Chapter 1 Sets

NCERT Solutions for Class 11 Maths Chapter 1 Sets: Download PDF

NCERT Solutions for Class 11 Maths Chapter 1 (Sets) are prepared by expert faculty to build strong conceptual clarity from fundamentals. The solutions are simple, structured, and exam-oriented, helping students understand set theory with ease and accuracy. The concept of a "Set" can be found in almost every area of mathematics (Probability, Relations & Functions). Chapter 1 NCERT Solutions helps students build a strong understanding of Sets and provides step-by-step solutions to all NCERT textbook questions, enabling them to learn and practise at their own pace.

The understanding of Sets is essential for competitive exams like JEE Main and NDA since they are part of the logical/critical thinking framework necessary to solve complex Algebraic/Statistical problems. Chapter 1 NCERT Solutions include representations of the sets, properties of the sets and relationships between sets with the use of Venn Diagrams as an aid in visualizing the relationships between the different sets.

1.0Class 11 Maths Chapter 1 Sets: Key Concepts

Sets is an important chapter in NCERT Class 11 Maths. It introduces the meaning of a set, different types of sets, ways to represent sets, and operations on sets. The main concepts covered in NCERT Class 11 Maths Chapter 1 – Sets are:

Roster or Tabular Form: The elements of a set are written inside curly brackets { } and separated by commas.
Set-Builder Form: A set is represented by stating the common property or condition that its elements satisfy.
Empty Set: A set that contains no elements is called an empty set. It is represented by ∅ or { }.
Finite and Infinite Sets: A finite set has a fixed number of elements, while an infinite set has an unlimited number of elements.
Equal Sets: Two sets are equal if they contain exactly the same elements.
Subsets: A set A is a subset of a set B if every element of A is also an element of B. It is written as A ⊆ B.
Intervals as Subsets of R: Intervals are used to represent subsets of real numbers. Examples include the open interval (a, b) and the closed interval [a, b].
Power Set: The collection of all subsets of a set A is called the power set of A. It is represented by P(A).
Universal Set: The universal set contains all the objects or elements being considered in a particular situation. It is generally represented by U.
Venn Diagrams: Venn diagrams use closed shapes, usually circles inside a rectangle, to show the relationship between sets.

Operations on Sets

Union (A ∪ B): The union of A and B contains all elements that are in A, in B, or in both.
Intersection (A ∩ B): The intersection of A and B contains the elements that are common to both sets.
Difference (A − B): The difference A − B contains the elements that are in A but not in B.
Complement of a Set (A′): The complement of A contains all the elements of the universal set U that are not in A.

De Morgan's Laws

De Morgan's laws describe the relationship between the complement, union, and intersection of sets:

(A ∪ B)′ = A′ ∩ B′
(A ∩ B)′ = A′ ∪ B′

Understanding these concepts helps students solve questions based on sets, subsets, Venn diagrams, intervals, and operations on sets in Class 11 Maths.

2.0Detailed NCERT Textbook Solutions : Class 11 Maths Chapter 1

EXERCISE - 1.1

Which of the following are sets? Justify your answer. (i) The collection of all 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.

Sol.

(i) The collection of all months of a year beginning with the letter J is a well-defined collection of objects because one can definitely identify a month that belongs to this collection. Hence, this collection is a set. (ii) The collection of ten most talented writers of India is not a well-defined collection because the criteria for determining a writer's talent may vary from person to person. Hence, this collection is not a set. (iii) A team of eleven best cricket batsmen of the world is not a well-defined collection because the criteria for determining a batsman's talent may vary from person to person. Hence, this collection is not a set. (iv) The collection of all boys in your class is a welldefined collection because you can definitely identify a boy who belongs to this collection. Hence, this collection is a set. (v) The collection of all natural numbers less than 100 is a well-defined collection because one can definitely identify a number that belongs to this collection. Hence, this collection is a set. (vi) A collection of novels written by the writer Munshi Prem Chand is a well-defined collection because one can definitely identify a book that belongs to this collection. Hence, this collection is a set. (vii) The collection of all even integers is a welldefined collection because one can definitely identify an even integer that belongs to this collection. Hence, this collection is a set. (viii) The collection of questions in this chapter is a well-defined collection because one can definitely identify a question that belongs to this chapter. Hence, this collection is a set. (ix) The collection of most dangerous animals of the world is not a well-defined collection because the criteria for determining the dangerousness of an animal can vary from person to person. Hence, this collection is not a set.

2. Let $A = {1, 2, 3, 4, 5, 6}$ . Insert the appropriate symbol $\in$ or $\in /$ in the blank spaces:

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

Sol. (i) $5 \in A$

(iii) $0 \in / A$ (iv) $4 \in A$ (v) $2 \in A$ (vi) $10 \in / A$

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.

Sol.

(i) $A = {x : x$ is an integer and $- 3 < x < 7}$ The elements of this set are -2, -1, 0, 1, 2, 3, 4, 5, and 6 only. Therefore, the given set can be written in roster form as $A = {- 2, - 1, 0, 1, 2, 3, 4, 5, 6}$ (ii) $B = {x : x$ is a natural number less than 6} The elements of this set are 1, 2, 3, 4, and 5 only. Therefore, the given set can be written in roster form as $B = {1, 2, 3, 4, 5}$ (iii) $C = {x : x$ is a two-digit natural number such that the sum of its digits is 8} The elements of this set are 17, 26, 35, 44, 53, 62, 71, and 80 only. Therefore, this set can be written in roster form as $C = {17, 26, 35, 44, 53, 62, 71, 80}$ (iv) $D = {x : x$ is a prime number which is a divisor of 60} $60 = 2 \times 2 \times 3 \times 5$ The elements of this set are 2, 3, and 5 only. Therefore, this set can be written in roster form as $D = {2, 3, 5}$ . (v) $E =$ The set of all letters in the word TRIGONOMETRY. There are 12 letters in the word TRIGONOMETRY, out of which letters T, R and O are repeated. Therefore, this set can be written in roster form as $E = {T, R, I, G, O, N, M, E, Y}$ (vi) $F =$ The set of all letters in the word BETTER. There are 6 letters in the word BETTER, out of which letters E and T are repeated. Therefore, this set can be written in roster form as $F = {B, E, T, R}$

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) ${2, 4, 6 \dots}$ (v) ${1, 4, 9 \dots 100}$

Sol.

