Class 11 Mathematics

Chapter 1 Solutions — Sets

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Step-by-step NCERT solutions for Sets (Chapter 1, NCERT Class 11 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Sets textbook chapter.

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All 49 questions in Sets are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise 1.1

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…
2.
Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces:
- (i) 5 ... A
- (ii) 8 ... A
- (iii) 0 ... A
- (iv) 4 ... A
- (v) 2 ... A
- (vi) 10 ... A
3.
Write the following sets in roster form:
- (i) A = {x : x is an integer and –3 ≤ x < 7}
- (ii) B = {x : x is a natural number less than 6}
- (iii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}
- (iv) D = {x : x is a prime number which is divisor of 60}
- (v) E = The set of all letters in the word TRIGONOMETRY
- (vi) F = The set of all letters in the word BETTER
4.
Write the following sets in the set-builder form:
- (i) {3, 6, 9, 12}
- (ii) {2, 4, 8, 16, 32}
- (iii) {5, 25, 125, 625}
- (iv) {2, 4, 6, ...}
- (v) {1, 4, 9, ..., 100}
5.
List all the elements of the following sets:
- (i) A = {x : x is an odd natural number}
- (ii) B = {x : x is an integer, –1/2 < x < 9/2}
- (iii) C = {x : x is an integer, x² ≤ 4}
- (iv) D = {x : x is a letter in the word "LOYAL"}
- (v) E = {x : x is a month of a year not having 31 days}
- (vi) F = {x : x is a consonant in the English alphabet which precedes k}
6.
Match each of the set on the left in the roster form with the same set on the right described in set-builder form:
- (i) {1, 2, 3, 6}
- (a) {x : x is a prime number and a divisor of 6}
- (ii) {2, 3}
- (b) {x : x is an odd natural number less than 10}
- (iii) {M, A, T, H, E, I, C, S}
- (c) {x : x is natural number and divisor of 6}
- (iv) {1, 3, 5, 7, 9}
- (d) {x : x is a letter of the word MATHEMATICS}

Exercise 1.2

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) {x : x is a natural number, x < 5 and x > 7}
- (iv) {y : y is a point common to any two parallel lines}
2.
Which of the following sets are finite or infinite?
- (i) The set of months of a year
- (ii) {1, 2, 3, ...}
- (iii) {1, 2, 3, ..., 99, 100}
- (iv) The set of positive integers greater than 100
- (v) The set of prime numbers less than 99
3.
State whether each of the following set 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)
4.
In the following, state whether A = B or not:
- (i) A = {a, b, c, d}, B = {d, c, b, a}
- (ii) A = {4, 8, 12, 16}, B = {8, 4, 16, 18}
- (iii) A = {2, 4, 6, 8, 10}, B = {x : x is positive even integer and x ≤ 10}
- (iv) A = {x : x is a multiple of 10}, B = {10, 15, 20, 25, 30, ...}
5.
Are the following pair of sets equal? Give reasons.
- (i) A = {2, 3}, B = {x : x is solution of x² + 5x + 6 = 0}
- (ii) A = {x : x is a letter in the word FOLLOW}, B = {y : y is a letter in the word WOLF}
From the sets given below, select equal sets: A = {2, 4, 8, 12}, B = {1, 2, 3, 4}, C = {4, 8, 12, 14}, D = {3, 1, 4, 2}, E = {–1, 1}, F = {0, a}, G = {1, –1}, H = {0, 1}

Exercise 1.3

1.
Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces:
- (i) {2, 3, 4} ... {1, 2, 3, 4, 5}
- (ii) {a, b, c} ... {b, c, d}
- (iii) {x : x is a student of Class XI of your school} ... {x : x is student of your school}
- (iv) {x : x is a circle in the plane} ... {x : x is a circle in the same plane with radius 1 unit}
- (v) {x : x is a triangle in a plane} ... {x : x is a rectangle in…
2.
Examine whether the following statements are true or false:
- (i) {a, b} ⊄ {b, c, a}
- (ii) {a, e} ⊂ {x : x is a vowel in the English alphabet}
- (iii) {1, 2, 3} ⊂ {1, 3, 5}
- (iv) {a} ⊂ {a, b, c}
- (v) {a} ∈ {a, b, c}
- (vi) {x : x is an even natural number less than 6} ⊂ {x : x is a natural number which divides 36}
3.
Let A = {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why?
- (i) {3, 4} ⊂ A
- (ii) {3, 4} ∈ A
- (iii) {{3, 4}} ⊂ A
- (iv) 1 ∈ A
- (v) 1 ⊂ A
- (vi) {1, 2, 5} ⊂ A
- (vii) {1, 2, 5} ∈ A
- (viii) {1, 2, 3} ⊂ A
- (ix) φ ∈ A
- (x) φ ⊂ A
- (xi) {φ} ⊂ A
4.
Write down all the subsets of the following sets:
- (i) {a}
- (ii) {a, b}
- (iii) {1, 2, 3}
- (iv) φ
5.
Write the following as intervals:
- (i) {x : x ∈ R, –4 < x ≤ 6}
- (ii) {x : x ∈ R, –12 < x < –10}
- (iii) {x : x ∈ R, 0 ≤ x < 7}
- (iv) {x : x ∈ R, 3 ≤ x ≤ 4}
6.
Write the following intervals in set-builder form:
- (i) (–3, 0)
- (ii) [6, 12]
- (iii) (6, 12]
- (iv) [–23, 5)
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.
8.
Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universal set(s) for all the three sets A, B and C?
- (i) {0, 1, 2, 3, 4, 5, 6}
- (ii) φ
- (iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
- (iv) {1, 2, 3, 4, 5, 6, 7, 8}

