**NCERT Solutions for Class 11 Maths Chapter 2 Relations And Functions**: In the previous chapter you have learnt about sets. This chapter is the continuation of chapter 1 sets. In this article, you will get NCERT solutions for class 11 maths chapter 2 relations and functions. You have many relations in your daily life, like father-son, mother-son, etc. In a similar way, you can form relations in mathematics also. When there is only one output for every input, the relation becomes a function. Important topics like domain, co-domain, and range of functions are covered in this chapter. In solutions of NCERT for class 11 maths chapter 2 relations and functions, questions related to these topics are covered. Also, you will learn different types of specific real-valued functions and their graphs. CBSE NCERT solutions for class 11 maths chapter 2 relations and functions will build your fundamentals of functions which will be helpful in the 12th board exam also. Check all **NCERT solutions** from class 6 to 12 to learn science and maths. There are three exercises and a miscellaneous exercise in this chapter which are explained below.

A={1,2,3,4,5}, B={2,6,12,20,30}, we can find a relation from A to B by inspection. That is set B is related to set A by the relation and this is one possible relation from set A to set B

The NCERT solutions for class 11 maths chapter 2 relations and functions start with cartesian products of sets. The main points to remember in these topics are:

1. Cartesian products two sets A and B are set of all ordered pair of the elements from A and B, for example, A={1,2,3}, B= {2,3} then

2. If set A contains n elements and set B contains m elements then Cartesian Products A and B contains mn elements.

3. Two ordered pairs are equal if corresponding first and second elements are equal. That is if (a,b) =(c,d) then a=c and b=d

2.1 Introduction

2.2 Cartesian Products of Sets

2.3 Relations

2.4 Functions

The second important topic of this chapter is a relation. A few points to remember are listed below

1. A relation from non-empty set A to non-empty set B is a subset of

2. The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain R and the set of all second elements in a relation R from a set A to a set B is called the range of R., For example, consider a relation R from set A ={1,2,3,4} to Set B={1,2,3.........20} of real numbers defined by R={(1,1),(2,4),(3,9),(4,16)}. Then the set {1,2,3,4} is the domain of R and {1,4,9,16} is the range of R and the whole set B is the codomain.

3. Let the set A has n element and set B has m element then the number of possible relation from A to B is

The last topic of this chapter is functions: A relation R from set A to set B is called as a function if every element of set A has only one image in set B.

**Algebra of real functions**: Let f and g be any real functions then

**NCERT solutions for class 11 maths chapter 2 relations and functions-Exercise: 2.1**

**Question:1 **If , find the values of x and y.

**Answer:**

It is given that

Since the ordered pairs are equal, the corresponding elements will also be equal

Therefore,

Therefore, values of x and y are ** 2 and 1 **respectively

**Question:2** If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in .

**Answer:**

It is given that set A has 3 elements and the elements in set B are 3 , 4 , and 5

Therefore, the number of elements in set B is 3

Now,

Number of elements in

= ( Number of elements in set A ) ( Number of elements in set B)

= 3 3

= 9

Therefore, number of elements in is **9**

**Question:3 ** If G = {7, 8} and H = {5, 4, 2}, find

**Answer:**

It is given that

G = {7, 8} and H = {5, 4, 2}

We know that the cartesian product of two non-empty sets P and Q is defined as

P Q = {(p,q) , where p P , q Q }

Therefore,

G H = {(7,5),(7,4),(7,2),(8,5),(8,4),(8,2)}

And

H G = {(5,7),(5,8),(4,7),(4,8),(2,7),(2,8)}

**Answer:**

**FALSE**

If P = {m, n} and Q = { n, m}

Then,

**Answer:**

It is a ** TRUE ** statement

If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs (x, y) such that and

**Answer:**

This statement is ** TRUE**

If A = {1, 2}, B = {3, 4}, then

There for

**Question:5 **If A = {–1, 1}, find

**Answer:**

It is given that

A = {–1, 1}

A is an non-empty set

Therefore,

Lets first find

Now,

**Question:6** If = {(a, x),(a , y), (b, x), (b, y)}. Find A and B.

**Answer:**

It is given that

= {(a, x),(a , y), (b, x), (b, y)}

We know that the cartesian product of two non-empty set P and Q is defined as

Now, we know that A is the set of all first elements and B is the set of all second elements

Therefore,

**Question:7 (i)** Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that .

**Answer:**

It is given that

A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}

Now,

Now,

And

Now,

From equation (i) and (ii) it is clear that

Hence,

**Question:7 (ii) **Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that is a subset of

**Answer:**

It is given that

A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}

Now,

And

We can clearly observe that all the elements of the set are the elements of the set

Therefore, is a subset of

**Question:8 **Let A = {1, 2} and B = {3, 4}. Write . How many subsets will have?

List them.