(i) ${3, 6, 9, 12} = {x : x = 3 n, n \in N$ and $1 \leq n \leq 4}$ (ii) ${2, 4, 8, 16, 32}$ It can be seen that $2 = 2^{1}, 4 = 2^{2}, 8 = 2^{3}$ , $16 = 2^{4}$ , and $32 = 2^{5}$ . $∴ {2, 4, 8, 16, 32} = {x : x = 2^{n}, n \in N$ and $1 \leq n \leq 5$ } (iii) ${5, 25, 125, 625}$ It can be seen that $5 = 5^{1}, 25 = 5^{2}, 125 = 5^{3}$ , and $625 = 5^{4}$ . $∴ {5, 25, 125, 625} = {x : x = 5^{n}, n \in N$ and $1 \leq n \leq 4}$ (iv) ${2, 4, 6 \dots}$ It is a set of all even natural numbers. $∴ {2, 4, 6 \dots} = {x : x$ is an even natural number $}$ (v) ${1, 4, 9 \dots 100}$ It can be seen that $1 = 1^{2}, 4 = 2^{2}, 9 = 3^{2} \dots 100$ $= 1 0^{2}$ . $∴ {1, 4, 9 \dots 100} = {x : x = n^{2}, n \in N$ and $1 \leq n \leq 10}$

5. List all the elements of the following sets: (i) $A = {x : x$ is an odd natural number $}$ (ii) $B = {x : x$ is an $- \frac{1}{2} < x < \frac{9}{2}$ integer $}$ (iii) $C = {x : x$ is an $x^{2} \leq 4$ integer $}$ (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 proceeds k }.

Sol.

(i) $A = {x : x$ is an odd natural number $} = {1, 3, 5$ , 7, 9 ...} (ii) B : ${x : x$ is an integer; $- \frac{1}{2} < x < \frac{9}{2}}$ It can be seen that $- \frac{1}{2} = (- 0.5)$ and $\frac{9}{2} = 4.5$ $∴ B = {0, 1, 2, 3, 4}$

(iii) $C = {x : x$ is an integer; $x^{2} \leq 4}$ It can be seen that

$(- 1)^{2} = 1 \leq 4; (- 2)^{2} = 4 \leq 4; (- 3)^{2} = 9 > 4 0^{2} = 0 \leq 4 1^{2} = 1 \leq 4 2^{2} = 4 \leq 4 3^{2} = 9 > 4$

$∴ C = {- 2, - 1, 0, 1, 2}$

(iv) $D = (x : x$ is a letter in the word "LOYAL")

$= {L, O, Y, A}$

(v) $E = {x : x$ is a month of a year nothaving 31 days $}$

$= {February, April, June, September, November}$

(vi) $F = {x : x$ is a consonant in the English alphabet which precedes k }

$= {b, c, d, f, g, h, j}$

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}

Sol.

(i) All the elements of this set are natural numbers as well as the divisors of 6 . Therefore, (I) matches with (C). (ii) It can be seen that 2 and 3 are prime numbers. They are also the divisors of 6. Therefore, (II) matches with (A). (iii) All the elements of this set are letters of the word MATHEMATICS. Therefore, (III) matches with (D). (iv) All the elements of this set are odd natural numbers less than 10. Therefore, (IV) matches with (B).

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

Sol.

(i) A set of odd natural numbers divisible by 2 is a null set because no odd number is divisible by 2 . (ii) A set of even prime numbers is not a null set because 2 is an even prime number. (iii) {x: x is a natural number, $x < 5$ and $x > 7$ } is a null set because a number cannot be simultaneously less than 5 and greater than 7. (iv) ${y : y$ is a point common to any two parallel lines} is a null set because parallel lines do not intersect. Hence, they have no common point.

2. Which of the following sets are finite or infinite (i) The set of months of a year (ii) ${1, 2, 3 \dots}$ (iii) ${1, 2, 3 \dots 99, 100}$ (iv) The set of positive integers greater than 100 (v) The set of prime numbers less than 99

Sol.

(i) The set of months of a year is a finite set because it has 12 elements. (ii) ${1, 2, 3 \dots}$ is an infinite set as it has infinite number of natural numbers. (iii) ${1, 2, 3 \dots 99, 100}$ is a finite set because the numbers from 1 to 100 are finite in number. (iv) The set of positive integers greater than 100 is an infinite set because positive integers greater than 100 are infinite in number. (v) The set of prime numbers less than 99 is a finite set because prime numbers less than 99 are finite in number.

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 multiple of 5 (iv) The set of animals living on the earth (v) The set of circles passing through the origin $(0, 0)$

Sol.

(i) The set of lines which are parallel to the x-axis is an infinite set because lines parallel to the x-axis are infinite in number. (ii) The set of letters in the English alphabet is a finite set because it has 26 elements. (iii) The set of numbers which are multiple of 5 is an infinite set because multiples of 5 are infinite in number. (iv) The set of animals living on the earth is a finite set because the number of animals living on the earth is finite (although it is quite a big number). (v) The set of circles passing through the origin $(0, 0)$ is an infinite set because infinite number of circles can pass through the origin.

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 $x \leq 10}$ (iv) $A = {x : x$ is a multiple of 10 $}; B = {10, 15, 20$ , 25, 30 ...}

Sol.

(i) $A = {a, b, c, d}; B = {d, c, b, a}$ The order in which the elements of a set are listed is not significant. $∴ A = B$ (ii) $A = {4, 8, 12, 16}; B = {8, 4, 16, 18}$ It can be seen that $12 \in A$ but $12 \in / B$ . $∴ A \neq = B$ (iii) $A = {2, 4, 6, 8, 10}$ $B = {x : x$ is a positive even integer and $x \leq 10}$ $= {2, 4, 6, 8, 10}$ $∴ A = B$ (iv) $A = {x : x$ is a multiple of 10 $} = {10, 20, 30$ , 40, ......? $B = {10, 15, 20, 25, 30 \dots}$ It can be seen that $15 \in B$ but $15 \in / A$ . $∴ A \neq = B$

5. Are the following pair of sets equal? Give reasons. (i) $A = {2, 3}$ ; $B = {x : x$ is solution of $x^{2} + 5 x + 6 = 0}$ (ii) $A = {x : x$ is a letter in the word FOLLOW $}$ ; $B = {y : y$ is a letter in the word WOLF $}$

Sol.

