Types of Relations

Sets and relation are interconnected with each other. The relation defines the relation between two given
sets. If there are two sets available, then to check if there is any connection between the two sets,
we use relations For example, An empty relation denotes none of the elements in the two sets is same.

There are 8 main types of relations which include:

1. Empty Relation

An empty relation (or void relation) is one in which there is no relation between any
elements of a set. For example, if set A = {1, 2, 3} then, one of the void relations can be R = {x, y}
where, |x – y| = 8. For empty relation,

R = φ ⊂ A × A

2. Universal Relation

A universal (or full relation) is a type of relation in which every element of a
set is related to each other. Consider set A = {a, b, c}. Now one of the universal
relations will be R = {x, y} where, |x – y| ≥ 0. For universal relation,

R = A × A

3. Identity Relation

In an identity relation, every element of a set is related to itself only.
For example,in a set A = {a, b, c}, the identity relation will be
I = {a, a}, {b, b}, {c, c}. For identity relation,

I = {(a, a), a ∈ A}

4. Inverse Relation

Inverse relation is seen when a set has elements which are inverse pairs of
another set. For example if set A = {(a, b), (c, d)}, then inverse relation will be
R^-1 = {(b, a), (d, c)}. So, for an inverse relation,

R^-1 = {(b, a): (a, b) ∈ R}

5. Reflexive Relation

In a reflexive relation, every element maps to itself. For example, consider a
set A = {1, 2,}. Now an example of reflexive relation will be
R = {(1, 1), (2, 2), (1, 2), (2, 1)}. The reflexive relation is given by-

(a, a) ∈ R

6. Symmetric Relation

In a symmetric relation, if a=b is true then b=a is also true. In other words,
a relation R is symmetric only if (b, a) ∈ R is true when (a,b) ∈ R. An example of symmetric
relation will be R = {(1, 2), (2, 1)} for a set A = {1, 2}. So, for a symmetric relation,

aRb ⇒ bRa, ∀ a, b ∈ A

7. Transitive Relation

For transitive relation, if (x, y) ∈ R, (y, z) ∈ R, then (x, z) ∈ R. For a transitive relation,

aRb and bRc ⇒ aRc ∀ a, b, c ∈ A

8. Equivalence Relation

If a relation is reflexive, symmetric and transitive at the same time it is known as an equivalence relation.

Representation of Types of Relations