NCERT Mathematics Chapter 1 -

Sets

(All Problems with answers)

Exercise 1.1

Question 1:
Which of the following are sets? Justify your answer.

(i) The collection of all the months of a year beginning with the letter J.
(ii) The collection of ten most talented writers of India.
(iii) A team of eleven best-cricket batsmen of the world.
(iv) The collection of all boys in your class.
(v) The collection of all natural numbers less than 100.
(vi) A collection of novels written by the writer Munshi Prem Chand.
(vii) The collection of all even integers.
(viii) The collection of questions in this Chapter.
(ix) A collection of most dangerous animals of the world.

Explanation:
A set is a well-defined collection of objects where it is possible to decide whether an object belongs to the collection or not.

Answer:
(i) Yes, it is a set because the months beginning with J are well-defined: {January, June, July}.
(ii) No, because “most talented” is subjective and varies from person to person, so it is not well-defined.
(iii) No, “best-cricket batsmen” is subjective and varies, so not well-defined.
(iv) Yes, it is a set because all boys in your class can be clearly identified.
(v) Yes, it is a set: {1, 2, 3, …, 99}.
(vi) Yes, it is a set because novels written by Munshi Prem Chand are well-defined.
(vii) Yes, it is a set: {…, -4, -2, 0, 2, 4, …}.
(viii) Yes, it is a set because the questions in the chapter are clearly defined.
(ix) No, “most dangerous animals” is subjective and varies, so not well-defined.

Question 2:
Let

. Insert the appropriate symbol

in the blanks:

(i) 5 __ A
(ii) 8 __ A
(iii) 0 __ A
(iv) 4 __ A
(v) 2 __ A
(vi) 10 __ A

Explanation:
An element

belongs to set

is listed as an element of

Answer:
(i)

(ii)

(iii)

(iv)

(v)

(vi)

Question 3:
Write the following sets in roster form:

(i)

(ii)

(iii)

(iv)

(v)

The set of all letters in the word “TRIGONOMETRY”
(vi)

The set of all letters in the word “BETTER”

Explanation:
Roster form lists all elements explicitly within braces.

Answer:
(i)

(ii)

(iii) Two-digit numbers whose digits sum to 8 are: 17, 26, 35, 44, 53, 62, 71, 80
So,

(iv) Prime divisors of 60 are 2, 3, 5
So,

(v) Letters in “TRIGONOMETRY” without repetition:

(vi) Letters in “BETTER” without repetition:

Question 4:
Write the following sets in set-builder form:

(i)

(ii)

(iii)

(iv)

(v)

Explanation:
Set-builder form describes the property that elements satisfy.

Answer:
(i)

(ii)

(iii)

(iv)

(v)

Question 5:
List all the elements of the following sets:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Explanation:
Identify elements based on the given conditions.

Answer:
(i)

(infinite set of odd natural numbers)
(ii) Integers between 0.5 and 4.5 are 1, 2, 3, 4
So,

(iii) Integers with square ≤ 4 are -2, -1, 0, 1, 2
So,

(iv) Letters in “LOYAL” are L, O, Y, A
So,

(v) Months not having 31 days: April, June, September, November
So,

(vi) Consonants before ‘k’ are b, c, d, f, g, h, j
So,

Question 6:
Match each of the sets on the left in roster form with the same set on the right described in set-builder form:

(i)

(ii)

(iii)

(iv)

with

(a)

(b)

(d)

Explanation:
Match sets by comparing elements and their defining properties.

Answer:
(i) matches © because 1, 2, 3, 6 are divisors of 6 (natural numbers).
(ii) matches (a) because 2 and 3 are prime divisors of 6.
(iii) matches (d) because the letters correspond to the word MATHEMATICS.
(iv) matches (b) because 1, 3, 5, 7, 9 are odd natural numbers less than 10.

e PDF in question, explanation, and answer format:

Exercise 1.2

Question 1:
Which of the following are examples of the null set?

(i) Set of odd natural numbers divisible by 2
(ii) Set of even prime numbers
(iii)

(iv)

Explanation:
A null set (empty set) is a set that contains no elements.

Answer:
(i) Null set, because no odd number is divisible by 2.
(ii) Null set, because 2 is the only even prime number, so no even prime number other than 2 exists.
(iii) Null set, because no natural number is both less than 5 and greater than 7 simultaneously.
(iv) Null set, because two parallel lines do not intersect, so no common point exists.

Question 2:
Which of the following sets are finite or infinite?