**Answer:**

It is given that

A = {1, 2} and B = {3, 4}

Then,

Now, we know that if C is a set with

Then,

Therefore,

The set has subsets.

**Answer:**

It is given that

n(A) = 3 and n(B) = 2 and If (x, 1), (y, 2), (z, 1) are in A × B.

By definition of Cartesian product of two non-empty Set P and Q:

Now, we can see that

P = set of all first elements.

And

Q = set of all second elements.

Now,

(x, y, z) are elements of A and (1,2) are elements of B

As n(A) = 3 and n(B) = 2

Therefore,

A = {x, y, z} and B = {1, 2}

**Answer:**

It is given that Cartesian product A × A having 9 elements among which are found (–1, 0) and (0,1).

Now,

Number of elements in (A× B) = (Number of elements in set A) × (Number of elements in B)

It is given that

Therefore,

Now,

By definition A × A = {(a, a): a ? A}

Therefore,

-1, 0 and 1 are the elements of set A

Now, because, n(A) = 3 therefore, A = {-1, 0, 1}

Therefore,

the remaining elements of set (A × A) are

(-1,-1), (-1,1), (0,0), (0, -1), (1,1), (1, -1) and (1, 0)

**Answer:**

It is given that

Now, the relation R from A to A is given as

Therefore,

the relation in roaster form is ,

Now,

We know that Domain of R = set of all first elements of the order pairs in the relation

Therefore,

Domain of

And

Codomain of R = the whole set A

i.e. Codomain of

Now,

Range of R = set of all second elements of the order pairs in the relation.

Therefore,

range of

**Answer:**

It is given that

is a natural number less than

As x is a natural number which is less than 4.

Therefore,

the relation in roaster form is,

As Domain of R = set of all first elements of the order pairs in the relation.

Therefore,

Domain of

Now,

Range of R = set of all second elements of the order pairs in the relation.

Therefore,

the range of

Therefore, domain and the range are respectively

**Question:3 **A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by . Write R in roster form.

**Answer:**

It is given that

A = {1, 2, 3, 5} and B = {4, 6, 9}

And

Now, it is given that the difference should be odd. Let us take all possible differences.

(1 - 4) = - 3, (1 - 6) = - 5, (1 - 9) = - 8(2 - 4) = - 2, (2 - 6) = - 4, (2 - 9) = - 7(3 - 4) = - 1, (3 - 6) = - 3, (3 - 9) = - 6(5 - 4) = 1, (5 - 6) = - 1, (5 - 9) = - 4

Taking the difference which are odd we get,

Therefore,

the relation in roaster form,

**Question:4 (i)** The Fig2.7 shows a relationship between the sets P and Q. Write this relation in set-builder form

**Answer:**

It is given in the figure that

P = {5,6,7}, Q = {3,4,5}

Therefore,

the relation in set builder form is ,

**OR**

**Question:4 (ii) **The Fig2.7 shows a relationship between the sets P and Q. Write this relation roster form. What is it domain and range?

**Answer:**

From the given figure. we observe that

P = {5,6,7}, Q = {3,4,5}

And the relation in roaster form is ,

As Domain of R = set of all first elements of the order pairs in the relation.

Therefore,

Domain of

Now,

Range of R = set of all second elements of the order pairs in the relation.

Therefore,

the range of

**Question:5 (i) **Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by Write R in roster form

**Answer:**

It is given that

A = {1, 2, 3, 4, 6}

And

Therefore,

the relation in roaster form is ,

**Question:5 (ii) **Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by Find the domain of R

**Answer:**

It is given that

A = {1, 2, 3, 4, 6}

And

Now,

As Domain of R = set of all first elements of the order pairs in the relation.

Therefore,

Domain of

**Question:5 (iii) **Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by Find the range of R.

**Answer:**

It is given that

A = {1, 2, 3, 4, 6}

And

Now,

As the range of R = set of all second elements of the order pairs in the relation.

Therefore,

Range of

**Question:6 **Determine the domain and range of the relation R defined by

**Answer:**

It is given that

Therefore,

the relation in roaster form is ,

Now,

As Domain of R = set of all first elements of the order pairs in the relation.

Therefore,

Domain of

Now,

As Range of R = set of all second elements of the order pairs in the relation.

Range of

Therefore, the domain and range of the relation R is respectively

**Question:7 **Write the relation in roster form.

**Answer:**

It is given that

Now,

As we know the prime number less than 10 are 2, 3, 5 and 7.

