Check sibling questions

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

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.