20

Algebra Std Ix Maharashtra Board

Embed Size (px)

DESCRIPTION

its the first lesson of allgebra with full solution

Citation preview

Page 1: Algebra Std Ix Maharashtra Board

 

Page 2: Algebra Std Ix Maharashtra Board

No part of this book may be reproduced or transmitted in any form or by any means, C.D. ROM/Audio Video Cassettes or electronic, mechanical including photocopying; recording or by any information storage and retrieval system without permission in writing from the Publisher.

 

 

STD. IX

Algebra  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Printed at: Kings Security Printers Pvt. Ltd., Valsad

Written as per the revised syllabus prescribed by the Maharashtra State Board

of Secondary and Higher Secondary Education, Pune.

TEID : 890

Fourth Edition: April 2015

Salient Features

• Written as per the new textbook.

• Exhaustive coverage of entire syllabus. 

• Topic-wise distribution of all textual questions and practice problems at the beginning of every chapter.

• Precise theory for every topic.

• Covers answers to all textual exercises and problem set.

• Includes additional problems for practice.

• Comprehensive solution to Question Bank.

• Attractive layout of the content.

• Self evaluative in nature.

Page 3: Algebra Std Ix Maharashtra Board

PREFACE Algebra is the branch of mathematics which deals with the study of rules of operations and relations, and the concepts arising from them. It has wide applications in different fields of science and technology. It deals with concepts like linear equations, quadratic equations etc. Its application in statistics deals with measures of central tendency, representation of statistical data etc. The study of Algebra requires a deep and intrinsic understanding of concepts, terms and formulae. Hence, to ease this task, we present “Std. IX: Algebra”, a complete and thorough guide, extensively drafted to boost the students confidence. The question answer format of this book helps the student to understand and grasp each and every concept thoroughly. The book is based on the new text book and covers the entire syllabus. At the beginning of every chapter, topic-wise distribution of all textual questions and practice problems has been provided for simpler understanding of different types of questions. It contains answers to textual exercises, problems sets and Question bank. It also includes additional questions for practice. Graphs are drawn with proper scale. Another feature of the book is its layout which is attractive and inspires the student to read. Lastly, I would like to thank all those who have helped me in preparing this book. There is always room for improvement and hence I welcome all suggestions and regret any errors that may have occurred in the making of this book. A book affects eternity; one can never tell where its influence stops.

Best of luck to all the aspirants!

Yours faithfully, Publisher  

 

No. Topic Name Page No.

1 Sets 1

2 Real Numbers 18

3 Algebraic Expressions 61

4 Linear Equations in Two Variables 95

5 Graphs 130

6 Ratio and Proportion 191

7 Statistics 220

8 Question Bank 254

Page 4: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Chapter 01: Sets

1

Type of Problems Exercise Q. Nos.

Definition of Sets 1.1 Q.1

Problem set-1 Q.1

Method of Writing Sets

1.1 Q.2, 3, 4

Practice Problems

(Based on Exercise 1.1) Q.1, 2

Problem set-1 Q.2, 3, 10

Types of Sets

1.2 Q.1, 2, 3, 4

Practice Problems

(Based on Exercise 1.2) Q.1, 2

Problem set-1 Q.4, 5, 9

Subset and Universal Set

1.3 Q.1, 2, 4, 5

Practice Problems

(Based on Exercise 1.3) Q.1, 3, 4

Problem set-1 Q.11, 12, 22

Operations on Sets and their Properties

1.4 Q.1, 2, 3, 4, 5

Practice Problems

(Based on Exercise 1.4) Q.1, 2, 3, 4

Problem set-1 Q.6, 7, 8, 13, 14, 21, 23

Number of elements in a the Set

1.5 Q.1, 2, 5

Practice Problems

(Based on Exercise 1.5) Q.2

Problem set-1 Q.15, 18

Word Problems on Sets

1.5 Q. 3, 4

Practice Problems

(Based on Exercise 1.5) Q.3, 4

Problem set-1 Q.16, 17, 19

Draw a Venn Diagram

1.3 Q.3

Practice Problems

(Based on Exercise 1.3) Q.2

Practice Problems

(Based on Exercise 1.5)

