# Types of Relation

## Types of Relation:

(1) Identity Realtion- Relation R on a set A defined as {(a, a) ∀ a ∈ A} is said to be an identity relation on set A and is denoted by IA.

For example- If A = {1, 2, 3} then IA = {(1, 1) (2, 2) (3, 3)}

(2) Reflexive Relation- A relation R on a set A is said to be reflexive if every element of A is related to itself i.e. R is reflexive ⇔ (iff) aRa ∀ a ∈ A. Considering the set A = {a, b, c, d}, if any one of the ordered pairs (a, a) (b, b) (c, c) (d, d) is absent in the relation R, then the relation cannot be reflexive.

For example- If A = {1, 2, 3} and R = {(1, 1) (1, 2) (2, 2) (3, 2) (3, 3)} then R is reflexive.

The universal relation on a non-void set is reflexive. The identity relation on a non-void set is always reflexive but a reflexive relation is not necessarily an identity relation.

Let A = set of lines in a plane and then the relation R is defined by a1Ra2 iff a1 || a2 is reflexive because every line is parallel to itself.

(3) Symmetric Relation- A relation R on a set A is said to be symmetric iff (a, b) ∈ R ⇒ (b, a) ∈ R.

For example- If A = {1, 2, 3} and R = {(1, 1) (1, 2) (2, 1) (3, 3)} then R is symmetric.

Let A = set of lines in a plane, then the relation R is defined by a1Ra2 iff a1 || a2 is symmetric because if a1 || a2 ⇒ a2 || a1.

The identity and the universal relations on a non-empty set A are symmetric relations on A.

(4) Transitive Relation- A relation R on a set A is said to be transitive iff (a, b) ∈ R, (b, c) ∈ R ⇒ (a, c) ∈ R. Let A be the set of natural numbers and a relation R on a set A is defined by aRb iff a divides b is transitive because if a divides b and b divides c then a divides c.

The identity and the universal relations on a non-empty set are transitive.

(5) Equivalence Relation- A relation R on a set A is said to be an equivalence relation if-

• R is reflexive.
• R is symmetric.
• R is transitive.