# NCERT Solutions for Exercise 1.1 Class 12 Maths Chapter 1 - Relations and Functions

NCERT Solutions for Class 12 maths chapter 1 deals with questions related to various concepts of Relations and Functions which includes types of relations, functions, binary operations etc. **Exercise 1.1 Class 12 Maths **will help students to grasp the basic concepts of sets and relations. It is highly recommended to students to practise the NCERT Solutions for Class 12 Maths chapter 1 exercise 1.1 to score well in CBSE class 12 board exam. In competitive exams also like JEE main ,some questions can be asked from Class 12 Maths chapter 1 exercise 1.1. Concepts related to functions discussed in Class 12th Maths chapter 1 exercise 1.1 are important for Board examination also. The NCERT chapter Relations and Functions also has the following exercise for practice.

Relations and Functions Exercise 1.2

Relations and Functions Exercise 1.3

Relations and Functions Exercise 1.4

Relations and Functions Miscellaneous Exercise

## ** NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions: Exercise 1.1 **

** Question1(i) ** . Determine whether each of the following relations are reflexive, symmetric and transitive:

(i) Relation in the set defined as

** Answer: **

Since, so is not reflexive.

Since, but so is not symmetric.

Since, but so is not transitive.

Hence, is neither reflexive nor symmetric and nor transitive.

** Question 1(ii) ** . Determine whether each of the following relations are reflexive, symmetric and

transitive:

(ii) Relation ** R ** in the set ** N ** of natural numbers defined as

** Answer: **

Since,

so is not reflexive.

Since, but

so is not symmetric.

Since there is no pair in such that so this is not transitive.

Hence, is neither reflexive nor symmetric and

nor transitive.

** Question1(iii) ** Determine whether each of the following relations are reflexive, symmetric and

transitive:

(iii) Relation R in the set as

** Answer: **

Any number is divisible by itself and .So it is reflexive.

but .Hence,it is not symmetric.

and 4 is divisible by 2 and 4 is divisible by 4.

Hence, it is transitive.

Hence, it is reflexive and transitive but not symmetric.

** Question.1(iv) ** Determine whether each of the following relations are reflexive, symmetric and

transitive:

(iv). Relation R in the set Z of all integers defined as

** Answer: **

For , as which is an integer.

So,it is reflexive.

For , and because are both integers.

So, it is symmetric.

For , as are both integers.

Now, is also an integer.

So, and hence it is transitive.

Hence, it is reflexive, symmetric and transitive.

** Question:1(v) ** Determine whether each of the following relations are reflexive, symmetric and

transitive:

(v) Relation R in the set A of human beings in a town at a particular time given by

(a)

** Answer: **

,so it is reflexive

means .

i.e. so it is symmetric.

means also .It states that i.e. .So, it is transitive.

Hence, it is reflexive, symmetric and transitive.

** Question:1(v) ** Determine whether each of the following relations are reflexive, symmetric and

transitive:

(v) Relation R in the set A of human beings in a town at a particular time given by

(b)

** Answer: **

as and is same human being.So, it is reflexive.

means .

It is same as i.e. .

So,it is symmetric.

means and .

It implies that i.e. .

Hence, it is reflexive, symmetric and

transitive.

** Question:1(v) ** ** ** Determine whether each of the following relations are reflexive, symmetric and

transitive:

(v) Relation R in the set A of human beings in a town at a particular time given by

(c)

** Answer: **

means but i.e. .So, it is not reflexive.

means but i.e .So, it is not symmetric.

means and .

i.e. .

Hence, it is not reflexive,not symmetric and

not transitive.

** Question:1(v) ** Determine whether each of the following relations are reflexive, symmetric and

transitive:

(v). Relation R in the set A of human beings in a town at a particular time given by

(d)

** Answer: **

means but i.e. .

So, it is not reflexive.

means but i.e. .

So, it is not symmetric.

Let, means and .

This case is not possible so it is not transitive.

Hence, it is not reflexive, symmetric and

transitive.

