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RD Sharma solutions for Class 11 Mathematics chapter 1 - Sets

Mathematics Class 11

RD Sharma Mathematics Class 11 Chapter 1: Sets

Ex. 1.10Ex. 1.20Ex. 1.30Ex. 1.40Ex. 1.50Ex. 1.60Ex. 1.70Ex. 1.80Ex. 1.90Others

Chapter 1: Sets Exercise 1.10 solutions [Page 2]

Ex. 1.10 | Q 1 | Page 2

What is the difference between a collection and a set? Give reasons to support your answer?

Ex. 1.10 | Q 2.01 | Page 2

Which of the following collection are sets? Justify your answer:

A collection of all natural numbers less than 50.

Ex. 1.10 | Q 2.02 | Page 2

Which of the following collection are sets? Justify your answer:

The collection of good hockey players in India.

Ex. 1.10 | Q 2.03 | Page 2

Which of the following collection are sets? Justify your answer:

The collection of all girls in your class.

Ex. 1.10 | Q 2.04 | Page 2

Which of the following collection are sets? Justify your answer:

The collection of most talented writers of India.

Ex. 1.10 | Q 2.05 | Page 2

Which of the following collection are sets? Justify your answer:

The collection of difficult topics in mathematics.

Ex. 1.10 | Q 2.06 | Page 2

Which of the following collection are sets? Justify your answer:

The collection of all months of a year beginning with the letter J.

Ex. 1.10 | Q 2.07 | Page 2

Which of the following collection are sets? Justify your answer:

A collection of novels written by Munshi Prem Chand.

Ex. 1.10 | Q 2.08 | Page 2

Which of the following collection are sets? Justify your answer:

The collection of all question in this chapter.

Ex. 1.10 | Q 2.09 | Page 2

Which of the following collection are sets? Justify your answer:

A collection of most dangerous animals of the world.

Ex. 1.10 | Q 2.1 | Page 2

Which of the following collection are sets? Justify your answer:

The collection of prime integers.

Ex. 1.10 | Q 3.1 | Page 2

If A = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], then insert the appropriate symbol ∈ or ∉ in each of the following blank space:

4 ...... A

Ex. 1.10 | Q 3.2 | Page 2

If A = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], then insert the appropriate symbol ∈ or ∉ in each of the following blank space:

−4 ...... A

Ex. 1.10 | Q 3.3 | Page 2

If A = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], then insert the appropriate symbol ∈ or ∉ in each of the following blank space:

12 ...... A

Ex. 1.10 | Q 3.4 | Page 2

If A = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], then insert the appropriate symbol ∈ or ∉ in each of the following blank space:

9 ...... A

Ex. 1.10 | Q 3.5 | Page 2

If A = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], then insert the appropriate symbol ∈ or ∉ in each of the following blank space:

0 ...... A

Ex. 1.10 | Q 3.6 | Page 2

If A = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], then insert the appropriate symbol ∈ or ∉ in each of the following blank space:

−2 ...... A

Chapter 1: Sets Exercise 1.20 solutions [Pages 6 - 7]

Ex. 1.20 | Q 1.1 | Page 6

Describe the following sets in Roster form:

{x : x is a letter before e in the English alphabet}

Ex. 1.20 | Q 1.2 | Page 6

Describe the following sets in Roster form:

{x ∈ N : x2 < 25};

Ex. 1.20 | Q 1.3 | Page 6

Describe the following sets in Roster form:

{x ∈ N : x is a prime number, 10 < x < 20};

Ex. 1.20 | Q 1.4 | Page 6

Describe the following sets in Roster form:

{x ∈ N : x = 2nn ∈ N};

Ex. 1.20 | Q 1.5 | Page 6

Describe the following sets in Roster form:

{x ∈ R : x > x}.

Ex. 1.20 | Q 1.6 | Page 6

Describe the following sets in Roster form:

{x : x is a prime number which is a divisor of 60}

Ex. 1.20 | Q 1.7 | Page 6

Describe the following sets in Roster form:

The set of all letters in the word 'Trigonometry'

Ex. 1.20 | Q 1.8 | Page 6

Describe the following sets in Roster form:

The set of all letters in the word 'Better'.

Ex. 1.20 | Q 1.9 | Page 6

Describe the following sets in Roster form:

The set of all letters in the word 'Better'.

Ex. 1.20 | Q 2.1 | Page 6

Describe the following sets in set-builder form:

A = {1, 2, 3, 4, 5, 6}

Ex. 1.20 | Q 2.2 | Page 6

Describe the following sets in set-builder form:

B={1,1/2 ,1/3, 1/4,1/5,...........};

Ex. 1.20 | Q 2.3 | Page 6

Describe the following sets in set-builder form:

C = {0, 3, 6, 9, 12, ...}

Ex. 1.20 | Q 2.4 | Page 6

Describe the following sets in set-builder form:

D = {10, 11, 12, 13, 14, 15};

Ex. 1.20 | Q 2.5 | Page 6

Describe the following sets in set-builder form:

E = {0}

Ex. 1.20 | Q 2.6 | Page 6

Describe the following sets in set-builder form:

{1, 4, 9, 16, ..., 100}

Ex. 1.20 | Q 2.7 | Page 6

Describe the following sets in set-builder form:

{2, 4, 6, 8 .....}

Ex. 1.20 | Q 2.8 | Page 6

Describe the following sets in set-builder form:

{5, 25, 125 625}

Ex. 1.20 | Q 3.1 | Page 6

List all the elements of the following sets:

$A = \left\{ x: x^2 \leq 10, x \in Z \right\}$

Ex. 1.20 | Q 3.2 | Page 6

List all the elements of the following set:

$B = \left\{ x: x = \frac{1}{2n - 1}, 1 \leq n \leq 5 \right\}$

Ex. 1.20 | Q 3.3 | Page 6

List all the elements of the following set:

$C = \left\{ x: x \text{ is an integer }, - \frac{1}{2} < x < \frac{9}{2} \right\}$

Ex. 1.20 | Q 3.4 | Page 6

List all the elements of the following set:

D = {x : x is a vowel in the word "EQUATION"}

Ex. 1.20 | Q 3.5 | Page 6

List all the elements of the following set:

E = {x : x is a month of a year not having 31 days}

Ex. 1.20 | Q 3.6 | Page 6

List all the elements of the following set:

F = {x : x is a letter of the word "MISSISSIPPI"}

Ex. 1.20 | Q 4 | Page 7

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

 (i) {A, P, L, E} (i) x : x + 5 = 5, x ∈ Z (ii) {5, −5} (ii) {x : x is a prime natural number and a divisor of 10} (iii) {0} (iii) {x : x is a letter of the word "RAJASTHAN"} (iv) {1, 2, 5, 10,} (iv) {x: x is a natural number and divisor of 10} (v) {A, H, J, R, S, T, N} (v) x : x2 − 25 = 0 (vi) {2, 5} (vi) {x : x is a letter of the word "APPLE"}
Ex. 1.20 | Q 5 | Page 7

Write the set of all vowels in the English alphabet which precede q.