(i) The set of months of a year
(ii)

(iii)

(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99

Explanation:
A finite set has a definite number of elements; an infinite set has infinitely many elements.

Answer:
(i) Finite (12 months)
(ii) Infinite (natural numbers)
(iii) Finite (100 elements)
(iv) Infinite (all positive integers greater than 100)
(v) Finite (prime numbers less than 99 are finite in number)

Question 3:
State whether each of the following sets is finite or infinite:

(i) The set of lines which are parallel to the x-axis
(ii) The set of letters in the English alphabet
(iii) The set of numbers which are multiples of 5
(iv) The set of animals living on the earth
(v) The set of circles passing through the origin (0,0)

Explanation:
Countability or size of the sets determines finiteness or infiniteness.

Answer:
(i) Infinite (infinitely many lines parallel to x-axis)
(ii) Finite (26 letters)
(iii) Infinite (multiples of 5 continue indefinitely)
(iv) Infinite (large but finite in reality, but considered infinite in mathematics)
(v) Infinite (infinitely many circles can pass through a point)

Question 4:
In the following, state whether

or not:

(i)

(ii)

(iii)

(iv)

Explanation:
Two sets are equal if they contain exactly the same elements.

Answer:
(i)

(order does not matter in sets)
(ii)

(18 is in

but not in

)
(iii)

(both represent even numbers up to 10)
(iv)

(15 and 25 are not multiples of 10)

Question 5:
Are the following pairs of sets equal? Give reasons.

(i)

(ii)

Explanation:
Find elements of each set and compare.

Answer:
(i) The equation factors as

, so solutions are

.
Thus,

. So, not equal.
(ii) Letters in FOLLOW: {F, O, L, L, O, W} → {F, O, L, W}
Letters in WOLF: {W, O, L, F}
So,

Question 6:
From the sets given below, select equal sets:

Explanation:
Compare sets to find pairs with exactly the same elements.

Answer:

(both have {1, 2, 3, 4}),

(both have {1, -1}),

Exercise 1.3

Question 1:
Make correct statements by filling in the symbols

in the blanks:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

Explanation:
A set

is a subset of

(denoted

) if every element of

is also in

Answer:
(i)

(all elements of {2,3,4} are in {1,2,3,4,5})
(ii)

(a is not in {b,c,d})
(iii)

(students of Class XI are students of the school)
(iv)

(circles with radius 1 are a subset of all circles, not vice versa)
(v)

(triangles are not rectangles)
(vi)

(equilateral triangles are a subset of triangles)
(vii)

(even natural numbers are integers)

Question 2:
Examine whether the following statements are true or false:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Explanation:
Check if every element of the first set is in the second set; also distinguish between element and subset.

Answer:
(i) False (actually

)
(ii) True (a and e are vowels)
(iii) False (2 is not in the second set)
(iv) True (a is in the second set)
(v) False ({a} is a set, not an element of the second set)
(vi) True (even natural numbers less than 6 are 2 and 4, both divide 36)

Question 3:
Let

. Which of the following statements are incorrect and why?

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

Explanation:
Remember the difference between element (

) and subset (

) and that elements inside sets can themselves be sets.

Answer:
(i) Incorrect.

is an element of

, not a subset.
(ii) Correct.

.
(iii) Correct.

because

.
(iv) Correct.

.
(v) Incorrect.

is an element, not a set, so cannot be a subset.
(vi) Correct.

because 1, 2, and 5 are elements of

.
(vii) Incorrect.

is not an element of

.
(viii) Incorrect. 3 is not an element of

(only

is).
(ix) Incorrect.

is not an element of

.
(x) Correct. The empty set is a subset of every set.
(xi) Incorrect.

is not a subset of

because

Question 4:
Write down all the subsets of the following sets:

(i)

(ii)

(iii)

(iv)

Explanation:
The number of subsets of a set with

elements is

Answer:
(i) Subsets of

(ii) Subsets of

(iii) Subsets of

(iv) Subsets of

only

Question 5:
Write the following as intervals:

(i)

(ii)

(iii)

(iv)

Explanation:
Intervals are written using parentheses for open ends and brackets for closed ends.

Answer:
(i)

(ii)

(iii)

(iv)

Question 6:
Write the following intervals in set-builder form:

(i)

(ii)

(iii)

(iv)

Explanation:
Set-builder form describes the set of all

satisfying inequalities corresponding to the interval.

Answer:
(i)

(ii)

(iii)

(iv)

Question 7:
What universal set(s) would you propose for each of the following:

(i) The set of right triangles.
(ii) The set of isosceles triangles.