** Question:1(v) ** Determine whether each of the following relations are reflexive, symmetric and

transitive:

(v) Relation R in the set A of human beings in a town at a particular time given by

(e)

** Answer: **

means than i.e. .So, it is not reflexive..

means than i.e. .So, it is not symmetric.

Let, means and than i.e. .

So, it is not transitive.

Hence, it is neither reflexive nor symmetric and nor transitive.

** Question:2 ** Show that the relation R in the set R of real numbers defined as

is neither reflexive nor symmetric nor transitive.

** Answer: **

Taking

and

So, R is not reflexive.

Now,

because .

But, i.e. 4 is not less than 1

So,

Hence, it is not symmetric.

as

Since because

Hence, it is not transitive.

Thus, we can conclude that it is neither reflexive, nor symmetric, nor transitive.

** Question:3 ** Check whether the relation R defined in the set as

is reflexive, symmetric or transitive.

** Answer: **

R defined in the set

Since, so it is not reflexive.

but

So, it is not symmetric

but

So, it is not transitive.

Hence, it is neither reflexive, nor symmetric, nor transitive.

** Question:4 ** Show that the relation R in R defined as , is reflexive and

transitive but not symmetric.

** Answer: **

As so it is reflexive.

Now we take an example

as

But because .

So,it is not symmetric.

Now if we take,

Than, because

So, it is transitive.

Hence, we can say that it is reflexive and transitive but not symmetric.

** Question:5 ** Check whether the relation R in R defined by is reflexive,

symmetric or transitive.

** Answer: **

because

So, it is not symmetric

Now, because

but because

It is not symmetric

as .

But, because

So it is not transitive

Thus, it is neither reflexive, nor symmetric, nor transitive.

** Question:6 ** Show that the relation R in the set given by is

symmetric but neither reflexive nor transitive.

** Answer: **

Let A=

We can see so it is not reflexive.

As so it is symmetric.

But so it is not transitive.

Hence, R is symmetric but neither reflexive nor transitive.

** Question:7 ** Show that the relation R in the set A of all the books in a library of a college,

given by is an equivalence

relation.?

** Answer: **

A = all the books in a library of a college

because x and x have the same number of pages so it is reflexive.

Let means x and y have same number of pages.

Since y and x have the same number of pages so .

Hence, it is symmetric.

Let means x and y have the same number of pages.

and means y and z have the same number of pages.

This states,x and z also have the same number of pages i.e.

Hence, it is transitive.

Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence

relation.?

** Question:8 ** Show that the relation R in the set given by , is an equivalence relation. Show that all the elements of are related to each other and all the elements of are related to each other. But no element of is related to any element of .

** Answer: **

Let there be then as which is even number. Hence, it is reflexive

Let where then as

Hence, it is symmetric

Now, let

are even number i.e. are even

then, is even (sum of even integer is even)

So, . Hence, it is transitive.

Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence relation.

The elements of are related to each other because the difference of odd numbers gives even number and in this set all numbers are odd.

The elements of are related to each other because the difference of even number is even number and in this set, all numbers are even.

The element of is not related to because a difference of odd and even number is not even.

** Question:9(i) ** Show that each of the relation R in the set , given by

(i) is an equivalence relation. Find the set of all elements related to 1 in each case.

** Answer: **

For , as which is multiple of 4.

Henec, it is reflexive.

Let, i.e. is multiple of 4.

then is also multiple of 4 because = i.e.

Hence, it is symmetric.

Let, i.e. is multiple of 4 and i.e. is multiple of 4 .

is multiple of 4 and is multiple of 4

is multiple of 4

is multiple of 4 i.e.

Hence, it is transitive.

Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence relation.

The set of all elements related to 1 is

is multiple of 4.

is multiple of 4.

is multiple of 4.

** Question:9(ii) ** Show that each of the relation R in the set , given by

(ii) is an equivalence relation. Find the set of all elements related to 1 in each case.

** Answer: **

For , as

Henec, it is reflexive.

Let, i.e.

i.e.

Hence, it is symmetric.

Let, i.e. and i.e.

i.e.

Hence, it is transitive.

Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence relation.