Ex. 1.20 | Q 6 | Page 7

Write the set of all positive integers whose cube is odd.

Ex. 1.20 | Q 7 | Page 7

Write the set $\left\{ \frac{1}{2}, \frac{2}{5}, \frac{3}{10}, \frac{4}{17}, \frac{5}{26}, \frac{6}{37}, \frac{7}{50} \right\}$  in the set-builder form.

Chapter 1: Sets Exercise 1.30 solutions [Pages 9 - 10]

Ex. 1.30 | Q 1.1 | Page 9

Which of the following are examples of empty set?

Set of all even natural numbers divisible by 5

Ex. 1.30 | Q 1.2 | Page 9

Which of the following are examples of empty set?

Set of all even prime numbers

Ex. 1.30 | Q 1.3 | Page 9

Which of the following are examples of empty set?

{x : x2 −2 = 0 and x is rational}

Ex. 1.30 | Q 1.4 | Page 9

Which of the following are examples of empty set?

{x : x is a natural number, x < 8 and simultaneously x > 12};

Ex. 1.30 | Q 1.5 | Page 9

Which of the following are examples of empty set?

{x : x is a point common to any two parallel lines}.

Ex. 1.30 | Q 2.1 | Page 9

Which of the following sets are finite and which are infinite?

Set of concentric circles in a plane

Ex. 1.30 | Q 2.2 | Page 9

Which of the following sets are finite and which are infinite?

Set of letters of the English Alphabets

Ex. 1.30 | Q 2.3 | Page 9

Which of the following sets are finite and which are infinite?

{x ∈ N : x > 5}

Ex. 1.30 | Q 2.4 | Page 9

Which of the following sets are finite and which are infinite?

{x = ∈ N : x < 200}

Ex. 1.30 | Q 2.5 | Page 9

Which of the following sets are finite and which are infinite?

{x ∈ Z : x < 5};

Ex. 1.30 | Q 2.6 | Page 9

Which of the following sets are finite and which are infinite?

{x ∈ R : 0 < x < 1}.

Ex. 1.30 | Q 3 | Page 9

Which of the following sets are equal?

(i) $A = \left\{ 1, 2, 3 \right\};$

(ii) $B = \left\{ x \in R : x^2 - 2x + 1 = 0 \right\};$

(iii) $C = \left\{ 1, 2, 2, 3 \right\};$

(iv) $D = \left\{ x \in R : x^3 - 6 x^2 + 11x - 6 = 0 \right\}$

Ex. 1.30 | Q 4 | Page 10

Are the following sets equal?
A = {x : x is a letter in the word reap}:
B = {x : x is a letter in the word paper};
C = {x : x is a letter in the word rope}.

Ex. 1.30 | Q 5 | Page 10

From the sets given below, pair the equivalent sets:

$A = \left\{ 1, 2, 3 \right\}, B = \left\{ t, p, q, r, s \right\}, C = \left\{ \alpha, \beta, \gamma \right\}, D = \left\{ a, e, i, o, u \right\} .$

Ex. 1.30 | Q 6.1 | Page 10

Are the following pairs of sets equal? Give reasons.

A = {2, 3}, B = {x : x is a solution of x2 + 5x + 6 = 0}

Ex. 1.30 | Q 6.2 | Page 10

Are the following pairs of sets equal? Give reasons.

A = {x : x is a letter of the word " WOLF"};
B = {x : x is a letter of the word " FOLLOW"}.

Ex. 1.30 | Q 7 | Page 10

From the sets given below, select equal sets and equivalent sets.
A = {0, a}, B = {1, 2, 3, 4} C = {4, 8, 12}, D = {3, 1, 2, 4},
E = {1, 0}, F = {8, 4, 12} G = {1, 5, 7, 11}, H = {ab}.

Ex. 1.30 | Q 8 | Page 10

Which of the following sets are equal?
A = {x : x ∈ Nx, < 3},
B = {1, 2}
C = {3, 1}
D = {x : x ∈ Nx is odd, x < 5},
E = {1, 2, 1, 1} F = {1, 1, 3}.

Ex. 1.30 | Q 9 | Page 10

Show that the set of letters needed to spell "CATARACT" and the set of letters needed to spell "TRACT" are equal.

Chapter 1: Sets Exercise 1.40 solutions [Pages 16 - 17]

Ex. 1.40 | Q 1.1 | Page 16

Which of the following statements are true? Give reason to support your answer.
(i) For any two sets A and B either $A \subseteq B o\text{ or } B \subseteq A;$

Ex. 1.40 | Q 1.2 | Page 16

Which of the following statements are true? Give reason to support your answer.

Every subset of an infinite set is infinite

Ex. 1.40 | Q 1.3 | Page 16

Which of the following statements are true? Give reason to support your answer.

Every subset of a finite set is finite

Ex. 1.40 | Q 1.4 | Page 16

Which of the following statements are true? Give reason to support your answer.

Every set has a proper subset

Ex. 1.40 | Q 1.5 | Page 16

Which of the following statements are true? Give reason to support your answer.

{ababab, ...} is an infinite set

Ex. 1.40 | Q 1.6 | Page 16

Which of the following statements are true? Give reason to support your answer.

{abc} and {1, 2, 3} are equivalent sets

Ex. 1.40 | Q 1.7 | Page 16

Which of the following statements are true? Give reason to support your answer.