Q.1, 5

Problem set-1 Q.20

Sets 01 

Page 5: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Std. IX : Algebra

2

Introduction Consider the following examples: i. Collection of books in a library. ii. Collection of cloths in a shop. Objects in each of these examples can be seen clearly. Such collections are well defined collections. Consider the following examples: i. Brilliant students in a class. ii. Happy people in the city. The term “brilliant” and “happy” are relative terms. A person may be brilliant or happy according to one person but he may not be so according to the other person. It is important to determine whether a given collection is well defined or not. Well defined collections or groups are termed as “Sets”. George Cantor, (1845-1918) a German Mathematician is a creator of “Set theory” which has become a fundamental theory in Mathematics. 1.1 Definition of Sets Set: A well defined collection of objects is called a

“set”. Example: i. Collection of odd natural numbers. ii. Collection of whole numbers. Each object in the set is called as an “element” or a “member” of the set. Example: i. For a set containing odd natural numbers,

elements are 1, 3, 5, 7, … ii. For a set of whole numbers, elements are

0, 1, 2, 3, … Collection of elements which are not well defined, do not form a set. Such sets usually contain relative terms like easy, good, favourite, etc. Example: The collection of good books in a library. Here, ‘good’ is a relative term whose meaning will vary from person to person. Important Points to Remember: 1. Sets are denoted by capital alphabets. e.g. A,

B, C, X, Y, Z, etc. But the elements of a set are denoted by small

alphabets e.g. a, b, p, q, r, etc. 2. If ‘r’ is an element of set P, then it is written

as r P and is read as: i. ‘r’ belongs to set P or ii. ‘r’ is a member of set P or iii. ‘r’ is an element of set P.

Symbol ‘’ stands for ‘belongs to’, ‘is a member of’ or ‘is an element of’.

3. If ‘r’ is not an element of set P, then it is written as r P and it is read as:

i. ‘r’ does not belong to set P or ii. ‘r’ is not a member of set P or iii. ‘r’ is not an element of set P . The symbol stands for ‘does not belong to’

or ‘not a member of’ or ‘not an element of’. 4. The set of Natural numbers, Whole numbers,

Integers, Rational numbers, Real numbers are denoted by N, W, I, Q, R respectively.

1.2 Methods of Writing Sets There are two methods of writing a set: a. Listing method or Roster form b. Rule method or Set builder form a. Listing method or Roster form In this method: i. Elements of the set are enclosed within

curly brackets. ii. Each element is written only once. iii. Elements are separated by commas. iv. The order of writing the elements in a

set is not important. Example: A = {a, b, c, d, e} or A = {b, d, a, c, e} are

same or equal sets that represent first five letters of the English alphabet.

Few examples of writing a set by listing

method are: i. L is a set of letters of the word “fatal”. L = {f, a, t, l} ii. M is a set of integers less than 5. M = {… , 3, 2, 1, 0, 1, 2, 3, 4} iii. O is a set of even natural numbers from

1 to 100. O = {2, 4, 6, 8, … , 100} b. Rule method or Set builder form In this method, elements of the set are

described by specifying the property or rule that uniquely determines the elements of a set.

Example: i. Y = {x|x is a vowel in the English

alphabet} In the above notation, curly brackets

denotes ‘set of’, vertical line (|) denotes ‘such that’.

Set Y is read as: “Y is a set of all ‘x’ such that ‘x’ is a

vowel in the English alphabet”.

Page 6: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Chapter 01: Sets

3

ii. B = {x|x W, x < 10} Set B is read as: “B is a set of all ‘x’ such that ‘x’ is a

whole number less than 10”. Note: Instead of ‘|’ sometimes two vertical dots ‘:’ are also used. Exercise 1.1 1. Which of the following collections are sets? i. The collection of prime numbers. ii. The collection of easy sub topics in

this chapter. iii. The collection of good teachers in

your school. iv. The collection of girls in your class. v. The collection of odd natural numbers. Solution: i. It is a set. ii. Meaning of ‘easy sub topics’ may vary from

person to person, as it is a relative term. Therefore, it is not a set.

iii. Choice of good teachers varies from student to student as ‘good’ is a relative term. Therefore, it is not a set.

