To prove relation reflexive, transitive, symmetric and equivalent

Chapter 1 Class 12 Relation and Functions
Concept wise

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)}

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, 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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