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The RD Sharma Solutions for Class 12 Maths are the best to buy. The chapters and exercises in RD Sharma Class 12 are what a student might find difficult to understand. RD Sharma Class 12th Exercise 1.2 Solution will make it easy for them. Students can now solve every problem of Relation with the help of these solutions.

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The questions and answers are formulated in a manner that the students easily understand. RD Sharma Solution Class 12th Chapter 1 Exercise 1.2 solutions build the foundation of students on Mathematics. The students can now learn the concepts without any difficulty**.**

Relation Exercise 1.2 Question 1

Answer: R is an equivalence relation on Z.Hint: To prove equivalence relation it is necessary that the given relation should be reflexive, symmetric and transitive.

Explanation:

Let us check the properties on R.

Reflexivity:

Let a be an arbitrary element of R.

Then

So, R is reflexive on Z.

Symmetry:

Here,

So, R is symmetric on Z.

Transitivity:

Here,

So, R is symmetric on Z.

Adding eq. (i) and (ii)

Here,

So, R is transitive on Z

Therefore, R is reflective, symmetric and transitive.

Hence, R is an equivalence relation on Z.

Relation Exercise1.2 Question 2

Answer: R is an equivalence relation on Z.Hint: To prove equivalence relation it is necessary that the given relation should be reflexive, symmetric and transitive.

Explanation:

Let us check these properties on R.

Reflexivity:

Let a be an arbitrary element of the set Z

So, R is reflexive on Z.

Symmetry:

So, R is symmetric on Z

Transitivity:

Adding eq. (i) and (ii)

So, R is transitive on Z

Therefore, R is reflexive, symmetric and transitive.

Hence, R is an equivalence relation on Z.

Relation Exercise 1 Question 3

Answer:R is an equivalence relation on Z.Hint: To prove equivalence relation it is necessary that the given relation should be reflexive,symmetric and transitive..

Explanation:

Let us check these properties on R.

Reflexivity:

Let a be an arbitrary element of the set Z

So, R is reflexive on Z.

Symmetry:

So, R is symmetric on Z

Transitivity:

Adding eq. (i) and (ii)

(

Therefore R is reflexive, symmetric and transitive.

Hence, R is an equivalence relation on Z.

Relation Exercise1.2 Question 4

Let us check these properties on R

So, R is reflexive on Z

So, R is symmetric on Z.

Adding eq. (i) and (ii)

So, R is transitive on Z.

Therefore, R is reflexive, symmetric and transitive.

Hence, R is an equivalence relation on Z.

Relation Exercise1.2 Question 5

Let us check these properties on R.

Let a be an arbitrary element of Z.

So, R is reflexive on Z.

So, R is symmetric on Z.

Adding (i) and (ii)

So, R is transitive on Z

Therefore, R is reflexive, symmetry and transitive.

Hence, R is an equivalence relation on Z.

Relation Exercise 1.2 Question 6

Answer: R is an equivalence relation on ZHint: To prove equivalence relation it is necessary that the given relation should be reflexive, symmetry and transitive.

Explanation:

Let us check these properties on R.

Reflexivity:

Let m be an arbitrary element of Z.

Hence, R is reflexive on Z.

Symmetry:

So, R is symmetric on Z.

Transitivity:

Adding (i) and (ii)

So, R is transitive on Z

Therefore, R is reflexive, symmetric and transitive.

Hence, R is an equivalence relation on Z.

Relation Exercise 1.2 Question 7

Explanation:

Thus, R is reflexive on A.

So, R is symmetric on A.

Multiply eq. (i) and (ii)

cancelling out vu it is common on both sides

So, R is transitive on A.

Therefore, R is reflexive, symmetric and transitive.

Hence, R is an equivalence relation on A.

Relation Exercise 1.2 Question 8

Let a be an arbitrary element of A.

Then,

So, R is reflexive on A

So, R is symmetric on A.

multiplying eqn (i) and (ii), we get

So, R is transitive on A.

Therefore, R is reflexive, symmetric and transitive.

Hence, R is an equivalence relation on A.

Thus, the set of all elements related to 1 is 1.

Relation Exercise 1.2 Question 9

Answer: R is an equivalence relation.Hint:To prove equivalence relation it is necessary that the given relation should be reflexive, symmetric and transitive.

Given: We have, L is the set of lines

Explanation:

Now,

Reflexivity:

Since, one line is always parallel to itself.

R is reflexive.

Symmetric:

R is symmetric

Transitive:

R is transitive

Therefore, R is reflexive, symmetric and transitive.

Hence, R is an equivalence relation.

The set of lines parallel to the line is

where R is the set of real numbers.

Relation Exercise 1.2 Question 10

Answer: R is an equivalence relation.The set of all elements in A related to triangle T is the set of all triangles.

Hint: To prove equivalence relation it is necessary that the given relation should be reflexive, symmetric and transitive.

Explanation:

Reflexive:

Symmetric:

R is symmetric

Transitive:

.

R is transitive.

Therefore, R is reflexive, symmetric and transitive.

Hence, R is an equivalence relation.

The elements in A related to the right-angle triangle (T) with sides 3, 4, and 5 are those polygons that have 3 sides (since T is a polygon with 3 sides).

Hence, the set of all elements in A related to triangle T is the set of all triangles.

Relation Exercise 1.2 Question 11

Answer: R is an equivalence relation on AHint: To prove equivalence relation it is necessary that the given relation should be reflexive, symmetric and transitive.

.

Explanation:

Let A be a set of points on a plane.

Reflexivity:

R is reflexive.