iv. It is a set. v. It is a set. 2. Write the following sets in the roster form:

i. A = {x|x is a month of the Gregorian year not having 30 days}

ii. B = {y|y is a colour in the rainbow} iii. C = {x|x is an integer and 4 < x < 4} iv. D = {x|x I, 3 < x 3} v. E = {x|x = (n 1)3, n < 3, n W}

Solution: i. A = {January, February, March, May, July,

August, October, December} ii. B = {violet, indigo, blue, green, yellow, orange,

red} iii. C = {3, 2, 1, 0, 1, 2, 3} iv. D = {2, 1, 0, 1, 2, 3} v. Putting n = 0, 1, 2, we have, E = {1, 0, 1} 3. Write the following sets in the set builder

form: i. F = {5, 10, 15, 20} ii. G = {9, 16, 25, 36, … , 81} iii. H = {5, 52, 53, 54} iv. X = {8, 8}

v. Y = 1 1 1 1

1, , , ,8 27 64 125

Solution: i. F = {x|x = 5n, n N, n 4} ii. G = {x|x = n2, n N, 3 n < 10} iii. H = {x|x = 5n, n N, n 4} iv. X = {x| square of x is 64} or X = {x|x is a square root of 64}

v. Y = {x|x = 3

1

n, n N, n 5}

4. Write the set of first five positive integers

whose square is odd. Solution: P = {1, 3, 5, 7, 9} 1.3 Venn Diagrams L. Euler, a great Mathematician, introduced the idea of diagrammatic representation of sets. Later, British logician, John-Venn (1834-1923) used and developed the idea of the above concept to study sets. Such representations are called Venn Diagrams. A set is represented by a ‘closed’ figure in a Venn Diagram, where the elements of the set are represented by points in the closed figure. Some of the closed figures used to represent Venn Diagrams are: rectangle, circle, triangle, etc. Examples: 1.4 Types of Sets i. Singleton set: A set containing exactly one

element is called as a singleton set. Example: a. A = {5} b. B = {x|x + 3 = 0} Set B having only one element i.e., 3 ii. Empty set: A set which does not contain any

element is called as an empty or a null set. It is represented as {} or (phi).

A . a . e . i

. o . u

. a

. b . c

. d . e

C

. 0

. 2

. 1

. 2

. 4

. 6

. 8

B

D

Page 7: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Std. IX : Algebra

4

Example: a. A = {a|a is a natural number, 5 < a < 6}

A = { } or A =

b. B = {x|x is a natural number, x < 1} B = iii. Finite set: If counting of elements in a set

terminates at a certain stage, the set is called as finite set.

Example: A = {1, 2, 3, 4, 5, 6, 7} B = {x|x is days in a week} The above sets A and B have finite elements. Set A and set B are finite sets. iv. Infinite set: If counting of elements in a set

does not terminate at any stage, the set is called as infinite set.

Example: P = {1, 2, 3, 4, 5, 6, …} W = {x|x is a whole number} The above sets P and W have elements that

cannot be counted. They are sets that do not terminate at any stage. Therefore, P and W are infinite sets.

Note: i. X = {0} is not a null set as ‘0’ is an element of

set X. ii. An empty set is a finite set. iii. Sets of Natural numbers, Whole numbers,

Integers, Rational numbers and Real numbers are all infinite sets.

Exercise 1.2 1. State which of the following sets are

singleton sets:

i. A = 16x x

ii. B = {y|y2 = 36} iii. C = {p|p I, p3 = 8} iv. D = {q|(q 4)2 = 0} v. E = {x|1 + 2x = 3x, x W} Solution: i. x = 16 x = 256 A = {256} It is a singleton set. ii. y2 = 36 y = 6 B = {–6, +6} It is not a singleton set.

iii. p3 = 8

p3 = (2)3

p = 2

C = {2}

It is a singleton set. iv. (q 4)2 = 0

q 4 = 0

q = 4

D = {4}

It is a singleton set. v. 1 + 2x = 3x

1 = 3x 2x

1 = x

x = 1

E = {1}

It is a singleton set. 2. Which of the following sets are empty? i. A set of all even prime numbers ii. B = {x|x is a capital of India}

iii. F = {y|y is a point of intersection of two parallel lines}

iv. G = {z|z N, 3 < z < 4} v. H = {t|t is a triangle having four sides} Solution: i. A = {2} It is not an empty set. ii. B = {Delhi} It is not an empty set. iii. Parallel lines do not intersect each other. F = { } It is an empty set. iv. z is a natural number. There is no natural

number between 3 and 4. G = { } It is an empty set. v. A triangle is a three-sided figure. H = { } It is an empty set.

