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)}
**

####
**
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)}
**

####
**
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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