A set can have infinitely many subsets.

Ex. 1.40 | Q 2.1 | Page 16

State whether the following statements are true or false:

$1 \in \left\{ 1, 2, 3 \right\}$

Ex. 1.40 | Q 2.2 | Page 16

State whether the following statements are true or false:

$a \subset {b, c, a}$

Ex. 1.40 | Q 2.3 | Page 16

State whether the following statements are true or false:

$\left\{ a \right\} \in \left\{ a, b, c \right\}$

Ex. 1.40 | Q 2.4 | Page 16

State whether the following statements are true or false:

$\left\{ a, b \right\} = \left\{ a, a, b, b, a \right\}$

Ex. 1.40 | Q 2.5 | Page 16

State whether the following statements are true or false:

The set {x ; x + 8 = 8} is the null set.

Ex. 1.40 | Q 3 | Page 16

Decide among the following sets, which are subsets of which:

$A = {x : x \text{ satisfies } x^2 - 8x + 12 = 0},$

$B = \left\{ 2, 4, 6 \right\}, C = \left\{ 2, 4, 6, 8, . . . \right\}, D = \left\{ 6 \right\} .$

Ex. 1.40 | Q 4.1 | Page 16

Write which of the following statements are true? Justify your answer.

The set of all integers is contained in the set of all set of all rational numbers.

Ex. 1.40 | Q 4.2 | Page 16

Write which of the following statement are true? Justify your answer.

The set of all crows is contained in the set of all birds.

Ex. 1.40 | Q 4.3 | Page 16

Write which of the following statement are true? Justify your answer.

The set of all rectangle is contained in the set of all squares.

Ex. 1.40 | Q 4.4 | Page 16

Write which of the following statement are true? Justify your answer.

The set of all real numbers is contained in the set of all complex numbers.

Ex. 1.40 | Q 4.5 | Page 16

Write which of the following statement are true? Justify your answer.

The sets P = {a} and B = {{a}} are equal.

Ex. 1.40 | Q 4.6 | Page 16

Write which of the following statement are true? Justify your answer.

The sets A = {x : x is a letter of the word "LITTLE"} and,B = {x : x is a letter of the word "TITLE"} are equal.

Ex. 1.40 | Q 5.1 | Page 16

Which of the following statement are correct?
Write a correct form of each of the incorrect statements.

$a \subset \left\{ a, b, c \right\}$

Ex. 1.40 | Q 5.2 | Page 16

Which of the following statement are correct?
Write a correct form of each of the incorrect statement.

$\left\{ a \right\} \in \left\{ a, b, c \right\}$

Ex. 1.40 | Q 5.3 | Page 16

Which of the following statements are correct?
Write a correct form of each of the incorrect statement.

$a \in {\left\{ a \right\}, b}$

Ex. 1.40 | Q 5.4 | Page 16

Which of the following statement are correct?
Write a correct form of each of the incorrect statement.

$\left\{ a \right\} \subset \left\{ \left\{ a \right\}, b \right\}$

Ex. 1.40 | Q 5.5 | Page 16

Which of the following statement are correct?
Write a correct form of each of the incorrect statement.

$\left\{ b, c \right\} \subset \left\{ a, \left\{ b, c \right\} \right\}$

Ex. 1.40 | Q 5.6 | Page 16

Which of the following statemen are correct?
Write a correct form of each of the incorrect statement.

$\left\{ a, b \right\} \subset \left\{ a, \left\{ b, c \right\} \right\}$

Ex. 1.40 | Q 5.7 | Page 16

Which of the following statement are correct?
Write a correct form of each of the incorrect statement.

$\left\{ a, b \right\} \subset \left\{ a, \left\{ b, c \right\} \right\}$

Ex. 1.40 | Q 5.8 | Page 16

Which of the following statement are correct?
Write a correct form of each of the incorrect statement.

$\left\{ a, b \right\} \subset \left\{ a, \left\{ b, c \right\} \right\}$

Ex. 1.40 | Q 5.9 | Page 16

Which of the following statement are correct?
Write a correct form of each of the incorrect statement.

$\phi \subset \left\{ a, b, c \right\}$

Ex. 1.40 | Q 6.01 | Page 17

Let A = {ab, {cd}, e}. Which of the following statement are false and why?

$\left\{ c, d \right\} \subset A$

Ex. 1.40 | Q 6.02 | Page 17

Let A = {ab, {cd}, e}. Which of the following statement are false and why?

$\left\{ c, d \right\} \in A$

Ex. 1.40 | Q 6.03 | Page 17

Let A = {ab, {cd}, e}. Which of the following statement are false and why?

$\left\{ \left\{ c, d \right\} \right\} \subset A$

Ex. 1.40 | Q 6.04 | Page 17

Let A = {ab, {cd}, e}. Which of the following statement are false and why?

$a \in A$

Ex. 1.40 | Q 6.05 | Page 17

Let A = {ab, {cd}, e}. Which of the following statement are false and why?

$a \subset A$

Ex. 1.40 | Q 6.06 | Page 17

Let A = {ab, {cd}, e}. Which of the following statement are false and why?

$\left\{ a, b, e \right\} \subset A$

Ex. 1.40 | Q 6.07 | Page 17

Let A = {ab, {cd}, e}. Which of the following statements are false and why?

$\left\{ a, b, e \right\} \in A$

Ex. 1.40 | Q 6.08 | Page 17

Let A = {ab, {cd}, e}. Which of the following statement are false and why?

$\left\{ a, b, c \right\} \subset A$

Ex. 1.40 | Q 6.09 | Page 17

Let A = {ab, {cd}, e}. Which of the following statement are false and why?

$\phi \in A$

Ex. 1.40 | Q 6.1 | Page 17

Let A = {ab, {cd}, e}. Which of the following statement are false and why?