Exercise 1.4

1.
Find the union of each of the following pairs of sets:
- (i) X = {1, 3, 5}, Y = {1, 2, 3}
- (ii) A = {a, e, i, o, u}, B = {a, b, c}
- (iii) A = {x : x is a natural number and multiple of 3}, B = {x : x is a natural number less than 6}
- (iv) A = {x : x is a natural number and 1 < x ≤ 6}, B = {x : x is a natural number and 6 < x < 10}
- (v) A = {1, 2, 3}, B = φ
Let A = {a, b}, B = {a, b, c}. Is A ⊂ B? What is A ∪ B?
If A and B are two sets such that A ⊂ B, then what is A ∪ B?
4.
If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find
- (i) A ∪ B
- (ii) A ∪ C
- (iii) B ∪ C
- (iv) B ∪ D
- (v) A ∪ B ∪ C
- (vi) A ∪ B ∪ D
- (vii) B ∪ C ∪ D
Find the intersection of each pair of sets of question 1 above.
6.
If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find
- (i) A ∩ B
- (ii) B ∩ C
- (iii) A ∩ C ∩ D
- (iv) A ∩ C
- (v) B ∩ D
- (vi) A ∩ (B ∪ C)
- (vii) A ∩ D
- (viii) A ∩ (B ∪ D)
- (ix) (A ∩ B) ∩ (B ∪ C)
- (x) (A ∪ D) ∩ (B ∪ C)
7.
If A = {x : x is a natural number}, B = {x : x is an even natural number}, C = {x : x is an odd natural number} and D = {x : x is a prime number}, find
- (i) A ∩ B
- (ii) A ∩ C
- (iii) A ∩ D
- (iv) B ∩ C
- (v) B ∩ D
- (vi) C ∩ D
8.
Which of the following pairs of sets are disjoint?
- (i) {1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6}
- (ii) {a, e, i, o, u} and {c, d, e, f}
- (iii) {x : x is an even integer} and {x : x is an odd integer}
9.
If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}; find
- (i) A – B
- (ii) A – C
- (iii) A – D
- (iv) B – A
- (v) C – A
- (vi) D – A
- (vii) B – C
- (viii) B – D
- (ix) C – B
- (x) D – B
- (xi) C – D
- (xii) D – C
10.
If X = {a, b, c, d} and Y = {f, b, d, g}, find
- (i) X – Y
- (ii) Y – X
- (iii) X ∩ Y
If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?
12.
State whether each of the following statement is true or false. Justify your answer.
- (i) {2, 3, 4, 5} and {3, 6} are disjoint sets.
- (ii) {a, e, i, o, u} and {a, b, c, d} are disjoint sets.
- (iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.
- (iv) {2, 6, 10} and {3, 7, 11} are disjoint sets.

Exercise 1.5

1.
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find
- (i) A′
- (ii) B′
- (iii) (A ∪ C)′
- (iv) (A ∪ B)′
- (v) (A′)′
- (vi) (B – C)′
2.
If U = {a, b, c, d, e, f, g, h}, find the complements of the following sets:
- (i) A = {a, b, c}
- (ii) B = {d, e, f, g}
- (iii) C = {a, c, e, g}
- (iv) D = {f, g, h, a}
3.
Taking the set of natural numbers as the universal set, write down the complements of the following sets:
- (i) {x : x is an even natural number}
- (ii) {x : x is an odd natural number}
- (iii) {x : x is a positive multiple of 3}
- (iv) {x : x is a prime number}
- (v) {x : x is a natural number divisible by 3 and 5}
- (vi) {x : x is a perfect square}
- (vii) {x : x is a perfect cube}
- (viii) {x : x + 5 = 8}…
4.
If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that
- (i) (A ∪ B)′ = A′ ∩ B′
- (ii) (A ∩ B)′ = A′ ∪ B′
5.
Draw appropriate Venn diagram for each of the following:
- (i) (A ∪ B)′
- (ii) A′ ∩ B′
- (iii) (A ∩ B)′
- (iv) A′ ∪ B′
Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A′?
7.
Fill in the blanks to make each of the following a true statement:
- (i) A ∪ A′ = ...
- (ii) φ′ ∩ A = ...
- (iii) A ∩ A′ = ...
- (iv) U′ ∩ A = ...

