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Solutions that are accurate and beneficial for exam prepared Sharma is one of the most well-known books in the country. RD Sharma books are detailed, informative, and contain step-by-step solutions for their problems.

RD Sharma Class 12th Exercise 1.1 deals with also need to solve exercise problems. This is where Career360 comes to help. It contains expert-created solutions of the chapter 'Relations.' They have plenty of example problems which the students can practice to develop their skills.

Relations Exercise 1.1 Question 1 (i)

If is reflexive

If is symmetric

If is transitive

Thus, is reflexive

A relation on set A is said to be symmetric relation if for all

i.e.

A relation on set A is said to be transitive relation

If

i.e and for all

Now, let's get back to the actual problem

and work at same place

Similarly, and work at the same place.

So, is reflexive.

It is given that and work at the same place.

So, we can say that and work at the same place.

So, is symmetric.

Let be a person such that and work at the same place.

And we know that and work at the same place.

So, we can say that and work at the same place.

So, is Transitive.

Relations Exercise 1.1 Question 1 (ii)

If is reflexive

If is symmetric

If is transitive

A relation on set A is said to be reflexive if every element of A is related to itself.

Thus, is reflexive

A relation on set A is said to be symmetric relation if for all

I.e. for all

A relation on set A is said to be transitive relation if for all

i.e

and live in the same locality.

Similarly, and live in same locality

So, is reflexive.

and live in same locality.

So, we can easily say that and live in same locality.

So, is symmetric.

Let be a person; and and live in same locality

And it is given that and live in same locality

So, we can say that and live in the same locality.

So, is Transitive.

Relations Exercise 1.1 Question 1 (iii)

Neither reflexive, nor symmetric, not transitive.

If is reflexive

If is symmetric

If is transitive

A relation on set is said to be reflexive if every element of is related to itself.

Thus, is reflexive .

A relation on set is said to be symmetric relation if for all

i.e;

A relation on set is said to be transitive relation if and for all

i.e; and for all

is not wife of and is not wife of

So, is not reflexive.

is the wife of but is not the wife of .

So, is not symmetric.

Let be a person; such that is the wife of .

And it is given that is the wife of but this case is not possible. Also, here we can’t show x is the wife of z.

So, R is not transitive.

Relations Exercise 1.1 Question 1 (iv)

Neither reflexive, nor symmetric nor transitive.

If is reflexive

If is symmetric

If is transitive

A relation on set is said to be reflexive if every element of is related to itself.

Thus, is reflexive

A relation on set is said to be symmetric relation if for all

i.e for all

A relation on set is said to be transitive relation if and for all

i.e and for all

is not father of and is not father of

So, is not reflexive

It is given that is the father of .

But we can say that is not the father of .

So is not symmetric.

Let be a person; such that is father of and it is given that x is a father of y.

Then is grandfather of

So, is not Transitive.

Relations Exercise 1.1 Question 2

is reflexive and transitive but not symmetric

is reflexive, symmetric, and transitive.

is transitive but neither reflexive nor symmetric

is neither reflexive nor symmetric nor transitive

Hint :

A relation R on set A is

If for every

If

If

Given :

Set

Consider

Reflexive :

Given and

So, is reflexive.

For Symmetric:

We see that the ordered pairs obtained by interchanging the components of are not in .

For ex :

So, is not symmetric.

For Transitive:

Here, but

So, is transitive

(ii) Consider

Reflexive:

clearly

So, is reflexive.

Symmetric:

Clearly

So, is symmetric.

Transitive:

is a transitive relation, since there is only one element in it.

(iii) Consider

Reflexive:

Here neither nor

So, is not reflexive

Symmetric:

Here neither nor

So, is not symmetric.

Transitive:

has only one element

Hence is transitive.

(iv) Consider

Reflexive:

Here

So, is not reflexive

Symmetric:

Here

So, is not symmetric

Transitive:

Here but

Hence is not transitive.