Therefore,

the relation in roaster form is ,

**Question:8 **Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.

**Answer:**

It is given that

A = {x, y, z} and B = {1, 2}

Now,

Therefore,

Then, the number of subsets of the set

Therefore, the number of relations from A to B is

**Question:9 **Let R be the relation on Z defined by

Find the domain and range of R.

**Answer:**

It is given that

Now, as we know that the difference between any two integers is always an integer.

And

As Domain of R = set of all first elements of the order pairs in the relation.

Therefore,

The domain of R = Z

Now,

Range of R = set of all second elements of the order pairs in the relation.

Therefore,

range of R = Z

Therefore, the domain and range of R is **Z **and **Z ** respectively

**Answer:**

Since, 2, 5, 8, 11, 14 and 17 are the elements of domain R having their unique images. Hence, this relation R is a function.

Now,

As Domain of R = set of all first elements of the order pairs in the relation.

Therefore,

Domain of

Now,

As Range of R = set of all second elements of the order pairs in the relation.

Therefore,

Range of

Therefore, domain and range of R are respectively

**Answer:**

Since, 2, 4, 6, 8, 10,12 and 14 are the elements of domain R having their unique images. Hence, this relation R is a function.

Now,

As Domain of R = set of all first elements of the order pairs in the relation.

Therefore,

Domain of

Now,

As Range of R = set of all second elements of the order pairs in the relation.

Therefore,

Range of

Therefore, domain and range of R are respectively

**Question:1 (iii)** Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range. {(1,3), (1,5), (2,5)}.

**Answer:**

Since the same first element 1 corresponds to two different images 3 and 5. Hence, this relation is not a function.

**Question:2** **(i)**Find the domain and range of the following real functions:

**Answer:**

Given function is

_{Now, we know that}

Now, for a function f(x),

**Domain: The values that can be put in the function to obtain real value. For example f(x) = x, now we can put any value in place of x and we will get a real value. Hence, the domain of this function will be Real Numbers.**

**Range: The values that we obtain of the function after putting the value from domain. For Example: f(x) = x + 1, now if we put x = 0, f(x) = 1. This 1 is a value of Range that we obtained.**

Since f(x) is defined for , **the domain of f is R.**

It can be observed that the range of f(x) = -|x| is all real numbers except positive real numbers. Because will always get a negative number when we put a value from the domain.

Therefore, the range of f is

**Question:2 (ii) ** Find the domain and range of the following real functions:

**Answer:**

Given function is

Now,

Domain: These are the values of x for which f(x) is defined.

for the given f(x) we can say that, f(x) should be real and for that,9 - x^{2} ≥ 0 [Since a value less than 0 will give an imaginary value]

Therefore,

The domain of f(x) is

Now,

If we put the value of x from we will observe that the value of function varies from **0 to 3**

Therefore,

Range of f(x) is

**Question:3 (i) ** A function f is defined by f(x) = 2x –5. Write down the values of f (0),

**Answer:**

Given function is

Now,

Therefore,

Value of **f(0) **is **-5**

**Question:3 (ii)** A function f is defined by f(x) = 2x –5. Write down the values of f (7)

**Answer:**

Given function is

Now,

Therefore,

Value of **f(7) **is **9**

**Question:3** **(iii)**A function f is defined by f(x) = 2x –5. Write down the values of f (-3)

**Answer:**

Given function is

Now,

Therefore,

Value of **f(-3) **is **-11**

**Question:4(i) **The function ‘t’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t (0)

**Answer:**

Given function is

Now,

Therefore,

Value of **t(0) **is **32**

**Question:4(ii) **The function ‘t’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t (28)

**Answer:**

Given function is

Now,

Therefore,

Value of **t(28) **is ** **

**Question:4(iii) **The function ‘t’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t (-10)

**Answer:**

Given function is

Now,

Therefore,

Value of **t(-10) **is ** 14**

**Answer:**

Given function is

Now,

Therefore,

When t(C) = 212 , value of C** **is ** 100 **

**Question:5 (i) ** Find the range of each of the following functions.

**Answer:**

Given function is

It is given that

Now,

Add 2 on both the sides

Therefore,

Range of function is

**Question:5 (ii) **Find the range of each of the following functions

**Answer:**

Given function is

It is given that **x **is a real number

Now,

Add 2 on both the sides

Therefore,

Range of function is

**Question:5 (iii)** Find the range of each of the following functions.

**Answer:**

Given function is

It is given that **x **is a real number

Therefore,

Range of function is **R**

**Answer:**

It is given that

Now,

And

At x = 3,

Also, at x = 3,

We can see that for , f(x) has unique images.