(i) $A = {2, 3}; B = {x : x$ is a solution of $x^{2} + 5 x + 6 = 0}$ The equation $x^{2} + 5 x + 6 = 0$ can be solved as:

$x (x + 3) + 2 (x + 3) = 0$

$(x + 2) (x + 3) = 0; x = - 2$ or $x = - 3$ $∴ A = {2, 3}; B = {- 2, - 3}$ $∴ A \neq = B$ (ii) $A = {x : x$ is a letter in the word FOLLOW $}$

$= {F, O, L, W}$

$B = {y : y$ is a letter in the word WOLF $}$

$= {W, O, L, F}$

The order in which the elements of a set are listed is not significant. $∴ A = B$

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

$A = {2, 4, 8, 12}, C = {4, 8, 12, 14}, E = {- 1, 1}, G = {1, - 1}, B = {1, 2, 3, 4}, D = {3, 1, 4, 2}, F = {0, a}, H = {0, 1}$

Sol. $A = {2, 4, 8, 12}; B = {1, 2, 3, 4}$ ;

$C = {4, 8, 12, 14} E = {- 1, 1}; G = {1, - 1}; D = {3, 1, 4, 2}; F = {0, a} A = {0, 1}$

It can be seen that

$8 \in A, 8 \in / F, 8 \in / B, 8 \in / G, 8 \in / D, 8 \in / E, 8 \in / H$

$\Rightarrow A \neq = B, A \neq = D, A \neq = E, A \neq = F$ ,

$A \neq = G, A \neq = H$

Also,

$2 \in A, 2 \in / C$

$∴ A \neq = C$

$3 \in B, 3 \in / G, 3 \in / C, 3 \in / E, 3 \in / F, 3 \in / H$

$∴ B \neq = C, B \neq = E, B \neq = F, B \neq = G$ ,

$B \neq = H$

$12 \in C, 12 \in / G, 12 \in / D, 12 \in / E, 12 \in / F, 12 \in / H$

$∴ C \neq = D, C \neq = E, C \neq = F, C \neq = G$ ,

$C \neq = H$

$4 \in D, 4 \in / E, 4 \in / F, 4 \in / G,$

$∴ D \neq = E, D \neq = F, D \neq = G, D \neq = H$ Similarly,

$E \neq = F, F \neq = H, E \neq = G, E \neq = H, F \neq = G, G \neq = H$

The order in which the elements of a set are listed is not significant. $∴ B = D$ and $E = G$ Hence, among the given sets, $B = D$ and $E = G$ .

EXERCISE - 1.3

Make correct statements by filling in the symbols $\subset$ or $\neq \subset$ in the blank spaces: (i) ${2, 3, 4} \dots {1, 2, 3, 4, 5}$ (ii) ${a, b, c} \dots {b, c, d}$ (iii) {x : x is a student of Class XI of your school}

$\dots {x : x student of your school}$

(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 the plane} (vi) {x : x is an equilateral triangle in a plane}... {x: x is a triangle in the same plane} (vii) ${x : x$ is an even natural number $}$ ... {x: x is an integer}

Sol.

(i) ${2, 3, 4} \subset {1, 2, 3, 4, 5}$ (ii) ${a, b, c} \neq \subset {b, c, d}$ (iii) ${x : x$ is a student of class XI of your school $} \subset {x : x$ is student of your school $}$ (iv) ${x : x$ is a circle in the plane $} \neq \subset {x : x$ is a circle in the same plane with radius 1 unit} (v) ${x : x$ is a triangle in a plane $} \neq \subset {x : x$ is a rectangle in the plane} (vi) {x : x is an equilateral triangle in a plane} $\subset {x : x$ in a triangle in the same plane $}$ (vii) ${x : x$ is an even natural number $}$ $\subset {x : x$ is an integer $}$

2. Examine whether the following statements are true or false: (i) ${a, b} \neq \subset {b, c, a}$ (ii) ${a, e} \subset {x : x$ is a vowel in the English alphabet} (iii) ${1, 2, 3} \subset {1, 3, 5}$ (iv) ${a} \subset {a . b, c}$ (v) ${a} \in (a, b, c)$ (vi) {x: x is an even natural number less than 6} $\subset {x : x$ is a natural number which divides 36 $}$

Sol.

(i) False. Each element of ${a, b}$ is also an element of ${b, c, a}$ . (ii) True. a, e are two vowels of the English alphabet. (iii) False. $2 \in {1, 2, 3}$ ; however, $2 \in / {1, 3, 5}$ (iv) True. Each element of ${a}$ is also an element of {a, b, c}. (v) False. The elements of {a, b, c} are a, b, c. Therefore, ${a} \subset {a, b, c}$ (vi) True. {x:x is an even natural number less than $6} = {2, 4}$ {x:x is a natural number which divides 36} $= {1, 2, 3, 4, 6, 9, 12, 18, 36}$

Let $A = {1, 2, {3, 4}, 5}$ . Which of the following statements are incorrect and why? (i) ${3, 4} \subset A$ (ii) ${3, 4} \in A$ (iii) ${{3, 4}} \subset A$ (iv) $1 \in A$ (v) $1 \subset A$ (vi) ${1, 2, 5} \subset A$ (vii) ${1, 2, 5} \in A$ (viii) ${1, 2, 3} \subset A$ (ix) $ϕ \in A$ (x) $ϕ \subset$ A (xi) ${ϕ} \subset A$

Sol. $A = {1, 2, {3, 4}, 5}$

(i) The statement ${3, 4} \subset A$ is incorrect because $3 \in {3, 4}$ ; however, $3 \in / A$ . (ii) The statement ${3, 4} \in A$ is correct because {3, 4} is an element of A. (iii) The statement ${{3, 4}} \subset A$ is correct because ${3, 4} \in {{3, 4}}$ and ${3, 4} \in A$ . (iv) The statement $1 \in A$ is correct because 1 is an element of A. (v) The statement $1 \subset A$ is incorrect because an element of a set can never be a subset of itself. (vi) The statement ${1, 2, 5} \subset A$ is correct because each element of ${1, 2, 5}$ is also an element of A. (vii) The statement ${1, 2, 5} \in A$ is incorrect because {1, 2, 5} is not an element of A. (viii) The statement ${1, 2, 3} \subset A$ is incorrect because $3 \in {1, 2, 3}$ ; however, $3 \in / A$ . (ix) The statement $ϕ \in A$ is incorrect because $ϕ$ is not an element of A. (x) The statement $ϕ \subset A$ is correct because $ϕ$ is a subset of every set. (xi) The statement ${ϕ} \subset A$ is incorrect because $ϕ \in {ϕ}$ ; however, $ϕ \in A$ .

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

Sol.

(i) The subsets of ${a}$ are $ϕ$ and ${a}$ . (ii) The subsets of ${a, b}$ are $ϕ, {a}, {b}$ , and ${a, b}$ . (iii) The subsets of ${1, 2, 3}$ are $ϕ, {1}, {2}, {3}$ , ${1, 2}, {2, 3}, {1, 3}$ and ${1, 2, 3}$ (iv) The only subset of $ϕ$ is $ϕ$ .