Page 8: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Chapter 01: Sets

5

3. Classify the following sets into finite or infinite:

i. A = {1, 3, 5, 7, …} ii. B = {101, 102, 103, … , 1000} iii. C = {x|x Q, 3 < x < 5} iv. D = {y|y = 3n, n N} Solution: i. Here, counting of elements do not terminate at

any stage. A is an infinite set. ii. Here, counting of elements terminate at 1000. B is a finite set. iii. There is infinite number of rational elements

between 3 and 5. C is an infinite set. iv. Here, counting of elements do not terminate at

any stage. D is an infinite set. 4. Let G = {x|x is a boy of your class} and

H = {y|y is a girl of your class}. What type of sets G and H are?

Solution: Set G and set H are finite sets. 1.5 Subset If every element of set Y is an element of set X, then Y is said to be subset of set X. Symbolically, it is represented as Y X If we have say ‘a’, an element which belongs to set Y, we can say that, it (‘a’) also belongs to set X. But if a Y and a X then it is said that set Y is not a subset of X or Y X. Example: If Y = {b, z} and X = {b, l, z} then we say that Y X. If Y is a subset of X and set X contains atleast one element which is not in set Y, then set Y is the proper subset of set X. It is denoted as Y X. Set X is said to be the ‘super set’ of set Y and is denoted as X Y. If X = {a, b} and Y = {b, a}, then set X is a subset of set Y and Y is also subset of set X. In this case set X is the improper subset of the set Y. It is denoted as X Y and it is read as “X is an improper subset of Y.” Also set Y is the improper subset of the set X.

It is denoted as Y X and it is read as “Y is an improper subset of X.” Note: i. Every set is a subset of itself i.e. Y Y. ii. Empty set is a subset of every set i.e., X. 1.6 Universal Set A suitably chosen non-empty set of which all the sets under consideration are the subsets of that set is called the Universal set. It is denoted by ‘U’. Example: A = {x|x is Physics laboratory in your school} B = {y|y is Chemistry laboratory in your school} C = {z|z is Biology laboratory in your school} U = {l|l is laboratories in your school} It can be seen that A U, B U, C U. Set U is the universal set of sets A, B and C. Note: Universal set is a set that cannot be changed

once fixed for a particular solution. In Venn diagram, generally universal set is represented by a rectangle. Exercise 1.3 1. Observe the following sets and answer the

questions given below: A = The set of all residents in Mumbai B = The set of all residents in Bhopal C = The set of all residents in Maharashtra D = The set of all residents in India E = The set of all residents in Madhya Pradesh i. Write the subset relation between the

sets A and C. ii. Write the subset relation between the

sets E and D. iii. Which set can be chosen suitably as

the universal set? Solution: i. All residents of Mumbai are residents of

Maharashtra. A C ii. All residents of Madhya Pradesh are residents

of India. E D iii. Mumbai, Maharashtra, Bhopal, Madhya

Pradesh are parts of India. Set D can be chosen as the universal set.

Page 9: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Std. IX : Algebra

6

2. Let A = {a, b, c}, B = {a}, C = {a, b}, then i. Which sets given above are the proper

subsets of the set A? ii. Which set is the super set of set C? Solution: i. Elements of set B and set C are the elements

of set A. Also, there exists an element viz. c which is not an element of set B and set C but is in set A.

Set B and set C are the proper subsets of set A.

ii. Set A is the super set of set C i.e. A C. 3. Draw a Venn diagram, showing sub set

relations of the following sets: A = {2, 4} B = {x|x = 2n, n < 5, n N} C = {x|x is an even natural number 16} Solution: A = {2, 4} B = {2, 4, 8, 16} C = {2, 4, 6, 8, 10, 12, 14, 16} A B C 4. Prove that, if A B and B C, then A C. (Hint: Start with an arbitrary element

x A and show that x C) Solution: Let us assume that x A ….(i) But, A B x B

B C

x C ….(ii) From (i) and (ii), A C 5. If X = {1, 2, 3}, write all possible subsets of X. Solution: All possible subsets of X are as follows: i. { } or .…[a null set is a subset of every set] ii. {1} iii. {2} iv. {3}

v. {1, 2} vi. {1, 3} vii. {2, 3} viii. {1, 2, 3} .…[every set is a subset of itself] 1.7 Operations on Sets a. Equality: If A is a subset of B and B is a subset of A,

then A and B are said to be equal sets and are denoted by A = B.