The set of all elements related to 1 is {1}

** Question:10(i) ** Give an example of a relation.

(i) Which is Symmetric but neither reflexive nor transitive.

** Answer: **

Let

so it is not reflexive.

and so it is symmetric.

but so it is not transitive.

Hence, symmetric but neither reflexive nor transitive.

** Question:10(ii) ** Give an example of a relation.

(ii) Which is transitive but neither reflexive nor symmetric.

** Answer: **

Let

Now for , so it is not reflexive.

Let i.e.

Then is not possible i.e. . So it is not symmetric.

Let i.e. and i.e.

we can write this as

Hence, i.e. . So it is transitive.

Hence, it is transitive but neither reflexive nor symmetric.

** Question:10(iii) ** Give an example of a relation.

(iii) Which is Reflexive and symmetric but not transitive.

** Answer: **

Let

Define a relation R on A as

If , i.e. . So it is reflexive.

If , and i.e. . So it is symmetric.

and i.e. . and

But So it is not transitive.

Hence, it is Reflexive and symmetric but not transitive.

** Question:10(iv) ** Give an example of a relation.

(iv) Which is Reflexive and transitive but not symmetric.

** Answer: **

Let there be a relation R in R

because

Let i.e.

But i.e.

So it is not symmetric.

Let i.e. and i.e.

This can be written as i.e. implies

Hence, it is transitive.

Thus, it is Reflexive and transitive but not symmetric.

** Question:10(v) ** Give an example of a relation.

(v) Which is Symmetric and transitive but not reflexive.

** Answer: **

Let there be a relation A in R

So R is not reflexive.

We can see and

So it is symmetric.

Let and

Also

Hence, it is transitive.

Thus, it Symmetric and transitive but not reflexive.

** Question:11 ** Show that the relation R in the set A of points in a plane given by

, is an equivalence relation. Further, show that the set of

all points related to a point is the circle passing through P with origin as

centre.

** Answer: **

The distance of point P from the origin is always the same as the distance of same point P from origin i.e.

R is reflexive.

Let i.e. the distance of the point P from the origin is the same as the distance of the point Q from the origin.

this is the same as distance of the point Q from the origin is the same as the distance of the point P from the origin i.e.

R is symmetric.

Let and

i.e. the distance of point P from the origin is the same as the distance of point Q from the origin, and also the distance of point Q from the origin is the same as the distance of the point S from the origin.

We can say that the distance of point P, Q, S from the origin is the same. Means distance of point P from the origin is the same as the distance of point S from origin i.e.

R is transitive.

Hence, R is an equivalence relation.

The set of all points related to a point are points whose distance from the origin is the same as the distance of point P from the origin.

In other words, we can say there be a point O(0,0) as origin and distance between point O and point P be k=OP then set of all points related to P is at distance k from the origin.

Hence, these sets of points form a circle with the centre as the origin and this circle passes through the point.

** Question:12 ** Show that the relation R defined in the set A of all triangles as , is equivalence relation. Consider three right angle triangles T _{ 1 } with sides 3, 4, 5, T _{ 2 } with sides 5, 12, 13 and T _{ 3 } with sides 6, 8, 10. Which triangles among T _{ 1 } , T _{ 2 } and T _{ 3 } are related?

** Answer: **

All triangles are similar to itself, so it is reflexive.

Let,

i.e.T _{ 1 } is similar to T2

T _{ 1 } is similar to T2 is the same asT2 is similar to T _{ 1 } i.e.

Hence, it is symmetric.

Let,

and i.e. T _{ 1 } is similar to T2 and T2 is similar toT _{ 3 } .

T _{ 1 } is similar toT _{ 3 } i.e.

Hence, it is transitive,

Thus, , is equivalence relation.

Now, we see the ratio of sides of triangle T _{ 1 } andT _{ 3 } are as shown

i.e. ratios of sides of T _{ 1 } and T _{ 3 } are equal.Hence, T _{ 1 } and T _{ 3 } are related.

** Question:13 ** Show that the relation R defined in the set A of all polygons as , is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

** Answer: **

The same polygon has the same number of sides with itself,i.e. , so it is reflexive.