5. Write the following as intervals: (i) ${x : x \in R, - 4 < x \leq 6}$ (ii) ${x : x \in R, - 12 < x < - 10}$ (iii) ${x : x \in R, 0 \leq x < 7}$ (iv) ${x : x \in R, 3 \leq x \leq 4}$

Sol.

(i) ${x : x \in R, - 4 < x \leq 6} = (- 4, 6]$ (ii) ${x : x \in R, - 12 < x < - 10} = (- 12, - 10)$ (iii) ${x : x \in R, 0 \leq x < 7} = [0, 7)$ (iv) ${x : x \in R, 3 \leq x \leq 4} = [3, 4]$

6. Write the following intervals in set-builder form: (i) (-3, 0) (ii) [6, 12] (iii) (6, 12] (iv) [-23, 5)

Sol.

(i) $(- 3, 0) = {x : x \in R, - 3 < x < 0}$ (ii) $[6, 12] = {x : x \in R, 6 \leq x \leq 12}$ (iii) $(6, 12] = {x : x \in R, 6 < x \leq 12}$ (iv) $[- 23, 5) = {x : x \in R, - 23 \leq x < 5}$

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

Sol.

(i) For the set of right triangles, the universal set can be the set of triangles or the set of polygons. (ii) For the set of isosceles triangles, the universal set can be the set of triangles or the set of polygons or the set of two-dimensional figures.

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 universals 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) ${1, 2, 3, 4, 5, 6, 7, 8}$

Sol.

(i) It can be seen that $A \subset {0, 1, 2, 3, 4, 5, 6}$ $B \subset {0, 1, 2, 3, 4, 5, 6}$ However, $C \neq \subset {0, 1, 2, 3, 4, 5, 6}$ Therefore, the set ${0, 1, 2, 3, 4, 5, 6}$ cannot be the universal set for the sets A, B, and C. (ii) $A \neq \subset ϕ, B \neq \subset ϕ, C \neq \subset ϕ$ Therefore, $ϕ$ cannot be the universal set for the sets A, B, and C. (iii) $A \subset {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ $B \subset {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ $C \subset {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ Therefore, the set ${0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ is the universal set for the sets A, B, and C. (iv) $A \subset {1, 2, 3, 4, 5, 6, 7, 8}$ $B \subset {1, 2, 3, 4, 5, 6, 7, 8}$ However, $C \neq \subset {1, 2, 3, 4, 5, 6, 7, 8}$ Therefore, the set ${1, 2, 3, 4, 5, 6, 7, 8}$ cannot be the universal set for the sets A, B, and C.

EXERCISE - 1.4

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 $1 < x \leq 6}$ $B = {x : x$ is a natural number and $6 < x < 10}$ (v) $A = {1, 2, 3}; B = ϕ$

Sol.

(i) $X = {1, 3, 5}, Y = {1, 2, 3}$ $X \cup Y = {1, 2, 3, 5}$ (ii) $A = {a, e, i, o, u} B = {a, b, c}$ $A \cup B = {a, b, c, e, i, o, u}$ (iii) $A = {x : x$ is a natural number and multiple of 3 $}$ $= {3, 6, 9 \dots}$ $B = {x : x$ is a natural number less than 6 $}$ $= {1, 2, 3, 4, 5, 6}$ $A \cup B = {1, 2, 4, 5, 3, 6, 9, 12 \dots .$ . $∴ A \cup B = {x : x = 1, 2, 4, 5$ or a multiple of 3 $}$ (iv) $A = {x : x$ is a natural number and $1 < x \leq 6}$ $= {2, 3, 4, 5, 6}$ $B = {x : x$ is a natural number and $6 < x < 10}$ $= {7, 8, 9}$ $A \cup B = {2, 3, 4, 5, 6, 7, 8, 9}$ $∴ A \cup B = {x : x \in N$ and $1 < x < 10}$ (v) $A = {1, 2, 3}, B = ϕ$ $A \cup B = {1, 2, 3}$

2. Let $A = {a, b}, B = {a, b, c}$ . Is $A \subset B$ ? What is $A \cup B$ ? Sol. Here, $A = {a, b}$ and $B = {a, b, c}$ Yes, $A \subset B$ . $A \cup B = {a, b, c} = B$

3. If A and B are two sets such that $A \subset B$ , then what is $A \cup B$ ? Sol. If A and B are two sets such that $A \subset B$ , then $A \cup B = B$ .

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) $A \cup B$ (ii) $A \cup C$ (iii) $B \cup C$ (iv) $B \cup D$ (v) $A \cup B \cup C$ (vi) $A \cup B \cup D$ (vii) $B \cup C \cup D$ Sol. $A = {1, 2, 3, 4], B = {3, 4, 5, 6}$ , $C = {5, 6, 7, 8}$ and $D = {7, 8, 9, 10}$ (i) $A \cup B = {1, 2, 3, 4, 5, 6}$ (ii) $A \cup C = {1, 2, 3, 4, 5, 6, 7, 8}$ (iii) $B \cup C = {3, 4, 5, 6, 7, 8}$ (iv) $B \cup D = {3, 4, 5, 6, 7, 8, 9, 10}$ (v) $A \cup B \cup C = {1, 2, 3, 4, 5, 6, 7, 8}$ (vi) $A \cup B \cup D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ (vii) $B \cup C \cup D = {3, 4, 5, 6, 7, 8, 9, 10}$

5. Find the intersection of each pair 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 $1 < x \leq 6}$ , $B = {x : x$ is a natural number and $6 < x < 10}$ (v) $A = {1, 2, 3}, B = ϕ$

Sol.

(i) $X = {1, 3, 5}, Y = {1, 2, 3}$ $X \cap Y = {1, 3}$ (ii) $A = {a, e, i, o, u}, B = {a, b, c}$ $A \cap B = {a}$ (iii) $A = {x : x$ is a natural number and multiple of 3} $= (3, 6, 9 \dots)$ $B = {x : x$ is a natural number less than 6 $}$ $= {1, 2, 3, 4, 5}$

$∴ A \cap B = {3}$

(iv) $A = {x : x$ is a natural number and $1 < x \leq 6}$ $= {2, 3, 4, 5, 6}$ $B = {x : x$ is a natural number and $6 < x < 10}$ $= {7, 8, 9}$ $A \cap B = ϕ$ (v) $A = {1, 2, 3}, B = ϕ$ . So, $A \cap B = ϕ$

6. If $A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}$ , $C = {11, 13, 15}$ and $D = {15, 17}$ ; find (i) $A \cap B$ (ii) $B \cap C$ (iii) $A \cap C \cap D$ (iv) $A \cap C$ (v) $B \cap D$ (vi) $A \cap (B \cup C)$ (vii) $A \cap D$ (viii) $A \cap (B \cup D)$ (ix) $(A \cap B) \cap (B \cup C)$ (x) $(A \cup D) \cap (B \cup C)$

Sol.

