**Types of Relations**

**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., (a, a) ∈ R for all a ∈ A.**Symmetric Relation:**A relation R on a set A is said to be symmetric if (a, b) ∈ R implies (b, a) ∈ R, for all a, b ∈ A.**Transitive Relation:**A relation R on a set A is said to be transitive if (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R, for all a, b, c ∈ A.**Equivalence Relation:**A relation R on a set A is said to be an equivalence relation if it is reflexive, symmetric, and transitive. Equivalence relations are used to partition a set into equivalence classes.

## Problems Based on Types of Relation for Class 12 Maths

Here we are providing problems based on types of relations for class 12 maths.

Q 1. Let A = {3, 5} and {7,11}.

Let R = {(a, b) : a ∈ A, b ∈ B, a – b is odd}.

Show that R is an empty relation from A into B.

Q 2. Let A = {3, 5}, B ={7,11}.

Let R = {(a, b): a ∈ A , b ∈ B, a – b is even}.

Show that R is an universal relation from a A to B.

Q 3. Let A = {1, 2, 3} and R = {(a, b) :a ,b ∈ A, a divides b and b divides a}

Show that R is an identity relation on A.

Q 4. Let R be a relation from N into N defined by

R = {(a, b): a, ∈ Î N and a = b^{2}}. Are the following true ?

- (a,b) ∈ R, for all a ∈ N.
- (a,b) ∈ R implies (b, a) ∈ R.
- (a, b) ∈ R, (a,c) ∈ R implies (a, c) ∈ R ?

Justify your answer in each case.

Q 5. Prove that the relation R defined on the set N of natural numbers by xRy ⇔ 2x^{2} – 3xy + y^{2 }= 0 i.e., by R = {( x, y) : x,y ∈ N and 2x^{2} – 3xy + y^{2 }= 0} is not symmetric but it is reflexive.

Q 6. Let N be the set of natural numbers and relation R on N be defined as xRy ⇔ x divides y

I,e ., as R = { x,y ) : x , y ∈ N and x divides y }

Examine whether R is reflexive, symmetric, antisymmetric or transitive.

Q 7. Let S be any nonempty set and P(S) be its power set. We define a relation R on P (S) by ARB to mean A ⊆ B ; A ⊆ B ; i.e,. R = {(A, B) : A ⊆ B}

Examine whether R is (i) reflexive (ii) symmetric (iii) antisymmetric (iv) transitive.

Q 8. Prove that a relation R on a set A is

- Reflexive ⇔ I
_{A}⊆ R, where I_{A }= {(x,x) : x ∈ A}. - Symmetric ⇔ R
^{-1}= R.

Q 9. Give example of relation which are

_{ }Neither reflexive nor symmetric nor transitive.- Symmetric and reflexive but not transitive.
- Reflexive and transitive but not symmetric.

## Show Answer

Q,4.

(i) not true

(ii) not true

(iii) does not imply

(iv) not true

Q.6.

R is transitive

Q.7.

(i) R is not symmetric

(ii) R is antisymmetric

(iii) R is transitive relation

Q.9.

(i) R = {1, 2, 3}

(ii) R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}

(iii) R = {(1, 1), (2, 2), (3, 3), (1, 2)}