$\phi \in A$

Ex. 1.40 | Q 7.1 | Page 17

Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:

$1 \in A$

Ex. 1.40 | Q 7.2 | Page 17

Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:

$\left\{ 1, 2, 3 \right\} \subset A$

Ex. 1.40 | Q 7.3 | Page 17

Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:

$\left\{ 6, 7, 8 \right\} \in A$

Ex. 1.40 | Q 7.4 | Page 17

Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:

$\left\{ \left\{ 4, 5 \right\} \right\} \subset A$

Ex. 1.40 | Q 7.5 | Page 17

Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:

$\phi \in A$

Ex. 1.40 | Q 7.6 | Page 17

Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:

$\phi \in A$

Ex. 1.40 | Q 8.1 | Page 17

Let$A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}$Which of the following are true?

Ex. 1.40 | Q 8.2 | Page 17

Let$A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}$ Which of the following are true?

$\left\{ \phi \right\} \in A$

Ex. 1.40 | Q 8.3 | Page 17

Let $A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}$ Which of the following are true? $\left\{ 1 \right\} \in A$

Ex. 1.40 | Q 8.4 | Page 17

Let $A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}$Which of the following are true? $\left\{ 1 \right\} \in A$

Ex. 1.40 | Q 8.5 | Page 17

Let $A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}$ Which of the following are true? $2 \subset A$

Ex. 1.40 | Q 8.6 | Page 17

Let $A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}$Which of the following are true?$\left\{ 2 \left\{ 1 \right\} \right\} \not\subset A$

Ex. 1.40 | Q 8.7 | Page 17

Let $A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}$Which of the following are true? $\left\{ \left\{ 2 \right\}, \left\{ 1 \right\} \right\} \not\subset A$

Ex. 1.40 | Q 8.8 | Page 17

Let $\left\{ \left\{ 2 \right\}, \left\{ 1 \right\} \right\} \not\subset A$ Which of the following are true? $\left\{ \left\{ 2 \right\}, \left\{ 1 \right\} \right\} \not\subset A$

Ex. 1.40 | Q 8.9 | Page 17

Let $A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}$ Which of the following are true?$\left\{ \left\{ \phi \right\} \right\} \subset A$

Ex. 1.40 | Q 9.1 | Page 17

Write down all possible subsets of each of the following set:

{a

Ex. 1.40 | Q 9.2 | Page 17

Write down all possible subsets of each of the following set:

{0, 1},

Ex. 1.40 | Q 9.3 | Page 17

Write down all possible subsets of each of the following set:

{abc},

Ex. 1.40 | Q 9.4 | Page 17

Write down all possible subsets of each of the following set:

{1, {1}},

Ex. 1.40 | Q 9.5 | Page 17

Write down all possible subsets of each of the following set:

$\left\{ \phi \right\}$

Ex. 1.40 | Q 10.1 | Page 17

Write down all possible proper subsets each of the following set:

{1, 2},

Ex. 1.40 | Q 10.2 | Page 17

Write down all possible proper subsets each of the following set:

{1, 2, 3}

Ex. 1.40 | Q 10.3 | Page 17

Write down all possible proper subsets each of the following set:

{1}.

Ex. 1.40 | Q 11 | Page 17

What is the total number of proper subsets of a set consisting of n elements?

Ex. 1.40 | Q 12 | Page 17

If A is any set, prove that: $A \subseteq \phi \Leftrightarrow A = \phi .$

Ex. 1.40 | Q 13 | Page 17

Prove that:

$A \subseteq B, B \subseteq C \text{ and } C \subseteq A \Rightarrow A = C .$

Ex. 1.40 | Q 14 | Page 17

How many elements has $P \left( A \right), \text{ if } A = \phi$

Ex. 1.40 | Q 15.1 | Page 17

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

The set of right triangles.

Ex. 1.40 | Q 15.2 | Page 17

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

The set of isosceles triangles.

Ex. 1.40 | Q 16 | Page 17

If $X = \left\{ 8^n - 7n - 1: n \in N \right\} \text{ and } Y = \left\{ 49\left( n - 1 \right): n \in N \right\}$ $X \subseteq Y .$

Chapter 1: Sets Exercise 1.50 solutions [Page 21]

Ex. 1.50 | Q 1.1 | Page 21

If A and B are two set such that $A \subset B$then find:

$A \cap B$

Ex. 1.50 | Q 1.2 | Page 21

If A and B are two sets such that $A \subset B$ then find:

$A \cup B$

Ex. 1.50 | Q 2.01 | Page 21

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find:

$A \cup B$

Ex. 1.50 | Q 2.02 | Page 21

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find:

$A \cup C$

Ex. 1.50 | Q 2.03 | Page 21

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find:

$B \cup C$

Ex. 1.50 | Q 2.04 | Page 21

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find$B \cup D$

Ex. 1.50 | Q 2.05 | Page 21

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find:

$A \cup B \cup C$

Ex. 1.50 | Q 2.06 | Page 21

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find:

$A \cup B \cup D$

Ex. 1.50 | Q 2.07 | Page 21

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find:

$B \cup C \cup D$

Ex. 1.50 | Q 2.08 | Page 21

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find:

$A \cap \left( B \cup C \right)$

Ex. 1.50 | Q 2.09 | Page 21

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find:

$\left( A \cap B \right) \cap \left( B \cap C \right)$

Ex. 1.50 | Q 2.1 | Page 21

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find:

$\left( A \cup D \right) \cap \left( B \cup C \right)$

Ex. 1.50 | Q 3.1 | Page 21

Let $A = \left\{ x: x \in N \right\}, B = \left\{ x: x - 2n, n \in N \right\}, C = \left\{ x: x = 2n - 1, n \in N \right\}$and D = {x : x is a prime natural number}. Find: $A \cap B$

Ex. 1.50 | Q 3.2 | Page 21

Let $A = \left\{ x: x \in N \right\}, B = \left\{ x: x - 2n, n \in N \right\}, C = \left\{ x: x = 2n - 1, n \in N \right\}$ and D = {x : x is a prime natural number}. Find: $A \cap C$

Ex. 1.50 | Q 3.3 | Page 21

Let $A = \left\{ x: x \in N \right\}, B = \left\{ x: x - 2n, n \in N \right\}, C = \left\{ x: x = 2n - 1, n \in N \right\}$ and D = {x : x is a prime natural number}. Find: $A \cap D$

