ R for every a ∈ {1,2,3}

 

Since (1, 1) ∈ R ,(2, 2) ∈ R & (3, 3) ∈ R

∴ R is reflexive

 

Check symmetric

To check whether symmetric or not,

If (a, b) ∈ R, then (b, a) ∈ R

 

Here (1, 2) ∈ R , but (2, 1) ∉ R

∴ R is not symmetric

 

Check transitive

To check whether transitive or not,

If (a , b ) ∈ R  & (b , c ) ∈ R ,  then (a , c ) ∈ R

 

Here, (1, 2) ∈ R and (2, 2) ∈ R and (1, 2) ∈ R

∴  R is transitive

 

Hence, R is reflexive and transitive but not symmetric

 


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

 

View Answer

 


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

Check Reflexive

If the relation is reflexive,

then (a, a) ∈ R for every a ∈ {1,2,3}

 

Since (1, 1) ∈ R but (2, 2) ∉ R & (3, 3) ∉ R

∴ R is not reflexive

 

Check symmetric

To check whether symmetric or not,

If (a, b) ∈ R, then (b, a) ∈ R

 

Here (1, 2) ∈ R , and (2, 1) ∈ R

∴ R is  symmetric

 

Check transitive

To check whether transitive or not,

If (a , b ) ∈ R  & (b , c ) ∈ R ,  then (a , c ) ∈ R

 

Here, (1, 2) ∈ R and (2, 1) ∈ R and (1, 1) ∈ R

∴  R is transitive

 

Hence, R is symmetric and transitive but not reflexive

  1. Chapter 1 Class 12 Relation and Functions
  2. Concept wise
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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.