(i) $A \cap B = {7, 9, 11}$ (ii) $B \cap C = {11, 13}$ (iii) $A \cap C \cap D = {A \cap C} \cap D$ $= {11} \cap {15, 17} = ϕ$ (iv) $A \cap C = {11}$ (v) $B \cap D = ϕ$ (vi) $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ $= {7, 9, 11} \cup {11}$ $= {7, 9, 11}$ (vii) $A \cap D = ϕ$ (viii) $A \cap (B \cup D) = (A \cap B) \cup (A \cap D)$ $= {7, 9, 11} \cup ϕ$ $= {7, 9, 11}$ (ix) $(A \cap B) \cap (B \cup C)$ $= {7, 9, 11} \cap {7, 9, 11, 13, 15}$ $= {7, 9, 11}$ (x) $(A \cup D) \cap (B \cup C)$ $= {3, 5, 7, 9, 11, 15, 17) \cap {7, 9, 11, 13, 15}$ $= {7, 9, 11, 15}$

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) $A \cap B$ (ii) $A \cap C$ (iii) $A \cap D$ (iv) $B \cap C$ (v) $B \cap D$ (vi) $C \cap D$

Sol. $A = {x : x$ is a natural number $}$ $= {1, 2, 3, 4, 5 \dots}$ $B = {x : x$ is an even natural number $}$ $= {2, 4, 6, 8 \dots}$ $C = {x : x$ is an odd natural number $}$ $= {1, 3, 5, 7, 9 \dots}$ $D = {x : x$ is a prime number $}$ $= {2, 3, 5, 7 \dots}$

(i) $A \cap B = {x : x$ is a even natural number $} = B$ (ii) $A \cap C = {x : x$ is an odd natural number $} = C$ (iii) $A \cap D = {x : x$ is a prime number $} = D$ (iv) $B \cap C = ϕ$ (v) $B \cap D = {2}$ (vi) $C \cap D = {x : x$ is odd prime number $}$

8. Which of the following pairs of sets are disjoint (i) ${1, 2, 3, 4}$ and ${x : x$ is a natural number and $4 \leq x \leq 6}$ 10. If $X = {a, b, c, d}$ and $Y = {f, b, d, g}$ , find (i) $X - Y$ (ii) $Y - X$ (iii) $X \cap Y$ (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 $}$

Sol.

(i) Let $A = {1, 2, 3, 4}$

$B = = A {x : x is a natural number and 4 \leq x \leq 6} {4, 5, 6} \cap B \Rightarrow {1, 2, 3, 4} \cap {4, 5, 6} = {4} \Rightarrow A \cap B \neq = ϕ$

Therefore, this pair of sets is not disjoint. (ii) $A \cap B \Rightarrow {a, e, i, o, u} \cap {c, d, e, f} = {e}$

$\Rightarrow A \cap B \neq = ϕ$

Therefore, ${a, e, i, o, u}$ and (c, d, e, f } are not disjoint. (iii) $A \cap B \Rightarrow {x : x$ is an even integer $}$

$\Rightarrow \cap {x : x is an odd integer} = ϕ A \cap B = ϕ$

Therefore, this pair of sets is disjoint.

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 Sol. (i) $A - B = {3, 6, 9, 15, 18, 21}$ (ii) $A - C = {3, 9, 15, 18, 21}$ (iii) $A - D = {3, 6, 9, 12, 18, 21}$ (iv) $B - A = {4, 8, 16, 20}$ (v) $C - A = {2, 4, 8, 10, 14, 16}$ (vi) $D - A = {5, 10, 20}$ (vii) $B - C = {20}$ (viii) $B - D = {4, 8, 12, 16}$ (ix) $C - B = {2, 6, 10, 14}$ (x) $D - B = {5, 10, 15}$ (xi) $C - D = {2, 4, 6, 8, 12, 14, 16}$ (xii) $D - C = {5, 15, 20}$

If X = {a, b, c, d} and Y = {f, b, d, g}, find

(i) X – Y. (ii) Y – X. (iii) X ∩ Y

Sol. (i) $X - Y = {a, c}$ (ii) $Y - X = {f, g}$ (iii) $X \cap Y = {b, d}$

11. If R is the set of real numbers and Q is the set of rational numbers, then what is R-Q? Sol. R: set of real numbers Q: set of rational numbers Therefore, $R - Q$ is a set of irrational numbers.

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 Sol. (i) False. As $3 \in {2, 3, 4, 5}, 3 \in {3, 6}$

$\Rightarrow {2, 3, 4, 5} \cap {3, 6} = {3}$

(ii) False. As a $\in {a, e, i, o, u}, a \in {a, b, c, d}$

$\Rightarrow {a, e, i, o, u} \cap {a, b, c, d} = {a}$

(iii) True. As ${2, 6, 10, 14} \cap {3, 7, 11, 15} = ϕ$ (iv) True. As ${2, 6, 10} \cap {3, 7, 11} = ϕ$

EXERCISE - 1.5

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) $(A \cup C)^{'}$ (iv) $(A \cup B)^{'}$ (v) (A')' (vi) $(B - C)^{'}$ Sol. $U = {1, 2, 3, 4, 5, 6, 7, 8, 9}$ , $A = {1, 2, 3, 4}$ , $B = {2, 4, 6, 8}$ , $C = {3, 4, 5, 6}$ (i) $A^{'} = U - A$ $A^{'} = {5, 6, 7, 8, 9}$ (ii) $B^{'} = U - B$ $B^{'} = (1, 3, 5, 7, 9)$

(iii) $(A \cup C) = {1, 2, 3, 4, 5, 6}$ Now $(A \cup C)^{'} = U - (A \cup C)$ $∴ (A \cup C)^{'} = {7, 8, 9}$ (iv) $(A \cup B) = {1, 2, 3, 4, 6, 8}$ Now $(A \cup B)^{'} = U - (A \cup B)$ $∴ (A \cup B)^{'} = {5, 7, 9}$ (v) $(A)^{'} = A = {1, 2, 3, 4}$ (vi) $(B - C) = {2, 8}$ Now $(B - C)^{'} = U - (B - C)$ $∴ (B - C)^{'} = {1, 3, 4, 5, 6, 7, 9}$

