# Empty relation

*R*on a set

*A*is said to be an empty relation or a void relation if

*R*=

*φ*.

# Universal relation

*R*from

*A*to

*B*is said to be the universal relation if

*R*=

*A*×

*B*.

# Identity relation

*R*on a set

*A*is said to be the identity relation on

*A*if

*R*= {(

*a*,

*b*):

*a*∈

*A*,

*b*∈

*A*and

*a*=

*b*}. Thus identity relation

*R*= {(

*a*,

*a*) : ∀

*a*∈

*A*}. Identity relation on set

*A*is also denoted by

*I*.

_{A}# Reflexive relation

*R*on a set

*A*is said to be reflexive if

*aRa*∀

*a*∈

*A*, i.e., (

*a*,

*a*) ∈

*R*∀

*a*∈

*A*.

Let *A* = {1, 2, 3}

Let *R*_{1} = {(1, 1), (2, 2), (3, 3)}

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

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

Here *R*_{1} and *R*_{2} are reflexive relations on *A*. *R*_{3} is not a reflexive relation on *A* as (3,3) ∉ *A*, i.e., 3 *R* 3.

**The identity relation is always a reflexive relation but a reflexive relation may or may not be the identity relation. In the examples given above**

*Note:**R*

_{1}is both reflexive and identity relation on

*A*but

*R*

_{2}is a reflexive relation on A but not an identity relation on

*A*.

# Symmetric relation

*R*on a set

*A*is said to be a symmetric relation if

*aRb*⇒

*bRa*(whenever

*aRb*, then

*bRa*), i.e., (

*a*,

*b*) ∈

*R*⇒ (

*b*,

*a)*∈

*R*, ∀

*a*,

*b*∈

*A*.

Let *A* = {1,2,3}

Let *R*_{1} = {(1,2), (2,1)};

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

Here *R*_{1} and *R*_{2} are symmetric relations on *A*. *R*_{3} = {(2, 3), (3, 1), (1, 3)};

*R*_{3} is not a symmetric relation on *A* because (2, 3) ∈ *R*_{3} and (3, 2) ∉ *R*_{3}.

# Antisymmetric relation

*R*on a set

*A*is said to be antisymmetric if

*aRb*and

*bRa*⇒

_{ }

*a*=

*b*, i.e., (

*a*,

*b*) ∈

*R*and (

*b*,

*a*) ∈

*R*⇒

*a*=

*b*.

*R*is antisymmetric if

*a*≠

*b*, then both (

*a*,

*b*) and (

*b*,

*a*) cannot belong to

*R*. One of them may belong to

*R*.

Let *A* = {1, 2, 3}

Let *R*_{1} = {(1, 2), (1, 3), (1, 1)}

*R*_{2} = {(1, 2)}; *R*_{3} = {(1, 2), (2, 1)}

Here *R*_{1} and *R*_{2} are antisymmetric relations on *A* but *R*_{3} is not antisymmetric relation on *A*.

**A relation which is not symmetric is not necessarily antisymmetric.**

*Note:*Let *A* = {1, 2, 3}

Let *R* = {(1, 2), (2, 1), (2, 3)}; here *R* is not symmetric.

Also *R* is not antisymmetric because (1, 2) ∈ *R* and (2, 1) ∈ *R* but 1 ≠ 2.

# Transitive relation

*R*on a set

*A*is said to be a transitive relation if

*aRb*and

*bRc*⇒

*aRc*, i.e., (

*a*,

*b*) ∈

*R*and (

*b*,

*c*) ∈

*R*⇒ (

*a*,

*c*) ∈

*R*.

Let *A* = {1, 2, 3}; *R* = {(1, 2), (2, 3), (1, 3), (3, 2)}

*R*_{1 }= {(2, 3)}; *R*_{2} = {(1, 3), (3, 2), (1, 2)}

Then *R* is not transitive relation on *A* because (2, 3) ∈ *R* and (3, 2) ∈ *R* but (2, 2) ∉ *R*, i.e., 2. *R* _{2}. Again *R*_{1} is transitive relation on *A* because (2, 3) ∈ *R*_{1} and (3, *x*) ∈ *R*_{1} ⇒ (2, *x*) ∈ *R*_{1} cannot be considered false because there is no (3, *x*) ∈ *R*_{1} so no need to test whether (2, *x*) ∈ *R*_{1} or not. Finally *R*_{2} is a transitive relation.

# Inverse relation

*R*⊆

*A*

*x*

*B*be a relation from

*A*to

*B*. Then the inverse relation of

*R*denoted by

*R*

^{-1 }is a relation from

*B*to

*A*defined by

*R*

^{-1}= {(

*b*,

*a*) : (

*a*,

*b*) ∈

*R*}.

*a*,

*b*) ∈

*R*⇔ (

*b*,

*a*) ∈

*R*

^{-1}∀

*a*∈

*A*,

*b*∈

*B*

*R*

^{-1}= Range

*R*and Range

*R*

^{-1}= Domain

*R*.

# Equivalence relation

*A*be a non-empty set, then a relation

*R*on

*A*is said to be an equivalence relation if

*R*is reflexive, i.e.,*aR**a*∀*a*∈*A**a*,*a*) ∈*R*, ∀*a*∈*A**R*is symmetric, i.e., if*aRb*then*bRa**, (**a*,*b*) ∈*R*⇒ (*b*,*a*) ∈*R*, ∀*a*,*b*∉*A**R*is transitive, i.e., if*aRb*and*bRc*, then*aRc**, (*and*a*,*b*) ∈*R**(**b*,*c*) ∈*R*⇒ (*a*,*c*) ∈*R*

- Let A = {1, 2, 3}. Let a relation R be defined on
*A*as*R*= {(1, 2), (1, 1), (2, 1), (2, 2), (3, 3)}. Then*R*is reflexive, symmetric and transitive. So*R*is an equivalence relation on*A*. - Let
*A*be the set of all members in a family having male and female children and*R*be the relation “is the brother of” on*A*. Then*R*is not an equivalence relation on*A*. Here*R*is neither reflexive nor symmetric but it is transitive.