Problems Based on Types of Relation for Class 12 Maths

Types of Relations

1. 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.
2. 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.
3. 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.
4. 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 = b²}. 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 ⇔ 2x² – 3xy + y² = 0 i.e., by R = {( x, y) : x,y ∈ N and 2x² – 3xy + y² = 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.