# Types of Relations on the Set A

Let A be a set and R is a relation on A, i.e. R ⊆ A × A. Then we define

**Void Relation :**If R = φ , then R is called a void relation on A.**Universal Relation :**If R = A×A, then R is called an universal relation on A.**Identity Relation :**A relation R is defined as an identity relation if R = {(a, a) : a ∈ A}. Thus in an identity relation on A, every element of A is related to itself only. Identity relation on A is also denoted by I_{A}. Thus

I_{A}= {(a, a) : a ∈ A}

If A = {1, 2, 3}, then I*Example*:_{A}= {(1, 1), (2, 2), (3,3)}**Reflexive Relation :**A relation R is said to be a reflexive relation on A if every element of A is related to itself.

Thus R is reflexive ⇔ (a,a) ∈R, i.e. aRaa∈A

[The symbolis read as “for every element”]

**Example :**Let A = {1, 2, 3} be a set.

Then R = {(1, 1), (2, 2), (3, 3), (1, 3), (2, 1)} is a reflexive relation on A.**Symmetric Relation :**A relation R on a set A is defined as a symmetric relation if (a,b) ∈R ⇒ (b, a) ∈R That is, aRb ⇒ bRa, where a, b∈A.

**Example :**Let A = {1, 2, 3, 4} and let R_{1}be relation onA given by R_{1}= {(1, 3), (1, 4), (3, 1), (2, 2), (4, 1)} is a symmetric relation on A.**Transitive relation :**A relation R on a set A is defined as a transitive relation if (a,b) ∈R and (b,c) ∈R ⇒ (a,c) ∈R

That is, aRb and bRc ⇒ aRc, where a, b, c, ∈A.

**Example :**Let L be the set of all straight line in a plane. Then the relation ‘is parallel to on L is a transitive relation, because of any_{1},_{2},_{3}, ∈ L.

_{1}||_{2}and_{2}||_{3 }⇒_{1}||_{3}**Antisymmetric Relation :**A relation R on a set A is antisymmetric if (a,b) ∈R and (b,a) ∈R ⇒ a = b

If (a, b) ∈ R and (b, a) ∉R, then still R is an antisymmetric relation.

**Example :**Let R be a relation on the set N of natural numbers defined by

xRY ⇔ ‘x divides y’ for all x, y ∈ N

This relation is a antisymmetric relation on N.

Since for any two numbers a, b ∈ N.

a/b and b/a ⇒ a = b i.e. aRb and bRa ⇒ a = b