Symmetric:

R is symmetric

Transitive:

Putting (ii) in (i), we get

R is transitive

Therefore, R is reflexive, symmetric and transitive.

Thus, R is an equivalence relation on A.

Relation Exercise 1.2 Question 12

Answer: R is an equivalence relation on A.No number of the subset {1, 3, 5, 7} is related to any number of the subset {2, 4, 6}.

Hint: To prove an equivalence relation it is necessary that the given relation should be reflexive, symmetric and transitive.

Explanation:

We observe the following properties of R on A.

Reflexivity:

.

So, R is a reflexive relation in A

Symmetric:

Both a and b are either odd or even.

Both b and a are either odd or even.

So, R is a symmetric relation on A

Transitivity:

Both a and b are either odd or even.

⇒ Both b and c are odd or even.

⇒ Both a and c are even or odd.

R is a transitive relation on A

Thus, R is reflexive, symmetric and transitive.

Hence, R is an equivalence relation on A.

We observe that two numbers in A are related if both are odd or both are even.

Since, {1, 3, 5, 7} has all odd numbers of A

So, all the numbers of {1, 3, 5, 7} are related to each other

Similarly, all the numbers of {2, 4, 6} are related to each other as it contains all even numbers of set A.

An even, odd number in A is related to an even, odd number in A respectively.

So, no number of the subset {1, 3, 5, 7} is related to any number of the subset {2, 4, 6}

Relation Exercise 1.2 Question 13

Answer: Hence prove, S is not an equivalence relation on R.Hint: To prove an equivalence relation it is necessary that the given relation should be reflexive, symmetric and transitive.

Explanation:

Now,

Reflexivity:

S is not reflexive.

When one the three properties are not valid, relation not an equivalence.

Hence, S is not an equivalence relation on R.

Relation Exercise 1.2 Question 14

Answer: R is an equivalence relation onHint: To prove equivalence relation it is necessary that the given relation should be reflexive, symmetric and transitive.

Explanation:

Reflexivity:

R is reflexive

Symmetric:

R is symmetric

Transitive:

Using (ii) in (i), we get

R is transitive

Therefore, R is reflexive, symmetric and transitive.

Hence, R is an equivalence relation on

Relation Exercise 1.2 Question 15 (i)

Given:Hint: For symmetric relation.

Given: R and S are relations on a set A.

R and S are two symmetric relations on set A.

Explanation:

Relation Exercise 1.2 Question 15 (ii)

Given:

Hint:

Explanation:

Given: R and S are two relations on A such that R is reflexive.

To Prove:

Relation Exercise 1.2 Question 16

Answer:Hint: For transitive relation

Given:

To Prove:

Explanation:

We will prove this by means of an example.

Relation Exercise 1.2 Question 17

(i) Test for reflexivity

Hence, R is a reflexive relation.

(ii) Test for symmetric

Hence, R is a symmetric relation.

(iii) Test for transitivity

Dividing (i) and (ii), we get

From (i), (ii) & (iii)

- Chapter 1 - Relations
- Chapter 2 - Functions
- Chapter 3 - Inverse Trigonometric Functions
- Chapter 4 - Algebra of Matrices
- Chapter 5 - Determinants
- Chapter 6 - Adjoint and Inverse of a Matrix
- Chapter 7 - Solution of Simultaneous Linear Equations
- Chapter 8 - Continuity
- Chapter 9 - Differentiability
- Chapter 10 - Differentiation
- Chapter 11 - Higher Order Derivatives
- Chapter 12 - Derivative as a Rate Measurer
- Chapter 13 - Differentials, Errors and Approximations
- Chapter 14 - Mean Value Theorems
- Chapter 15 - Tangents and Normals
- Chapter 16 - Increasing and Decreasing Functions
- Chapter 17 - Maxima and Minima
- Chapter 18 - Indefinite Integrals
- Chapter 19 - Definite Integrals
- Chapter 20 - Areas of Bounded Regions
- Chapter 21 - Differential Equations
- Chapter 22 - Algebra of Vectors
- Chapter 23 - Scalar Or Dot Product
- Chapter 24 - Vector or Cross Product
- Chapter 25 - Scalar Triple Product
- Chapter 26 - Direction Cosines and Direction Ratios
- Chapter 27 - Straight Line in Space
- Chapter 28 - The Plane
- Chapter 29 - Linear programming
- Chapter 30- Probability
- Chapter 31 - Mean and Variance of a Random Variable

1. Why should I refer to RD Sharma Class 12 Maths Book?

RD Sharma Class 12 Maths Book is the best book for preparation for any exam. This book will take you to your goal if you want to give competitive exams like IIT JEE. RD Sharma Class 12 Maths Solutions Chapter 1 Exercise 1.2 has everything like examples, tips, tricks, and practice questions to grab a hold on the subject.

2. Are these solutions free of cost?

Yes, the solutions by Career360 are free, and you can avail the benefits by going to the website.

3. What are the benefits of these solutions?

The solutions are beneficial for every student because of the easy language and explained topics. For example, RD Sharma Class 12 Chapter 1 Exercise 1.2 Maths Solutions has everything a student requires to score well and compete better in exams.

4. What is a relation?

A relation between two sets is a collection of ordered pairs containing one object from each set. For example, if the object x is from the first set and object y is from the second set, the objects are said to be related in an ordered pair.

5. What are the different types of relations?

There are three types of relations, and these are void, universal, and identity relations.

You can learn more about these relations by RD Sharma Class 12 Chapter 1 Exercise 1.2.

Sep 11, 2024

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