Ex. 1.50 | Q 3.4 | Page 21

Let $A = \left\{ x: x \in N \right\}, B = \left\{ x: x - 2n, n \in N \right\}, C = \left\{ x: x = 2n - 1, n \in N \right\}$ and D = {x : x is a prime natural number}. Find: $B \cap C$

Ex. 1.50 | Q 3.5 | Page 21

Let $A = \left\{ x: x \in N \right\}, B = \left\{ x: x - 2n, n \in N \right\}, C = \left\{ x: x = 2n - 1, n \in N \right\}$ and D = {x : x is a prime natural number}. Find: $B \cap D$

Ex. 1.50 | Q 3.6 | Page 21

Let $A = \left\{ x: x \in N \right\}, B = \left\{ x: x - 2n, n \in N \right\}, C = \left\{ x: x = 2n - 1, n \in N \right\}$ and D = {x : x is a prime natural number}. Find: $C \cap D$

Ex. 1.50 | Q 4.1 | Page 21

Let A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}. Find: $A - B$

Ex. 1.50 | Q 4.2 | Page 21

Let A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}. Find: $A - C$

Ex. 1.50 | Q 4.3 | Page 21

Let A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}. Find: $A - D$

Ex. 1.50 | Q 4.4 | Page 21

Let A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}. Find:

$B - A$

Ex. 1.50 | Q 4.5 | Page 21

Let A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}. Find: $C - A$

Ex. 1.50 | Q 4.6 | Page 21

Let A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}.

Find: $D - A$

Ex. 1.50 | Q 4.7 | Page 21

Let A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}.

Find: $B - C$

Ex. 1.50 | Q 4.8 | Page 21

Let A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}.

Find: $B - D$

Ex. 1.50 | Q 5.1 | Page 21

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find $A'$

Ex. 1.50 | Q 5.2 | Page 21

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, = {2, 4, 6, 8} and C = {3, 4, 5, 6}.

Find $B'$

Ex. 1.50 | Q 5.3 | Page 21

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, = {2, 4, 6, 8} and C = {3, 4, 5, 6}.

Find $\left( A \cap C \right)'$

Ex. 1.50 | Q 5.4 | Page 21

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, = {2, 4, 6, 8} and C = {3, 4, 5, 6}.

Find $\left( A \cup B \right)'$

Ex. 1.50 | Q 5.5 | Page 21

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find $\left( A' \right)'$

Ex. 1.50 | Q 5.6 | Page 21

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, = {2, 4, 6, 8} and C = {3, 4, 5, 6}.

Find $\left( B - C \right)'$

Ex. 1.50 | Q 6.1 | Page 21

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that $\left( A \cup B \right)' = A' \cap B'$

Ex. 1.50 | Q 6.2 | Page 21

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that $\left( A \cap B \right)' = A' \cup B'$

Chapter 1: Sets Exercise 1.60 solutions [Page 27]

Ex. 1.60 | Q 1 | Page 27

We have to find the smallest set A such that$A \cup \left\{ 1, 2 \right\} = \left\{ 1, 2, 3, 5, 9 \right\}$

Ex. 1.60 | Q 2.1 | Page 27

Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identities:

$A \cup \left( B \cap C \right) = \left( A \cup B \right) \cap \left( A \cup C \right)$

Ex. 1.60 | Q 2.2 | Page 27

Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identitie:

$A \cap \left( B \cup C \right) = \left( A \cap B \right) \cup \left( A \cap C \right)$

Ex. 1.60 | Q 2.3 | Page 27

Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identitie:

$A \cap \left( B - C \right) = \left( A \cap B \right) - \left( A \cap C \right)$

Ex. 1.60 | Q 2.4 | Page 27

Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identitie:

$A - \left( B \cup C \right) = A\left( A - B \right) \cap \left( A - C \right)$

Ex. 1.60 | Q 2.5 | Page 27

Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identitie:

$A - \left( B \cap C \right) = \left( A - B \right) \cup \left( A - C \right)$

Ex. 1.60 | Q 2.6 | Page 27

Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identitie:

$A \cap \left( B ∆ C \right) = \left( A \cap B \right) ∆ \left( A \cap C \right)$

Ex. 1.60 | Q 3.1 | Page 27

If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that:

$\left( A \cup B \right)' = A' \cap B'$

Ex. 1.60 | Q 3.2 | Page 27

If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that:

$\left( A \cap B \right)' = A'B' .$

Ex. 1.60 | Q 4.2 | Page 27

For any two sets A and B, prove that

A ∩ ⊂ A

Ex. 1.60 | Q 4.2 | Page 27

For any two sets A and B, prove that

B ⊂ A ∪ B

Ex. 1.60 | Q 4.3 | Page 27

For any two sets A and B, prove that A ⊂ ⇒ A ∩

Ex. 1.60 | Q 5 | Page 27

For any two sets A and B, show that the following statements are equivalent:

(i) $A \subset B$

(ii) $A \subset B$=ϕ

(iii) $A \cup B = B$

(iv) $A \cap B = A .$

Ex. 1.60 | Q 6.1 | Page 27

For three sets AB and C, show that $A \cap B = A \cap C$

Ex. 1.60 | Q 6.2 | Page 27

For three sets AB and C, show that $A \subset B \Rightarrow C - B \subset C - A$

Ex. 1.60 | Q 7.1 | Page 27

For any two sets, prove that:

$A \cup \left( A \cap B \right) = A$

Ex. 1.60 | Q 7.2 | Page 27

For any two sets, prove that:

$A \cap \left( A \cup B \right) = A$

Ex. 1.60 | Q 8 | Page 27

Find sets AB and C such that $A \cap B, A \cap C \text{ and } B \cap C$are non-empty sets and$A \cap B \cap C = \phi$

Ex. 1.60 | Q 9 | Page 27

For any two sets A and B, prove that: $A \cap B = \phi \Rightarrow A \subseteq B'$

Ex. 1.60 | Q 10 | Page 27

If A and B are sets, then prove that  $A - B, A \cap B \text{ and } B - A$ are pair wise disjoint.