Both the sets A and B contain exactly the same elements.

If the elements of A and B are not same, then we write A B.

Note: To prove that sets A and B are equal, it is always necessary to prove that A B and B A.

i. Let A = {x|x = 2n, n N and x < 10} and B = {2, 4, 6, 8} A = {2, 4, 6, 8} A B and B A A = B ii. Let P = {x|x is an odd natural number,

x < 8} and Q = {y|y is an even natural number,

y <10} In roster form, P = {1, 3, 5, 7} Q = {2, 4, 6, 8} P Q and Q P P Q b. Intersection of Sets: If A and B are two sets then a set of common

elements in A and B is called intersection of set A and B. It is represented as A B and is read as ‘A intersection B’.

Example: Let A = {1, 3, 5, 7, 9} B = {3, 9, 12} A B = {3, 9} or A B = {x|x A and x B} Shaded part in the Venn diagram represents

intersection of sets A and B.

A B

.1 .3

.5

.7

.12 .9

A B

A B .12

.8

.6

.2

.4

.16

.14

.10

C

Page 10: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Chapter 01: Sets

7

Properties of Intersection of Sets: i. A B = B A [Commutative property] ii. A (B C) = (A B) C

[Associative property] iii. A B A; A B B iv. A P; B P then A B P v. If A B then A B = A If B A then A B = B vi. A = and A A = A c. Disjoint Sets: Let A = {2, 4, 6, 8} and B = {1, 3, 5, 7} A B = If there are no common elements in two sets,

then such sets are called disjoint sets. The Venn diagram represents two disjoint sets

A and B. A B = For disjoint sets i. A B = ii. A B iii. B A d. Union of Sets: If A and B are two sets then a set containing

all the elements of A and B together is called union of sets A and B.

Union of two sets A and B is denoted as ‘A B’ and is read as ‘A union B’. Let A = {1, 2, 3, 4, 5} and B = {3, 5, 7, 9} A B = {1, 2, 3, 4, 5, 7, 9} or A B = {x|x A or x B} The shaded portion in the Venn diagram

represents A B. Properties of Union of sets: i. A B = B A [Commutative property] ii. A (B C) = (A B) C [Associative property] iii. A A B and B A B

iv. If A B, then A B = B and if B A, then A B = A

v. A = A vi. A A = A Distributive Property: i. A (B C) = (A B) (A C) ii. A (B C) = (A B) (A C) e. Complement of a Set: If U is a universal set and set A is a subset of

the universal set, then set of all elements in U which are not in set A is called the complement of set A.

It is denoted by A or Ac. Let U = {x|x is a natural number, x 9} and A = {1, 3, 5, 7} In roster form, U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = { 2, 4, 6, 8, 9} or A = {x|x U and x A} In Venn diagram, complement of set A is

given as: Note: i. A A = ii. A A = U Properties of Complement of a Set: i. (A) = A ii. = U iii. U = iv. If A B, then B A v. A A = vi. A A = U If A and B are any two sets, then i. (A B) = A B and ii. (A B) = A B Exercise 1.4 1. Let P = {x|x is a letter in the word

‘CATARACT’} and Q = {y|y is a letter in the word ‘TRAC’}. Show that P = Q.