Relations Exercise 1.1 Question 3 (i)

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If and , then for every

defined by

Solution :

Reflexivity:

Let be an arbitrary element of

Then,

So, is not reflexive

Symmetry:

Let

Therefore, we can write 'a' as

Then

So, is symmetric.

For Transitive:

Let

So, is not transitive.

Relations Exercise 1.1 Question 3 (ii)

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

if is true then is also true for every

Transitive relation:

If and , then for every **Given:**

defined by **Solution :**

Reflexivity:

Let a be an arbitrary element of

Then,

On applying the given condition, we get

So, is reflexive

Symmetry :

Let

[Since ]

Then

So, is symmetric.

Transitivity :

Let

But,

So, is not transitive.

Relations Exercise 1.1 Question 3 (iii)

Answer:is reflexive but neither symmetric nor transitive

Hint :

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

if and , then for every

Given : on R defined by

Solution :

Reflexivity:

Let a be an arbitrary element of

Then,

So, is reflexive

Symmetry :

Let :

But for all

So, is not symmetric.

Transitivity:

Let

Adding (i) and (ii) we get,

So, is not transitive.

Relations Exercise 1.1 Question 4

Answer: is reflexive but neither symmetric nor transitiveis symmetric but neither reflexive nor transitive

is transitive but neither reflexive nor symmetric

Hint :

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

if and , then for every

Given :

Solution :

Consider

Reflexivity:

Here

So, is Reflexive

Symmetric :

Here

But,

So, is not symmetric

Transitivity:

So, is not transitive.

Now consider

Reflexivity :

Clearly,

So, is not reflexive.

Symmetric :

So, is symmetric.

Transitivity:

So, is not transitive.

Now consider

Reflexivity:

Clearly,

So, is not reflexive.

Symmetry:

Here

So, is not symmetric.

Transitivity:

Here

But

So, is transitive.

Relation Exercise 1.1 Question 5 (i)

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If and , then for every

Given :

Reflexivity:

Let be an arbitrary element of

Then,

But

So, this relation is not reflexive.

Symmetry :

Let

So, the given relation is not symmetric.

Transitivity :

Adding eq (i) & (ii), we get

So, the given relation is transitive.

Relation Exercise 1.1 Question 5 (ii)

Reflexive and symmetric but not transitive.

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If and , then for every

Reflexivity:

Let be an arbitrary element of

Then,

Since, Square of any number is positive.

So, the given relation is reflexive.

Symmetry:

Let

So, the given relation is symmetric.

Transitivity:

So, the given relation is not transitive.

Relation Exercise 1.1 Question 5 (iii)

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If and , then for every

Given :

Solution :

Reflexivity:

Let -a be an arbitrary element of R

Then

So, is not reflexive

Symmetry:

Let

So, is not symmetric.

Transitivity:

Multiplying the corresponding sides, we get

Thus, is transitive

Relation Exercise 1.1 Question 6

is neither reflexive nor symmetric nor transitive.

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If and , then for every

Given :

Let be an arbitrary element of set A.

Then,

cannot be true for all

So, is not reflexive on

Symmetry :

So, is not symmetric on

Transitivity :

Let

So, is not transitive on

Relation Exercise 1.1 Question 7

is neither reflexive nor symmetric nor transitive.

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If and , then for every

Solution :

Reflexive :

It is observed that

Symmetric :

Now

But

is not symmetric

Transitive:

We have

and

but

is not transitive

Hence, is neither reflexive, nor symmetric nor transitive

Relations Exercise 1.1 Question 8

Hence, prove every identity relation on a set is reflexive but the converse is not necessarily true.

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

Let be a set

Then,

Identity relation is reflexive, since

The converse of it need not be necessarily true.

Counter example:

Consider the set

Here,

Relation is reflexive on

However, is not an identity relation.

Relations Exercise 1.1 Question 9(i)

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

Solution:

The relation on having properties of being Reflexive, transitive but not symmetric is

Relation satisfies reflexivity and transitivity

And

However,

Relations Exercise 1.1 Question 9 (ii)

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

The relation on A having properties of being symmetric but neither transitive nor reflexive.