Explanation:
The universal set should be a set that contains all elements under consideration.

Answer:
(i) The universal set can be the set of all triangles in a plane.
(ii) The universal set can also be the set of all triangles in a plane.

Question 8:
Given the sets

, and

, which of the following may be considered as universal set(s) for all three sets

(i)

(ii)

(iii)

(iv)

Explanation:
A universal set must contain all elements of the sets

Answer:
(i) Contains all elements of

except 8 (missing), so not universal.
(ii) Empty set cannot be universal.
(iii) Contains all elements of

, so it can be universal.
(iv) Contains all elements of

, but missing 0 and 8 from

, so not universal.

Exercise 1.4

Question 1:
Find the union of each of the following pairs of sets:

(i)

(ii)

(iii)

(iv)

(v)

Explanation:
The union of two sets

and

, denoted

, is the set of all elements that belong to

or both.

Answer:
(i)

(ii)

(iii) Multiples of 3 are

, natural numbers less than 6 are

So,

(iv)

So,

(v)

Question 2:
Let

. Is

? What is

Explanation:

means every element of

is in

. The union is all elements in either set.

Answer:
Yes,

because

Question 3:
If

and

are two sets such that

, then what is

Explanation:
If

, then all elements of

are already in

Answer:

Question 4:
If

, and

, find:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

Explanation:
Union combines all unique elements from the sets.

Answer:
(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

Question 5:
Find the intersection of each pair of sets in Question 1.

Explanation:
The intersection

is the set of elements common to both

and

Answer:
(i)

(ii)

(iii)

(3 is the only natural number less than 6 and multiple of 3)
(iv)

(no common elements)
(v)

Question 6:
If

, and

, find:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

Explanation:
Find common elements for intersections and unions as needed.

Answer:
(i)

(ii)

(iii)

(no element common to all three)
(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

Question 7:
If

, and

, find:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Explanation:
Intersections of these sets correspond to numbers satisfying both conditions.

Answer:
(i)

(even natural numbers)
(ii)

(odd natural numbers)
(iii)

(prime numbers, all natural)
(iv)

(no number is both even and odd)
(v)

(2 is the only even prime)
(vi) Odd prime numbers (3, 5, 7, 11, …)

Question 8:
Which of the following pairs of sets are disjoint?

(i)

and

(ii)

and

(iii)

and

Explanation:
Two sets are disjoint if they have no elements in common.

Answer:
(i) Not disjoint (4 is common)
(ii) Not disjoint (e is common)
(iii) Disjoint (no integer is both even and odd)

Question 9:
If

, and

, find:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

(xii)

Explanation:
The difference

is the set of elements in

but not in

Answer:
(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(all elements of

are in

)
(viii)

(ix)

(x)

(xi)

(xii)

Question 10:
If

and

, find:

(i)

(ii)

(iii)

Explanation:
Find elements in one set but not the other, and common elements.

Answer:
(i)

(ii)

(iii)

Question 11:
If

is the set of real numbers and

is the set of rational numbers, then what is

Explanation:

is the set of real numbers that are not rational.

Answer:

is the set of irrational numbers.

Question 12:
State whether each of the following statements is true or false. Justify your answer.

(i)

and

are disjoint sets.
(ii)

and

are disjoint sets.
(iii)

and

are disjoint sets.
(iv)

and

are disjoint sets.

Explanation:
Two sets are disjoint if they have no common elements.

Answer:
(i) False (3 is common)
(ii) False (a is common)
(iii) True (no common elements)
(iv) True (no common elements)

Exercise 1.5

Question 1:
Let

, and

. Find:

(i)

(complement of

)
(ii)

(iii)

(iv)

(v)

(vi)

Explanation:
The complement

of a set

with respect to universal set

is the set of elements in

not in

. The difference

is elements in

not in

Answer:
(i)

(ii)

(iii)

, so

(iv)

, so

(v)

(vi)

, so

Question 2:
If

, find the complements of the following sets:

(i)

(ii)

(iii)

(iv)

Explanation:
Complement of a set is all elements in

not in the set.

Answer:
(i)

(ii)

(iii)

(iv)

Question 3:
Taking the set of natural numbers as the universal set, write down the complements of the following sets:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

Explanation:
Complement contains all natural numbers not in the given set.