Ex. 1.60 | Q 11 | Page 27

Using properties of sets, show that for any two sets A and B,$\left( A \cup B \right) \cap \left( A \cap B' \right) = A$

Ex. 1.60 | Q 12.1 | Page 27

For any two sets of A and B, prove that:

$A' \cup B = U \Rightarrow A \subset B$

Ex. 1.60 | Q 12.2 | Page 27

For any two sets of A and B, prove that:

$B' \subset A' \Rightarrow A \subset B$

Ex. 1.60 | Q 13 | Page 27

Is it true that for any sets A and $B, P \left( A \right) \cup P \left( B \right) = P \left( A \cup B \right)$? Justify your answer.

Ex. 1.60 | Q 14.1 | Page 27

Show that for any sets A and B, $A = \left( A \cap B \right) \cup \left( A - B \right)$

Ex. 1.60 | Q 14.2 | Page 27

Show that for any sets A and B, $A \cup \left( B - A \right) = A \cup B$

Ex. 1.60 | Q 15 | Page 27

Each set X, contains 5 elements and each set Y, contains 2 elements and $\cup^{20}_{r = 1} X_r = S = \cup^n_{r = 1} Y_r$ If each element of S belong to exactly 10 of the Xr's and to eactly 4 of Yr's, then find the value of n.

Chapter 1: Sets Exercise 1.70 solutions [Pages 34 - 35]

Ex. 1.70 | Q 1 | Page 34

For any two sets A and B, prove that :

$A' - B' = B - A$

Ex. 1.70 | Q 2.1 | Page 34

For any two sets A and B, prove the following:

$A \cap \left( A' \cup B \right) = A \cap B$

Ex. 1.70 | Q 2.2 | Page 34

For any two sets A and B, prove the following:

$A - \left( A - B \right) = A \cap B$

Ex. 1.70 | Q 2.3 | Page 34

For any two sets A and B, prove the following:

$A \cap \left( A \cup B \right)' = \phi$

Ex. 1.70 | Q 2.4 | Page 34

For any two sets A and B, prove the following:

$A - B = A \Delta\left( A \cap B \right)$

Ex. 1.70 | Q 3 | Page 34

If ABC are three sets such that $A \subset B$then prove that $C - B \subset C - A$

Ex. 1.70 | Q 4.1 | Page 35

For any two sets A and B, prove that $\left( A \cup B \right) - B = A - B$

Ex. 1.70 | Q 4.2 | Page 35

For any two sets A and B, prove that $A - \left( A \cap B \right) = A - B$

Ex. 1.70 | Q 4.3 | Page 35

For any two sets A and B, prove that $A - \left( A - B \right) = A \cap B$

Ex. 1.70 | Q 4.4 | Page 35

For any two sets A and B, prove that

$A \cup \left( B - A \right) = A \cup B$

Ex. 1.70 | Q 4.5 | Page 35

For any two sets A and B, prove that $\left( A - B \right) \cup \left( A \cap B \right) = A$

Chapter 1: Sets Exercise 1.80 solutions [Pages 46 - 47]

Ex. 1.80 | Q 1 | Page 46

If A and B are two sets such that $n \left( A \cup B \right) = 50, n \left( A \right) = 28 \text{ and } n \left( B \right) = 32$$n \left( A \cap B \right)$

Ex. 1.80 | Q 2 | Page 46

If P and Q are two sets such that P has 40 elements, $P \cup Q$has 60 elements and$P \cap Q$has 10 elements, how many elements does Q have?

Ex. 1.80 | Q 3 | Page 46

In a school there are 20 teachers who teach athematics or physics. Of these, 12 teach mathematics and 4 teach physics and mathematics. How many teach physics?

Ex. 1.80 | Q 4 | Page 46

In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many like both coffee and tea?

Ex. 1.80 | Q 5.1 | Page 47

Let A and B be two sets such that :$n \left( A \right) = 20, n \left( A \cup B \right) = 42 \text{ and } n \left( A \cap B \right) = 4$ Find$n\left( B \right)$

Ex. 1.80 | Q 5.2 | Page 47

Let A and B be two sets such that : $n \left( A \right) = 20, n \left( A \cup B \right) = 42 \text{ and } n \left( A \cap B \right) = 4$ $n \left( A - B \right)$

Ex. 1.80 | Q 5.3 | Page 47

Let A and B be two sets such that : $n \left( A \right) = 20, n \left( A \cup B \right) = 42 \text{ and } n \left( A \cap B \right) = 4$ $n \left( B - A \right)$

Ex. 1.80 | Q 6 | Page 47

A survey shows that 76% of the Indians like oranges, whereas 62% like bananas. What percentage of the Indians like both oranges and bananas?

Ex. 1.80 | Q 7.1 | Page 47

In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find:

how many can speak both Hindi and English:

Ex. 1.80 | Q 7.2 | Page 47

In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find: how many can speak Hindi only

Ex. 1.80 | Q 7.3 | Page 47

In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find:

how many can speak English only.

Ex. 1.80 | Q 8 | Page 47

In a group of 50 persons, 14 drink tea but not coffee and 30 drink tea. Find:
(i) how may drink tea and coffee both;
(ii) how many drink coffee but not tea.

Ex. 1.80 | Q 9 | Page 47

In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) the numbers of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper.

Ex. 1.80 | Q 10 | Page 47

Of the members of three athletic teams in a certain school, 21 are in the basketball team, 26 in hockey team and 29 in the football team, 14 play hockey and basket ball 15 play hockey and football, 12 play football and basketball and 8 play all the three games. How many members are there in all?

Ex. 1.80 | Q 11 | Page 47

In a group of 1000 people, there are 750 who can speak Hindi and 400 who can speak Bengali. How many can speak Hindi only? How many can speak Bengali? How many can speak both Hindi and Bengali?

Ex. 1.80 | Q 12 | Page 47

$\cap$A survey of 500 television viewers produced the following information; 285 watch football, 195 watch hockey, 115 watch basketball, 45 watch football and basketball, 70 watch football and hockey, 50 watch hockey and basketball, 50 do not watch any of the three games. How many watch all the three games? How many watch exactly one of the three games?

Ex. 1.80 | Q 13 | Page 47

In a survey of 100 persons it was found that 28 read magazine A, 30 read magazine B, 42 read magazine C, 8 read magazines A and B, 10 read magazines A and C, 5 read magazines B and C and 3 read all the three magazines. Find:
(i) How many read none of three magazines?
(ii) How many read magazine C only?