Let

Now,

So, it is symmetric.

Clearly is not transitive

Relations Exercise 1.1 Question 9(iii)

The relation is an equivalence relation on

A relation on set is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

Solution:

The relation on having properties of being

Symmetric, reflexive and transitive is

The relation is an equivalence relation on

Relations Exercise 1.1 Question 10

Domain of is x ∈ N where x∈and

range of is y ∈ N where

The relation having properties of being neither symmetric nor transitive nor reflexive.

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

The domain of

Since , largest value that can take corresponds to the smallest value that can take.

Range of is such that

Since

Since ,is not reflexive.

Also, since is not symmetric.

Finally, since is not transitive.

Relations Exercise 1.1 Question 11

No, it is not true that every relation which is symmetric and transitive is also reflexive.

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If and , then for every

Answer that, whether every relation which is symmetric and transitive is also reflexive.

No, it is not necessary that a relation that is symmetric and transitive is reflexive as well.

For example:

is symmetric and transitive but not reflexive.

Because

Relations Exercise 1.1 Question 13

Hence proved, the relation “” on the set of all real numbers is reflexive and transitive but not symmetric.

A relation on set is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If then for every

Relation is ””on the of all real numbers.

Reflexivity:

Let

" "is reflexive.

Transitive:

Let

Such that

Then

is transitive.

Symmetric: Let

Such that

not symmetric

Relations Exercise 1.1 Question 14(ii)

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

We have to give the example of a relation which is reflexive and transitive but not symmetric.

The relation having properties of being reflexive and transitive but not symmetric.

Define a relation in as:

Clearly

Therefore is reflexive.

Now

But

Therefore is not symmetric.

Now let

is transitive.

Hence, relation is reflexive and transitive but not symmetric.

Relations Exercise 1.1 Question 14(iii)

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

We have to give the example of a relation which is symmetric and transitive but not reflexive.

The relation having properties of being symmetric and transitive but not reflexive.

Let

Define a relation on as

Relation is not reflexive as

Relation is symmetric as

It is seen that

Also

The relation is transitive.

Hence relation is symmetric and transitive but not reflexive.

Relations Exercise 1.1 Question 14(iv)

Hint:

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If , then for every

Relations Exercise 1.1 Question 14(v)

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If

We have to give the example of a relation which is transitive but neither symmetric nor reflexive.

The relation having properties of being transitive but neither symmetric nor reflexive.

Consider a relation R in defined as:

For any we have , since can’t be strictly less than itself.

Infact

∴ Relation is not reflexive.

Now,

But 2 is not less than 1

∴

∴ is not symmetric

Now let

is transitive.

Relation Exercise 1.1 Question 15

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If and , then for every

To make reflexive we will add to get is reflexive.

Again, to make symmetric we will add and is reflexive and symmetric .

To make transitive we will add and is reflexive and symmetric and transitive.

Relation Exercise 1.1 Question 16

only 1 ordered pair maybe added to so that it may become a transitive relation on

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If and , then for every

be a relation on

To make transitive we shall add only

As we know,

Transitive relation

Relation Exercise 1.1 Question 17

At least 3 ordered pairs must be added for to be reflexive and transitive

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If and , then for every

A relation in is said to be reflexive if for all

is said to be transitive if and bRc then aRc for all

Hence for to be reflexive and must be there in set

Also for to be transitive must be in because

So, must be in

So, at least 3 ordered pairs must be added for to be reflexive and transitive.

Relation Exercise 1.1 Question 18 (i)

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If and then for every

This is not reflexive as are absents.

This is not symmetric as is present but is absent

This is transitive as

Hence, this relation is not satisfying reflexivity and symmetricity.

Relation Exercise 1.1 Question 18 (ii)

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If and then for every

This is not reflexive as are absent.

This only follows the condition of symmetry as

This is not transitive because is absent.