Miscellaneous Exercise

Decide, among the following sets, which sets are subsets of one and another: A = {x : x ∈ R and x satisfy x² – 8x + 12 = 0}, B = {2, 4, 6}, C = {2, 4, 6, 8, ...}, D = {6}.
2.
In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
- (i) If x ∈ A and A ∈ B, then x ∈ B
- (ii) If A ⊂ B and B ∈ C, then A ∈ C
- (iii) If A ⊂ B and B ⊂ C, then A ⊂ C
- (iv) If A ⊄ B and B ⊄ C, then A ⊄ C
- (v) If x ∈ A and A ⊄ B, then x ∉ B
- (vi) If A ⊂ B and x ∉ B, then x ∉ A
Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show that B = C.
4.
Show that the following four conditions are equivalent:
- (i) A ⊂ B
- (ii) A – B = φ
- (iii) A ∪ B = B
- (iv) A ∩ B = A
Show that if A ⊂ B, then C – B ⊂ C – A.
Show that for any sets A and B, A = (A ∩ B) ∪ (A – B) and A ∪ (B – A) = A ∪ B.
7.
Using properties of sets, show that
- (i) A ∪ (A ∩ B) = A
- (ii) A ∩ (A ∪ B) = A
Show that A ∩ B = A ∩ C need not imply B = C.
Let A and B be sets. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set X, show that A = B. (Hint: A = A ∩ (A ∪ X), B = B ∩ (B ∪ X) and use Distributive law)
Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty sets and A ∩ B ∩ C = φ.

Sample

Worked examples

3 questions from Sets, solved step by step.

Exercise 1.1

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces: (i) 5 ... A (ii) 8 ... A (iii) 0 ... A (iv) 4 ... A (v) 2 ... A (vi) 10 ... A

A = {1, 2, 3, 4, 5, 6}.

(i) 5 ∈ A, since 5 is an element of A.

(ii) 8 ∉ A, since 8 is not an element of A.

(iii) 0 ∉ A, since 0 is not an element of A.

(iv) 4 ∈ A, since 4 is an element of A.

(v) 2 ∈ A, since 2 is an element of A.

(vi) 10 ∉ A, since 10 is not an element of A.

Answer: (i) 5 ∈ A (ii) 8 ∉ A (iii) 0 ∉ A (iv) 4 ∈ A (v) 2 ∈ A (vi) 10 ∉ A

Exercise 1.2

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) {x : x is a natural number, x < 5 and x > 7} (iv) {y : y is a point common to any two parallel lines}

(i) No odd natural number is divisible by 2 (odd and even are mutually exclusive). This set has NO elements — it IS a null set.

(ii) The even prime numbers: 2 is the only even prime. So this set = {2}, which is NOT empty.

(iii) There is no natural number that is both less than 5 and greater than 7 simultaneously. This set has NO elements — it IS a null set.

(iv) Two parallel lines, by definition, never meet. So they have no common point. This set has NO elements — it IS a null set.

Answer: (i), (iii), and (iv) are null sets. (ii) is not a null set since it contains the element 2.

Exercise 1.3

Write down all the subsets of the following sets: (i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) φ

(i) Subsets of {a}: the empty set and {a} itself.

φ, {a}.

(ii) Subsets of {a, b}: φ, {a}, {b}, {a, b}.

(iii) Subsets of {1, 2, 3}: φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}.

(Total: 2³ = 8 subsets.)

(iv) Subsets of φ: only φ itself.

φ.

Answer: (i) φ, {a} (ii) φ, {a}, {b}, {a, b} (iii) φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} (iv) φ

All 49 questions in Sets are solved step by step in the Solutions PDF above.

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Chapter 1 Solutions — Sets

Overview

What these solutions cover

Exercise 1.1

Exercise 1.2

Exercise 1.3

Exercise 1.4

Exercise 1.5

Miscellaneous Exercise

Worked examples

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces: (i) 5 ... A (ii) 8 ... A (iii) 0 ... A (iv) 4 ... A (v) 2 ... A (vi) 10 ... A

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) {x : x is a natural number, x < 5 and x > 7} (iv) {y : y is a point common to any two parallel lines}

Write down all the subsets of the following sets: (i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) φ

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