Therefore, By definition of a function, the given relation is function.

Now,

It is given that

Now,

And

At x = 2,

Also, at x = 2,

We can clearly see that element 2 of the domain of relation g(x) corresponds to two different images i.e. 4 and 6. Thus, f(x) does not have unique images

Therefore, by definition of a function, the given relation is not a function

**Hence proved **

**Question:3 **Find the domain of the function

**Answer:**

Given function is

Now, we will simplify it into

Now, we can clearly see that

Therefore, the Domain of **f(x) **is

**Question:4** Find the domain and the range of the real function f defined by

**Answer:**

Given function is

We can clearly see that **f(x) is only defined** for the values of x ,

Therefore,

The domain of the function is

Now, as

take square root on both sides

Therefore,

Range of function is

**Question:5** Find the domain and the range of the real function f defined by

**Answer:**

Given function is

As the given function is defined of all real number

The domain of the function is **R**

Now, as we know that the mod function always gives only positive values

Therefore,

Range of function is all non-negative real numbers i.e.

**Question:6** Let R be a function from R into R. Determine the range of f.

**Answer:**

Given function is

Range of any function is the set of values obtained after the mapping is done in the domain of the function. So every value of the codomain that is being mapped is Range of the function.

Let's take

Now, 1 - y should be greater than zero and y should be greater than and equal to zero for x to exist because other than those values the x will be imaginary

Thus,

Therefore,

Range of given function is

**Question:7** Let f, g : R R be defined, respectively by f(x) = x + 1, g(x) = 2x – 3. Find f + g, f – g and f/g

**Answer:**

It is given that

Now,

Therefore,

Now,

Therefore,

Now,

Therefore, values of are respectively

**Answer:**

It is given that

And

Now,

At **x = 1 ,
**

Similarly,

At ,

Now, put this value of

we will get,

Therefore, values of

**Question:9 (i)** Let R be a relation from N to N defined by . Are the following true?

**Answer:**

It is given that

And

for all

Now, it can be seen that But,

Therefore, this statement is **FALSE **

**Question:9 (ii)** Let R be a relation from N to N defined by . Are the following true?

**Answer:**

It is given that

And

implies

Now , it can be seen that and , But

Therefore,

Therefore, given statement is **FALSE**

**Question:9 (iii)** Let R be a relation from N to N defined by . Are the following true?

(a,b) R, (b,c) R implies (a,c) R.

**Answer:**

It is given that

And

implies

Now, it can be seen that because and , But

Therefore,

Therefore, the given statement is **FALSE**

**Answer:**

It is given that

and

Now,

Now, a relation from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B

And we can see that f is a subset of

Hence f is a relation from A to B

Therefore, given statement is **TRUE**

**Answer:**

It is given that

and

Now,

As we can observe that same first element i.e. 2 corresponds to two different images that is 9 and 11.

Hence f is not a function from A to B

Therefore, given statement is **FALSE**

**Question:11** Let f be the subset of defined by . Is f a function from Z to Z? Justify your answer.

**Answer:**

It is given that

Now, we know that relation f from a set A to a set B is said to be a function only if every element of set A has a unique image in set B

Now, for value 2, 6, -2, -6

Now, we can observe that same first element i.e. 12 corresponds to two different images that are 8 and -8.

Thus, **f ** is not a function

**Answer:**

It is given that

A = {9,10,11,12,13}

*And
f : A*

Now,

Prime factor of 9 = 3

Prime factor of 10 = 2,5

Prime factor of 11 = 11

Prime factor of 12 = 2,3

Prime factor of 13 = 13

f(n) = the highest prime factor of n.

Hence,

f(9) = the highest prime factor of 9 = 3

f(10) = the highest prime factor of 10 = 5

f(11) = the highest prime factor of 11 = 11

f(12) = the highest prime factor of 12 = 3

f(13) = the highest prime factor of 13 = 13

As the range of f is the set of all f(n), where

Therefore, the range of f is: {3, 5, 11, 13}.

- NCERT solutions for class 11 maths chapter 2 relations and functions will build your basics of functions which will be helpful in 12th board exams also.
- All these questions are prepared and explained in a detailed manner so it will be very easy for you to understand the concepts
- CBSE NCERT solutions for class 11 maths chapter 2 relations and functions ll develop yous basic concept which will be helpful in further studies relational algebra, relational calculus, statistics, machine learning, etc.
- Tip- Only reading the solutions won't help, you should try to solve on your own. If you are not able to solve, you can take the help of NCERT solutions for class 11 maths chapter 2 relations and functions.

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