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) $D = {f, g, h, a}$

Sol. $U = {a, b, c, d, e, f, g, h}$

(i) $A = {a, b, c}$ $A^{'} = U - A ∴ A^{'} = {d, e, f, g, h}$ (ii) $B = {d, e, f, g}$ $B^{'} = U - B ∴ B^{'} = {a, b, c, h}$ (iii) $C = {a, c, e, g}$ $C^{'} = U - C ∴ C^{'} = {b, d, f, g}$ (iv) $D = {f, g, h, a}$ $D^{'} = U - D ∴ D^{'} = {b, c, d, e}$

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 perfect cube $}$ (viii) ${x : x + 5 = 8}$ (ix) ${x : 2 x + 5 = 9}$ (x) ${x : x \geq 7}$ (xi) ${x : x \in N$ and $2 x + 1 > 10}$

Sol. $U = N$ : Set of natural numbers

(i) ${x : x$ is an even natural number} $^{'}$ $= {x : x$ is an odd natural number $}$ (ii) {x : x is an odd natural number} $= {x : x$ is an even natural number $}$ (iii) ${x : x is a positive multiple of 3}^{'}$ $= {x : x \in N$ and x is not a multiple of 3 $}$ (iv) ${x : x is a prime number}^{'}$ $= {x : x$ is a positive composite number and $x =$ 1} (v) ${x : x is a natural number divisible by 3 and 5}^{'}$ $= {x : x$ is a natural number that is not divisible by 3 or 5} (vi) ${x : x is a perfect square}^{'}$ $= {x : x \in N$ and x is not a perfect square $}$ (vii) ${x : x is a perfect cube}^{'}$ $= {x : x \in N$ and x is not a perfect cube $}$ (viii) ${x : x + 5 = 8}^{'} = {x : x \in N$ and $x \neq = 3}$ (ix) ${x : 2 x + 5 = 9}^{'} = {x : x \in N$ and $x \neq = 2}$ (x) ${x : x \geq 7}^{'} = {x : x \in N$ and $x < 7}$ (xi) ${x : x \in N and 2 x + 1 > 10}^{'}$ $= {x : x \in N$ and $x \leq 9/2}$

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) $(A \cup B)^{'} = A^{'} \cap B^{'}$ (ii) $(A \cap B)^{'} = A^{'} \cup B^{'}$ Sol. $U = {1, 2, 3, 4, 5, 6, 7, 8, 9}$ , $A = {2, 4, 6, 8}, B = {2, 3, 5, 7}$ (i) $(A \cup B)^{'} = {2, 3, 4, 5, 6, 7, 8}^{'} = {1, 9}$ $A^{'} \cap B^{'} = {1, 3, 5, 7, 9} \cap {1, 4, 6, 8, 9}$ $= {1, 9}$ $∴ (A \cup B)^{'} = A^{'} \cap B^{'}$ (ii) $(A \cap B)^{'} = {2}^{'} = {1, 3, 4, 5, 6, 7, 8, 9}$ $A^{'} \cup B^{'} = {1, 3, 5, 7, 9} \cup {1, 4, 6, 8, 9}$ $= {1, 3, 4, 5, 6, 7, 8, 9}$ $∴ (A \cap B)^{'} = A^{'} \cup B^{'}$

Draw appropriate Venn diagram for each of the following: (i) $(A \cup B)^{'}$ (ii) $A^{'} \cap B^{'}$ (iii) $(A \cap B)^{'}$ (iv) $A^{'} \cup B^{'}$ Sol. (i) $(A \cup B)^{'} \Rightarrow U - (A \cup B)$

Class-11-chp-1-maths-exer-1.5-ques-5-(a)-ncert-sol

(ii) $A^{'} \cap B^{'} = (A \cup B)^{'}$

exer-1.5-ques-5-(b)-maths-chapter-1-class-11-ncert-sol

(iii) $(A \cap B)^{'} = U - (A \cap B)$

class-11-maths-chap-1-ncert-sol-exer-1.5-ques-5-(c)

(iv) $A^{'} \cup B^{'} = (A \cap B)^{'}$

ncert-sol-class-11-maths-exer-1.5-ques-5-(d)-chapter-1

6. 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'? Sol. A' is the set of all equilateral triangles. $A^{'} = U - A$ U = Set of all triangles A = consist of triangles that have at least are angle not equal to 60° Measuring non-equilateral triangle
7. Fill in the blanks to make each of the following a true statement: (i) $A \cup A^{'} =$ (ii) $ϕ^{'} \cap A =$ (iii) $A \cap A^{'} =$ (iv) $U^{'} \cup A =$ ..... Sol. (i) $A \cup A^{'} = A \cup (U - A) = U$ (ii) $ϕ^{'} \cap A = U \cap A = A ∴ ϕ^{'} \cap A = A$ (iii) $A \cap A^{'} = A \cap (U - A) = ϕ$ (iv) $U^{'} \cap A = ϕ \cap A = ϕ ∴ U^{'} \cap A = ϕ$

MISCELLANEOUS EXERCISE

Decide, among the following sets, which sets are subsets of one and another: $A = {x : x \in R$ and x satisfy $x^{2} - 8 x + 12 = 0}$ , $B = {2, 4, 6}, C = {2, 4, 6, 8 \dots}, D = {6}$ . Sol. $A = {x : x \in R$ and x satisfies $x^{2} - 8 x + 12 = 0} 2$ and 6 are the only solutions of $x^{2} - 8 x + 12 = 0$ . $∴ A = {2, 6}$ $B = {2, 4, 6}, C = {2, 4, 6, 8 \dots}, D = {6}$ $∴ D \subset A \subset B \subset C$ Hence, $A \subset B, A \subset C, B \subset C, D \subset A$ , $D \subset B, D \subset C$
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 $x \in A$ and $A \in B$ , then $x \in B$ (ii) If $A \subset B$ and $B \in C$ , then $A \in C$ (iii) If $A \subset B$ and $B \subset C$ , then $A \subset C$ (iv) If $A \neq \subset B$ and $B \neq \subset C$ , then $A \neq \subset C$ (v) If $x \in A$ and $A \neq \subset B$ , then $x \in B$ (vi) If $A \subset B$ and $x \in / B$ , then $x \in / A$

Sol.