Answer:
(i) Odd natural numbers
(ii) Even natural numbers
(iii) Natural numbers not multiples of 3
(iv) Natural numbers that are not prime
(v) Natural numbers not divisible by both 3 and 5
(vi) Natural numbers that are not perfect squares
(vii) Natural numbers that are not perfect cubes
(viii) All natural numbers except 3 (since

)
(ix) All natural numbers except 2 (since

)
(x) Natural numbers less than 7
(xi) Natural numbers

such that

, i.e.,

Question 4:
If

, and

, verify that:

(i)

(ii)

Explanation:
These are De Morgan’s laws for sets.

Answer:
(i)

So,

(ii)

So,

Question 5:
Draw appropriate Venn diagrams for each of the following:

(i)

(ii)

(iii)

(iv)

Explanation:
Venn diagrams visually represent these set operations and their complements.

Answer:
(i) and (ii) represent the same shaded region outside both

and

.
(iii) and (iv) represent the shaded region outside the intersection of

and

, which is the union of the complements.

Question 6:
Let

be the set of all triangles in a plane. If

is the set of all triangles with at least one angle different from 60°, what is

Explanation:

is the complement of

Answer:

is the set of all equilateral triangles (all angles 60°).

Question 7:
Fill in the blanks to make each of the following a true statement:

(i)

(ii)

(iii)

(iv)

Explanation:
Use properties of sets and complements.

Answer:
(i)

(universal set)
(ii)

(since complement of empty set is universal set)
(iii)

(empty set)
(iv)

(complement of universal set is empty set)

Miscellaneous Exercise on Chapter 1

Question 1:
Decide, among the following sets, which sets are subsets of one another:

Explanation:
Find the elements of each set and check subset relations.

Answer:

Solve :

So,
(even natural numbers starting from 2)

Subset relations:

(2 and 6 are in )
(all elements of are in )
(6 is in )
(6 is in )
(2 and 6 are in )
(6 is in )

Question 2:
In each of the following, determine whether the statement is true or false. If true, prove it; if false, give an example.

(i) If

and

, then

(ii) If

and

, then

(iii) If

and

, then

(iv) If

and

, then

(v) If

and

, then

(vi) If

and

, then

Explanation:
Analyze each statement logically.

Answer:
(i) False. Example: Let

. Then

and

, but

.
(ii) False. Example:

is false here, but even if

is false.
(iii) True. If every element of

is in

, and every element of

is in

, then every element of

is in

.
(iv) False. Counterexample:

. Neither

nor

, but

.
(v) False.

means some element of

is not in

, so

does not imply

.
(vi) True. If

and

, then

Question 3:
Let

be sets such that

and

. Show that

Explanation:
Use set identities and properties to prove equality.

Answer:
To prove

, show

and

Let .
If , then , so .
If , then and , so , so or . Since , .
Thus, , so .
Similarly, .

Hence,

Question 4:
Show that the following four conditions are equivalent:

(i)

(ii)

(iii)

(iv)

Explanation:
Prove equivalence by showing each implies the next.

Answer:

(i) (ii): If , no element of lies outside , so .
(ii) (iii): If , then all elements of are in , so .
(iii) (iv): If , then , so .
(iv) (i): If , then every element of is in , so .

Question 5:
Show that if

, then

Explanation:
Use the definition of set difference and subset.

Answer:
Let

. Then

and

. Since

. Hence,

.
Therefore,

Question 6:
Show that for any sets

and

(i)

(ii)

Explanation:
Use set operations and properties.

Answer:
(i) Every element of

is either in

or not. Those in

are in

, those not in

are in

. So,

.
(ii)

contains elements in

not in

. Adding these to

gives all elements in

, so

Question 7:
Using properties of sets, show that:

(i)

(ii)

Explanation:
Use absorption laws.

Answer:
(i)

because

, so union with

.
(ii)

because

, so intersection is

Question 8:
Show that

need not imply

Explanation:
Provide a counterexample.

Answer:
Let

.
Then

, but

Question 9:
Let

and

be sets. If

and

for some set

, show that

Explanation:
Use distributive laws and given conditions.

Answer:
Given

, so

. Similarly for

.
Since

Similarly,

.
Hence,

Question 10:
Find sets

such that

, and

are non-empty sets, but

Explanation:
Construct sets with pairwise intersections but no common triple intersection.

Answer:
Let

.
Then

,
but

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Lets Learn Study Guide

NCERT Class 11 Maths Chapter 1 Sets – All Problems with Detailed Solutions

NCERT Mathematics Chapter 1 -

Sets

(All Problems with answers)

Exercise 1.1

Exercise 1.2

Exercise 1.3

Exercise 1.4

Exercise 1.5

Miscellaneous Exercise on Chapter 1

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