Hence, this relation is not satisfying reflexivity and transitivity.

Relation Exercise 1.1 Question 18 (iii)

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If and then for every

This is reflexive as are present.

This is also symmetric because

This is transitive because if

This relation is reflexive, symmetric, and transitive.

Relation Exercise 1.1 Question 18 (iv)

A relation R on set A is

Reflexive relation:

If for every

Symmetric relation:

If is true then is also true for every

Transitive relation:

If and then for every

x

**Solution :**

This is not reflexive as are absent.

This is not symmetric as is present but is absent.

This is not transitive as there are only two elements in the set having no element in common.

This relation is neither symmetric nor reflexive nor transitive.

Void relation

Universal relation

Identity relation

Reflexive relation

Symmetric relation

Transitive relation

Antisymmetric relation

Equivalence relation

Theorems based on relations

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With such a vast number of topics, studying all of them at once gets pretty confusing. Therefore, the questions from RD Sharma Class 12th Exercise 1.1 are divided into two parts: Level 1 and Level 2. The best approach for this chapter it's to complete 20 Level 1 questions and 10 Level 2 questions per day to cover the entire chapter systematically. This will ensure that you cover a major part of the portion without the burden.

Why is it essential to study the RD Sharma Class 12 Chapter 1 Exercise-1.1 for the board exam?

Class 12 RD Sharma Chapter 1 Exercise 1.1 Solution is designed to help students grasp the principles behind the numerous questions that will be asked on the test.

The solutions pdf offers extensive and elaborate explanations to assist students in achieving a higher academic score.

Students will study for the board exam and score well if they refer to these solutions.

The solutions are given in plain English to assist students in their exam preparation.

RD Sharma class 12th exercise 1.1 solutions are well-known, and hundreds of students have trusted them and received good results.

Our subject specialists have solved all of the problems from the textbook. Therefore, students can rely on RD Sharma Class 12th Exercise 1.1 for test preparation. Students will write precise and elaborate answers to textbook questions by practicing the solutions offered in the solutions PDF and following the latest CBSE recommendations.

RD Sharma Class 12 Solution Chapter 1 Exercise 1.1 Relations have the following features:

Created by subject matter experts

Easily accessible online

Free of cost

Based on the most recent CBSE syllabus pattern

Chapter-by-chapter solutions

A simple tool for preparing for an exam

All ideas and definitions have been described in-depth in a lucid manner. They have also been explained with relevant graphic patterns in RD Sharma Class 12 Solutions Relations Chapter 1 Ex 1.1.

For Maths, RD Sharma solutions are chosen because they are provided chapter-by-chapter materials. Because the RD Sharma Class 12th Exercise 1.1 Maths syllabus is large, RD Sharma Maths Solutions are organized by chapter.

With RD Sharma Class 12th Exercise 1.1 solutions, students can read each chapter in depth. Most CBSE Class 12 students prefer our RD Sharma Maths Solutions since they were created by experts with extensive expertise teaching Maths.

- 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

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2. Are the RD Sharma solutions up to current with the most recent syllabus?

In the NCERT book, RD Sharma aims to include all of the answers to the questions. As a result, they are revised every year to reflect the most recent CBSE class 12 syllabus. Therefore, you can rest assured that the book will contain all of the information you require. All you have to do now is select the appropriate file for your exam year.

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For students who want to do well in their exams and have started studying ahead of time, the Class 12 RD Sharma chapter 1 exercise 1.1 solution book would be very helpful. It can also be used for self-evaluation, self-practice, and identifying your weak areas. Many students and teachers recommend the book, indicating that it can assist you in exam preparation.

5. Do the RD Sharma class 12 solutions Relations Ex 1.1 follow the most recent syllabus?

The NCERT book's RD Sharma 12 answers Relations Ex 1.1 is always updated with the latest syllabus. Likewise, the pdf of the RD Sharma 12 solutions chapter 1 Ex 1.1 is updated with each new edition of the NCERT book so that students may find answers to all of the questions in the book.

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