(i) False. Let $A = {1, 2}$ and $B = {1, {1, 2}, {3}}$ Now, $2 \in {1, 2}$ and ${1, 2} \in {{3}, 1, {1, 2}}$ $∴ A \in B$ However, $2 \in / {{3}, 1, {1, 2}}$ (ii) False. Let $A = {2}, B = {0, 2}$ and $C = {1, {0, 2}, {3}}$ As $A \subset B, B \in C$ However, $A \in / C$ (iii) True.Let $A \subset B, B \subset C$ Let $x \in A$ $\Rightarrow x \in B [∵ A \subset B]$ $\Rightarrow x \in C [∵ B \subset C]$ $∴ A \subset C$ (iv) False. Let $A = {1, 2}, B = {0, 6, 8}$ and $C = {0, 1, 2, 6, 9}$ Accordingly, $A \neq \subset B$ and $B \neq \subset C$ However, $A \subset C$

(v) False. Let $A = {3, 5, 7}$ and $B = {3, 4, 6}$ Now, $5 \in A$ and $A \neq \subset B$ However, $5 \in / B$ (vi) True. Let $A \subset B$ and $x \in / B$ To show : $x \in / A$ If possible, suppose $x \in A$ Then, $x \in B$ , which is a contradiction as $x \in / B$ $∴ x \in / A$

3. Let A, B and C be the sets such that $A \cup B = A \cup C$ and $A \cap B = A \cap C$ . show that $B = C$ . Sol. Let, $A, B$ and C be the sets such $A \cup B = A \cup C$ and $A \cap B = A \cap C$ . To show : $B = C$ Let $x \in B$

$\Rightarrow \Rightarrow \Rightarrow x \in A \cup B x \in A \cup C x \in A or x \in C [B \subset A \cup B] [A \cup B = A \cup C]$

Case I: $x \in A$ Also, $x \in B$

$∴ \Rightarrow ∴ ∴ ∴ x \in A \cap B x \in A \cap C x \in A and x \in C x \in C B \subset C [∵ A \cap B = A \cap C]$

Similarly, we can show that $C \subset B$ $∴ B = C$

4. Show that the following four conditions are equivalent: (i) $A \subset B$ (ii) $A - B = ϕ$ (iii) $A \cup B = B$ (iv) $A \cap B = A$ Sol. First, we have to show that (i) ⇔ (ii). Let $A \subset B$ To show: $A - B \neq = ϕ$ If possible, suppose $A - B \neq = ϕ$ This means that there exists $x \in A, x \neq = B$ , which is not possible as $A \subset B$ .

$∴ ∴ Let A - B = ϕ A - B = ϕ A \subset B \Rightarrow A - B = ϕ$

To show: $A \subset B$ Let $x \in A$ Clearly, $x \in B$ because if $x \in / B$ , then $A - B \neq = ϕ$

$∴ ∴ A - B = ϕ (i) \Leftrightarrow (ii) \Rightarrow A \subset B$

Let $A \subset B$ To show: $A \cup B = B$ Clearly, $B \subset A \cup B$ Let $x \in A \cup B \Rightarrow x \in A$ or $x \in B$ Case I: $x \in A \Rightarrow x \in B [∵ A \subset B]$ Case II : $x \in B$ Then, $A \cup B = B$ Conversely, let $A \cup B = B$ Let $x \in A$

$\Rightarrow \Rightarrow x \in A \cup B x \in B ∴ A \subset B [∵ A \subset A \cup B] [∵ A \cup B = B]$

Hence, (i) ⇔ (iii) Now, we have to show that (i) ⇔ (iv). Let $A \subset B$ Clearly $A \cap B \subset A$ Let $x \in A$ We have to show that $X \in A \cap B$ As $A \subset B, x \in B$

$∴ ∴ x \in A \cap B A \subset A \cap B$

Hence, $A = A \cap B$ Conversely, suppose $A \cap B = A$ Let $x \in A$

$\Rightarrow x \in A \cap B$

$\Rightarrow x \in A$ and $x \in B$

$\Rightarrow x \in B ∴ A \subset B$

Hence, (i) ⇔ (iv).

Show that if $A \subset B$ , then $C - B \subset C - A$ . Sol. Let $A \subset B$ To show: $C - B \subset C - A$ Let $x \in C - B$ $\Rightarrow x \in C$ and $x \in / B$ $\Rightarrow x \in C$ and $x \in / A [A \subset B]$ $\Rightarrow x \in C - A$ $∴ C - B \subset C - A$
Show that for any sets A and B,

$A = (A \cap B) \cup (A - B)$

and $A \cup (B - A) = (A \cup B)$ . Sol. To show: $A = (A \cap B) \cup (A - B)$ Let $x \in A$ We have to show that $x \in (A \cap B) \cup (A - B)$ Case I: $x \in A \cap B$ Then, $x \in (A \cap B) \subset (A \cup B) \cap (A - B)$ Case II: $x \in / A \cap B$ $\Rightarrow x \in / A$ or $x \in / B$ $∴ x \in / B [x \in / A]$ $∴ x \in / A - B \subset (A \cup B) \cup (A - B)$ $∴ A \subset (A \cap B) \cup (A - B)$ It is clear that $A \cap B \subset A$ and $(A - B) \subset A$ $∴ (A \cap B) \cup (A - B) \subset A$ From (1) and (2), we obtain $A = (A \cap B) \cup (A - B)$ To prove: $A \cup (B - A) \subset A \cup B$ Let $x \in A \cup (B - A)$ $\Rightarrow x \in A$ or $x \in (B - A)$ $\Rightarrow x \in A$ or $(x \in B$ and $x \in / A)$ $\Rightarrow (x \in A$ or $x \in B)$ and $(x \in A$ or $x \in / A)$ $\Rightarrow x \in (A \cup B)$ $∴ A \cup (B - A) \subset (A \cup B)$ Next, we show that $(A \cup B) \subset A \cup (B - A)$ . Let $y \in A \cup B$ $\Rightarrow y \in A$ or $y \in B$ $\Rightarrow (y \in A$ or $y \in B)$ and $(y \in A$ or $y \in / A)$ $\Rightarrow y \in A$ or $(y \in B$ and $y \in / A)$ $\Rightarrow y \in A \cup (B - A)$ $∴ A \cup B \subset A \cup (B - A)$ Hence, from (3) and (4), we obtain $A \cup (B - A) = A \cup B$ .

7. Using properties of sets show that (i) $A \cup (A \cap B) = A$ , (ii) $A \cap (A \cup B) = A$ .

Sol.