Let,

i.e.P _{ 1 } have same number of sides as P _{ 2 }

P _{ 1 } have the same number of sides as P _{ 2 } is the same as P _{ 2 } have same number of sides as P _{ 1 } i.e.

Hence,it is symmetric.

Let,

and i.e. P _{ 1 } have the same number of sides as P _{ 2 } and P _{ 2 } have same number of sides as P _{ 3 }

P _{ 1 } have same number of sides as P _{ 3 } i.e.

Hence, it is transitive,

Thus, , is an equivalence relation.

The elements in A related to the right angle triangle T with sides 3, 4 and 5 are those polygons which have 3 sides.

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

** Question:14 ** Let L be the set of all lines in XY plane and R be the relation in L defined as . Show that R is an equivalence relation. Find the set of all lines related to the line

** Answer: **

All lines are parallel to itself, so it is reflexive.

Let,

i.e.L _{ 1 } is parallel to L _{ 2 } .

_{ L1 } is parallel to L _{ 2 } is same as L _{ 2 } is parallel to L _{ 1 } i.e.

Hence, it is symmetric.

Let,

and i.e. _{ L1 } is parallel to L _{ 2 } and L _{ 2 } is parallel to L _{ 3 } .

L _{ 1 } is parallel to L _{ 3 } i.e.

Hence, it is transitive,

Thus, , is equivalence relation.

The set of all lines related to the line are lines parallel to

Here, Slope = m = 2 and constant = c = 4

It is known that the slope of parallel lines are equal.

Lines parallel to this ( ) line are ,

Hence, set of all parallel lines to are .

### ** Question:15 ** Let R be the relation in the set A= {1,2,3,4}

### given by . Choose the correct answer.

(A) R is reflexive and symmetric but not transitive.

(B) R is reflexive and transitive but not symmetric.

(C) R is symmetric and transitive but not reflexive.

(D) R is an equivalence relation.

### ** Answer: **

A = {1,2,3,4}

For every there is .

R is reflexive.

Given, but

R is not symmetric.

For there are

R is transitive.

Hence, R is reflexive and transitive but not symmetric.

The correct answer is option B.

### ** Question:16 ** Let R be the relation in the set N given by . Choose the correct answer.

(A)

(B)

(C)

(D)

### ** Answer: **

(A) Since, so

(B) Since, so

(C) Since, and so

(d) Since, so

The correct answer is option C.

Also Read| NCERT Notes For Class 12 Mathematics Chapter 1

**More About NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.1**

The NCERT Class 12 maths chapter **Relations and Function**s has a total of 5 exercises including miscellaneous. Exercise 1.1 Class 12 Maths covers solutions to 16 main questions and their sub-questions. The initial 10 questions are based on concepts like symmetric, reflexive and transitive relation and subsequent questions upto 15 are based in equivalence relation etc. NCERT Solutions for Class 12 Maths chapter 1 exercise 1.1 is good source to learn concepts related to symmetric relations, equivalence of a relation etc.

**Benefits of NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.1**

The Class 12th maths chapter 1 exercise provided here is in detail which is solved by subject matter experts .

Students are recommended to practice

**Exercise 1.1 Class 12 Maths**to prepare for exams, direct questions are asked in Board exams.These NCERT text book Class 12 Maths chapter 1 exercise 1.1 solutions can be referred by students to revise just before the exam.

NCERT Syllabus Class 12 Maths chapter 1 exercise 1.1 provided here are one stop solutions for students aspiring to score well in examinations.

**Also see-**

NCERT exemplar solutions class 12 maths chapter 1

NCERT solutions for class 12 maths chapter 1

**NCERT Solutions Subject Wise**

NCERT solutions class 12 chemistry

NCERT solutions for class 12 physics

NCERT solutions for class 12 biology

NCERT solutions for class 12 mathematics

**Subject wise NCERT Exemplar solutions**

- NCERT Exemplar Class 12th Maths
NCERT Exemplar Class 12th Physics

NCERT Exemplar Class 12th Chemistry

NCERT Exemplar Class 12th Biology

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