Ex. 1.80 | Q 14 | Page 47

In a survey of 100 students, the number of students studying the various languages were found to be : English only 18, English but not Hindi 23, English and Sanskrit 8, English 26, Sanskrit 48, Sanskrit and Hindi 8, no language 24. Find:
(i) How many students were studying Hindi?
(ii) How many students were studying English and Hindi?

Ex. 1.80 | Q 15 | Page 47

In a survey it was found that 21 persons liked product P1, 26 liked product P2 and 29 liked product P3. If 14 persons liked products P1 and P2; 12 persons liked product P3 and P1 ; 14 persons liked products P2 and P3 and 8 liked all the three products. Find how many liked product P3 only.

Chapter 1: Sets Exercise 1.90 solutions [Page 49]

Ex. 1.90 | Q 1 | Page 49

If a set contains n elements, then write the number of elements in its power set.

Ex. 1.90 | Q 2 | Page 49

Write the number of elements in the power set of null set.

Ex. 1.90 | Q 3 | Page 49

Let A = {x : x ∈ Nx is a multiple of 3} and B = {x : x ∈ N and x is a multiple of 5}. Write $A \cap B$

Ex. 1.90 | Q 4 | Page 49

Let A and B be two sets having 3 and 6 elements respectively. Write the minimum number of elements that $A \cup B$

Ex. 1.90 | Q 5 | Page 49

If A = {x ∈ C : x2 = 1} and B = {x ∈ C : x4 = 1}, then write A − B and B − A

Ex. 1.90 | Q 6 | Page 49

If A and B are two sets such that $A \subset B$, then write B' − A' in terms of A and B.

Ex. 1.90 | Q 7 | Page 49

Let A and B be two sets having 4 and 7 elements respectively. Then write the maximum number of elements that $A \cup B$ can have.

Ex. 1.90 | Q 8 | Page 49

If $A = \left\{ \left( x, y \right) : y = \frac{1}{x}, 0 \neq x \in R \right\}$and$B = \left\{ \left( x, y \right) : y = - x, x \in R \right\}$ then write$A \cap B$

Ex. 1.90 | Q 9 | Page 49

If $A = \left\{ \left( x, y \right) : y = e^x , x \in R \right\} and B = \left\{ \left( x, y \right) : y = e^{- x} , x \in R \right\}$write$A \cap B$

Ex. 1.90 | Q 10 | Page 49

If A and B are two sets such that $n \left( A \right) = 20, n \left( B \right) = 25$\text{ and } $n \left( A \cup B \right) = 40$, then write $n \left( A \cap B \right)$

Ex. 1.90 | Q 11 | Page 49

If A and B are two sets such that $n \left( A \right) = 115, n \left( B \right) = 326, n \left( A - B \right) = 47,$ then write $n \left( A \cup B \right)$

Chapter 1: Sets solutions [Pages 49 - 51]

Q 1 | Page 49

For any set A, (A')' is equal to

• (a) A'

• (b) A

• (c) ϕ

• (d) none of these.

Q 2 | Page 49

Let A and B be two sets in the same universal set. Then,$A - B =$

• (a) $A \cap B$

• (b)$A' \cap B$

• (c)$A \cap B'$

• (d) none of these.

Q 3 | Page 49

The number of subsets of a set containing n elements is

• (a) n

• (b) 2n − 1

• (c) n2

• (d) 2n

Q 4 | Page 50

For any two sets A and B,$A \cap \left( A \cup B \right) =$

• (a) A

• (b) B

• (c) ϕ

• (d) none of these.

Q 5 | Page 50

If A = {1, 3, 5, B} and B = {2, 4}, then

• (a)$4 \in A$

• (b)$\left\{ 4 \right\} \subset A$

• (c)$B \subset A$

• (d) none of these.

Q 6 | Page 50

The symmetric difference of A and B is

• (a)$\left( A - B \right) \cap \left( B - A \right)$

• (b)$\left( A - B \right) \cup \left( B - A \right)$

• (c) $\left( A \cup B \right) - \left( A \cap B \right)$

• (d) $\left\{ \left( A \cup B \right) - A \right\} \cup \left\{ \left( A \cup B \right) - B \right\}$

Q 7 | Page 50

The symmetric difference of A = {1, 2, 3} and B = {3, 4, 5} is

• (a) {1, 2}

• (b) {1, 2, 4, 5}

• (c) {4, 3}

• (d) {2, 5, 1, 4, 3}

Q 8 | Page 50

For any two sets A and B,$\left( A - B \right) \cup \left( B - A \right) =$

• (a) $\left( A - B \right) \cup A$

• (b)$\left( B - A \right) \cup B$

• (c)$\left( A \cup B \right) - \left( A \cap B \right)$

• (d)$\left( A \cup B \right) \cap \left( A \cap B \right)$

Q 9 | Page 50

Which of the following statements is false:

• $A - B = A \cap B'$

• $A - B = A - \left( A \cap B \right)$

• $A - B = A - B'$

• $A - B = \left( A \cup B \right) - B .$

Q 10 | Page 50

For any three sets A, B and C

• (a) $A \cap \left( B - C \right) = \left( A \cap B \right) - \left( A \cap C \right)$

• (b) $A \cap \left( B - C \right) = \left( A \cap B \right) - C$

• (c) $A \cup \left( B - C \right) = \left( A \cup B \right) \cap \left( A \cup C' \right)$

• (d) $A \cup \left( B - C \right) = \left( A \cup B \right) - \left( A \cup C \right) .$

Q 11 | Page 50

Let $A = \left\{ x : x \in R, x \geq 4 \right\} \text{ and } B = \left\{ x \in R : x < 5 \right\}$  Then, $n \left( A' \cap B' \right) =$

• (a) (4, 5]

• (b) (4, 5)

• (c) [4, 5)

• (d) [4, 5]

Q 12 | Page 50

Let U be the universal set containing 700 elements. If AB are sub-sets of U such that $n \left( A \right) = 200, n \left( B \right) = 300 \text{ and } \left( A \cap B \right) = 100$.Then $n \left( A' \cap B' \right) =$

• (a) 400

• (b) 600

• (c) 300

• (d) none of these.