(i) To show: $A \cup (A \cap B) = A$ We know that $A \subset A$ $A \cap B \subset A$ $∴ A \cup (A \cap B) \subset A$ Also, $A \subset A \cup (A \cap B)$ $∴$ From (1) and (2) $A \cup (A \cap B) = A$ (ii) To show: $A \cap (A \cup B) = A$

$A \cap (A \cup B) = (A \cap A) \cup (A \cap B) = A \cup (A \cap B) = A {from (1)}$

Show that $A \cap B = A \cap C$ need not imply $B = C$ . Sol. Let $A = {0, 1}, B = {0, 2, 3}$ , and $C = {0, 4, 5}$ Accordingly, $A \cap B = {0}$ and $A \cap C = {0}$ Here, $A \cap B = A \cap C = {0}$ However, $B \neq = C [2 \in B$ and $2 \in / C]$
Let A and B be sets. If $A \cap X = B \cap X = ϕ$ and $A \cup X = B \cup X$ for some set X , show that $A = B$ . (Hints $A = A \cap (A \cup X), B = B \cap (B \cup X)$ and use distributive law) Sol. Let A and B be two sets such that $A \cap X = B \cap X = f$ and $A \cup X = B \cup X$ for some set X. To show: $A = B$ It can be seen that

$A = A \cap (A \cup X) = A \cap (B \cup X) = (A \cap B) \cup (A \cap X)$

[Distributive law]

$= (A \cap B) \cup ϕ [A \cap X = ϕ] = A \cap B$

Now,

$B = B \cap (B \cup X) = B \cap (A \cup X) [A \cup X = B \cup X] = (B \cap A) \cup (B \cap X)$

[Distributive law]

$= (B \cap A) \cup ϕ [B \cap X = ϕ] = B \cap A = A \cap B$

Hence, from (1) and (2), we obtain $A = B$ .

Find sets $A, B$ and C such that $A \cap B$ , $B \cap C$ and $A \cap C$ are non-empty sets and $A \cap B \cap C = ϕ$ . Sol. Let $A = {0, 1}, B = {1, 2}$ , and $C = {2, 0}$ . Accordingly, $A \cap B = {1}, B \cap C = {2}$ , and $A \cap C = {0}$ . $∴ A \cap B, B \cap C$ and $A \cap C$ are non-empty. However, $A \cap B \cap C = ϕ$

3.0Class 11 Maths NCERT Solutions – Chapter-wise Links

Find complete NCERT Solutions for Class 11 Maths, organised chapter-wise to help students understand exercise questions, use the right formulas, and solve problems step by step.

Chapter Number	Chapterwise NCERT Solutions
Chapter 2	Relations and Functions
Chapter 3	Trigonometric Functions
Chapter 4	Complex Numbers and Quadratic Equations
Chapter 5	Linear Inequalities
Chapter 6	Permutations and Combinations
Chapter 7	Binomial Theorem
Chapter 8	Sequence and Series
Chapter 9	Straight Lines
Chapter 10	Conic Sections
Chapter 11	Introduction to Three-dimensional Geometry
Chapter 12	Limits and Derivatives
Chapter 13	Statistics
Chapter 14	Probability

4.0NCERT Solutions Class 11 Maths Chapter 1 Sets: Exercise-wise Questions and Topics

Exercise	Number of Questions	Important Topics Covered
Exercise 1.1	6 Questions & Solutions	Sets, roster form, set-builder form, empty set, finite and infinite sets, equal sets, subsets and power sets
Exercise 1.2	6 Questions & Solutions	Union, intersection, complement of sets, and laws of set operations
Exercise 1.3	8 Questions & Solutions	Venn diagrams, union and intersection of sets, complements, and set-based problems
Exercise 1.4	12 Questions & Solutions	Applications of sets, representation of data, and practical uses of sets
Exercise 1.5	7 Questions & Solutions	Cartesian product, ordered pairs, and applications of Cartesian products
Miscellaneous Exercise	10 Questions & Solutions	Mixed questions covering sets, set operations, properties, and applications

5.0Key Features of NCERT Solutions for Class 11 Maths Chapter 1

Clear Explanation of Set Concepts: The solutions explain important concepts from Sets in simple language. They help students understand the difference between “belongs to” (∈) and “is a subset of” (⊂) and use these symbols correctly.

Easy Venn Diagram Solutions: Step-by-step Venn diagrams are used to explain questions based on two or three sets. Students can understand concepts such as union, intersection, and difference of sets more easily with these diagrams.

Correct Use of Set Formulas: The solutions explain how to apply important formulas while solving questions based on sets. For example:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B),

The formula is useful for solving questions involving the union and intersection of two finite sets.

Simple Explanation of Interval Notation: The solutions explain how subsets of real numbers can be represented using interval notation. They also help students understand the use of brackets, endpoints, and inequalities.

Prepared by ALLEN Subject Experts: The NCERT Solutions for Class 11 Maths Chapter 1 are prepared by ALLEN subject experts and follow the NCERT syllabus. They are useful for school examinations and for building a strong foundation for competitive exams.

Step-by-Step Answers: Each question is solved in a clear and systematic manner. The steps help students understand how to reach the answer and avoid common mistakes while solving problems.

Complete Coverage of NCERT Questions: The solutions cover the NCERT Class 11 Maths Chapter 1 – Sets exercises and examples. Students can use them to understand concepts, check their answers, revise the chapter, and practise different types of questions.

Frequently Asked Questions

What topics are covered in Class 11 Maths Chapter 1 – Sets?

How can NCERT Solutions for Class 11 Maths Chapter 1 help students?

What are the important formulas and laws in Class 11 Maths Chapter 1?

What is the difference between union and intersection of sets?

What are the important concepts to study in Class 11 Maths Chapter 1 – Sets?

How do NCERT Solutions for Class 11 Maths Chapter 1 Sets help in JEE Main and NDA preparation?

Are NCERT Solutions for Class 11 Maths Chapter 1 Sets by ALLEN useful?

Join ALLEN!

Frequently Asked Questions

What topics are covered in Class 11 Maths Chapter 1 – Sets?

How can NCERT Solutions for Class 11 Maths Chapter 1 help students?

What are the important formulas and laws in Class 11 Maths Chapter 1?

What is the difference between union and intersection of sets?

What are the important concepts to study in Class 11 Maths Chapter 1 – Sets?

How do NCERT Solutions for Class 11 Maths Chapter 1 Sets help in JEE Main and NDA preparation?

Are NCERT Solutions for Class 11 Maths Chapter 1 Sets by ALLEN useful?

Join ALLEN!

NCERT Solutions for Class 11 Maths Chapter 1 Sets: Download PDF

1.0Class 11 Maths Chapter 1 Sets: Key Concepts

Operations on Sets

De Morgan's Laws

2.0Detailed NCERT Textbook Solutions : Class 11 Maths Chapter 1

EXERCISE - 1.1

EXERCISE - 1.2

EXERCISE - 1.3

EXERCISE - 1.4

EXERCISE - 1.5

MISCELLANEOUS EXERCISE

3.0Class 11 Maths NCERT Solutions – Chapter-wise Links

4.0NCERT Solutions Class 11 Maths Chapter 1 Sets: Exercise-wise Questions and Topics

5.0Key Features of NCERT Solutions for Class 11 Maths Chapter 1

Table of Contents