Q 13 | Page 50

Let A and B be two sets that $n \left( A \right) = 16, n \left( B \right) = 14, n \left( A \cup B \right) = 25$ Then, $n \left( A \cap B \right)$

• (a) 30

• (b) 50

• (c) 5

• (d) none of these

Q 14 | Page 50

If A = |1, 2, 3, 4, 5|, then the number of proper subsets of A is

• (a) 120

• (b) 30

• (c) 31

• (d) 32

Q 15 | Page 50

In set-builder method the null set is represented by

• (a) { }

• (b) Φ

• (c) $\left| x : x \neq x \right|$

• (d) $\left| x : x = x \right|$

Q 16 | Page 50

$\cap$ If A and B are two disjoint sets, then $n \left( A \cup B \right)$is equal to

• (a) $n \left( A \right) + n\left( B \right)$

• (b) $n \left( A \right) + n\left( B \right) - n\left( A \cap B \right)$

• (c)$n \left( A \right) + n \left( B \right) + n \left( A \cap B \right)$

• (d) $n \left( A \right) n \left( B \right)$

• (e) $n \left( A \right) - n \left( B \right)$

Q 17 | Page 50

For two sets [A \cup B = A\] iff

• (a) $B \subseteq A$

• (b) $A \subseteq B$

• (c) $A \neq B$

• (d) $A = B$

Q 18 | Page 50

If A and B are two sets such that $n \left( A \right) = 70, n \left( B \right) = 60, n \left( A \cup B \right) = 110$ then $n \left( A \cap B \right)$

• (a) 240

• (b) 50

• (c) 40

• (d) 20

Q 19 | Page 50

If A and B are two given sets, then $A \cap \left( A \cap B \right)^c$

• (a) A

• (b)

• (c) Φ

• (d)$A \cap B^c$

Q 20 | Page 50

If A = {x : x is a multiple of 3} and , B = {x : x is a multiple of 5}, then A − B is

• (a) $A \cap B$

• (b) $A \cap B$

• (c) $A \cap B$

• (d) $A \cap B$

Q 21 | Page 50

In a city 20% of the population travels by car, 50% travels by bus and 10% travels by both car and bus. Then, persons travelling by car or bus is

• (a) 80%

• (b) 40%

• (c) 60%

• (d) 70%

Q 22 | Page 51

If  $A \cap B - B$

• (a) $A \subset B$

• (b) $B \subset A$

• (c) $A = \Phi$

• (d) $B = \Phi$

Q 23 | Page 51

An investigator interviewed 100 students to determine the performance of three drinks: milk, coffee and tea. The investigator reported that 10 students take all three drinks milk, coffee and tea; 20 students take milk and coffee; 25 students take milk and tea; 12 students take milk only; 5 students take coffee only and 8 students take tea only. Then the number of students who did not take any of three drinks is

• 10

•  20

•  25

• 30

• N/A

Q 24 | Page 51

Two finite sets have m and n elements. The number of elements in the power set of first set is 48 more than the total number of elements in power set of the second set. Then, the values of m and n are:

•  7, 6

•  6, 3

• 7, 4

• 3, 7

Q 25 | Page 51

In a class of 175 students the following data shows the number of students opting one or more subjects. Mathematics 100; Physics 70; Chemistry 40; Mathematics and Physics 30; Mathematics and Chemistry 28; Physics and Chemistry 23; Mathematics, Physics and Chemistry 18. How many students have offered Mathematics alone?

• (a) 35

• (b) 48

• (c) 60

• (d) 22

• (e) 30

Q 26 | Page 51

Suppose $A_1 , A_2 , . . . , A_{30}$ are thirty sets each having 5 elements and $B_1 , B_2 , . . . , B_n$ are n sets each with 3 elements. Let $\cup^{30}_{i = 1} A_i = \cup^n_{j = 1} B_j = S$ and each element of S belong to exactly 10 of the $A_i 's$and exactly 9 of the$B_j 's$ then n is equal to

• (a) 15

•   (b) 3

•  (c) 45

•  (d) 35

Q 27 | Page 51

Two finite sets have m and n elements. The number of subsets of the first set is 112 more than that of the second. The values of m and n are respectively

• (a) 4, 7

•   (b) 7, 4

•  (c) 4, 4

•  (c) 4, 4

Q 28 | Page 51

For any two sets A and B, $A \cap \left( A \cup B \right)'$is equal to

• (a) A

•  (b) B

•  (c)$\phi$

•  (d)$A \cap B$

Q 29 | Page 51

The set  $\left( A \cup B' \right)' \cup \left( B \cap C \right)$ is equal to

• (a)  $A' \cup B \cup C$

•  (b)  $A' \cup B$

•  (c) $A' \cup c$

•  (d)  $A' \cap B$

• (e) n/a

Q 30 | Page 51

Let F1 be the set of all parallelograms, Fthe set of all rectangles, Fthe set of all rhombuses, F4 the set of all squares and Fthe set of trapeziums in a plane. Then F1may be equal to

• (a)  $F_2 \cap F_3$

•    (b) $F_3 \cap F_4$

• (c)  $F_2 \cup F_3$

•    (d) $F_2 \cup F_3 \cup F_4 \cup F_1$

Chapter 1: Sets

Ex. 1.10Ex. 1.20Ex. 1.30Ex. 1.40Ex. 1.50Ex. 1.60Ex. 1.70Ex. 1.80Ex. 1.90Others

RD Sharma Mathematics Class 11 RD Sharma solutions for Class 11 Mathematics chapter 1 - Sets

RD Sharma solutions for Class 11 Maths chapter 1 (Sets) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics chapter 1 Sets are Intersection of Sets, Difference of Sets, Proper and Improper Subset, Open and Close Intervals, Operation on Set - Disjoint Sets, Element Count Set, Universal Set, Venn Diagrams, Intrdouction of Operations on Sets, Union Set, Complement of a Set, Sets and Their Representations, The Empty Set, Finite and Infinite Sets, Equal Sets, Subsets, Power Set, Practical Problems on Union and Intersection of Two Sets.

Using RD Sharma Class 11 solutions Sets exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer RD Sharma Textbook Solutions to score more in exam.

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