Solution: Roster form of set P and set Q is as follows: P = {C, A, T, R} Q = {T, R, A, C} Set P and set Q are subset of each other. Also, the elements of set P and set Q are same. P = Q

A B

.2 .4

.6 .8

.1 .3

.5 .7

.1 .3

.5 .7

.2

.6

.4

.8

.9

A A

U

A B

.1 A

.2

.4

.3 .7

.9

B

.5

Page 11: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Std. IX : Algebra

8

2. Find the union of each of the following pairs of sets:

i. A = {2, 3, 5, 6, 7}, B = {4, 5, 7, 8} ii. C = {a, e, i, o, u}, D = {a, b, c, d} iii. E = {x|x N and x is a divisor of 12} F = {y|y N and y is a divisor of 18} Solution: i. A = {2, 3, 5, 6, 7}, B = {4, 5, 7, 8} A B = {2, 3, 4, 5, 6, 7, 8} ii. C = {a, e, i, o, u}, D = {a, b, c, d} C D = {a, b, c, d, e, i, o, u} iii. The Roster form of set E and set F is as

follows: E = {1, 2, 3, 4, 6, 12} F = {1, 2, 3, 6, 9, 18} E F = {1, 2, 3, 4, 6, 9, 12, 18} 3. Find the intersection of each of the

following pairs of sets: i. A = {1, 2, 4, 5, 7}, B = {2, 3, 4, 8} ii. C = {x|x N, 5 < x 10} D = {y|y W, 5 y < 10} iii. E = {x|x I, x < 0}, F = {y|y I, y > 0} Solution: i. A = {1, 2, 4, 5, 7}, B = {2, 3, 4, 8} A B = {2, 4} ii. The Roster form of set C and set D is as

follows: C = {6, 7, 8, 9, 10} D = {5, 6, 7, 8, 9} C D = {6, 7, 8, 9} iii. The Roster form of set E and set F is as

follows: E = {… , 4, 3, 2, 1} F = {1, 2, 3, 4, …} E F = { } or 4. Let U = {x|x = 2n, n W, n < 8} be the

universal set. A = {y|y = 4n, n W, n < 4}; B = {z|z = 8n, n W, n 2}. Then find:

i. A ii. B iii. (A B) iv. (A B) Solution: The Roster form of set U, set A and set B is as

follows: U = {20, 21, 22, 23, 24, 25, 26, 27} = {1, 2, 4, 8, 16, 32, 64, 128}

A = {40, 41, 42, 43} = {1, 4, 16, 64} B = {80, 81, 82} = {1, 8, 64} i. A = {2, 8, 32, 128} ii. B = {2, 4, 16, 32, 128} iii. A B = {1, 4, 8, 16, 64} (A B) = {2, 32, 128} iv. (A B) = {1, 64} (A B) = {2, 4, 8, 16, 32, 128} 5. Let A = {a|a is a letter in the word ‘college’}

and B = {b|b is a letter in the word ‘luggage’} and U = {a, b, c, d, e, f, g, l, o, u}. Verify: i. (A B) = A B ii. (A B) = A B Proof: i. In roster form, set A and set B can be written

as: A = {c, o, l, e, g} B = {l, u, g, a, e} U = {a, b, c, d, e, f, g, l, o, u} A = {a, b, d, f, u} B = {b, c, d, f, o} A B = {a, c, e, g, l, u, o} L.H.S. = (A B) = {b, d, f} .… (i) R.H.S. = A B = {b, d, f} .… (ii) From (i) and (ii), we get L.H.S. = R.H.S. (A B) = A B ii. A B = {l, g, e} L.H.S. = (A B) = {a, b, c, d, f, o, u} .… (iii) R.H.S. = A B = {a, b, c, d, f, o, u} …. (iv) From (iii) and (iv), we get L.H.S. = R.H.S. (A B) = A B 1.8 Number of Elements in the Set If A is any set then the number of elements in set A is denoted by n (A). Illustrations: i. Let A = {x|x N, 7 < x 12} A = {8, 9, 10, 11, 12} n(A) = 5 ii. For an empty set, n() = 0 iii. n(A B) = n(A) + n(B) n(A B) To verify this identity, let us consider the following example, A = {2, 3, 4} and B = {3, 4, 5, 6} A B = {2, 3, 4, 5, 6} and A B = {3, 4}

Page 12: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Chapter 01: Sets

9

n(A) = 3, n(B) = 4, n(A B) = 5 and n(A B) = 2

L.H.S. = n(A B) = 5 .... (i) R.H.S. = n(A) + n(B) n(A B) = 3 + 4 2 = 5 .... (ii) n(A B) = n(A) + n(B) n(A B) .... [From (i) and (ii)] Exercise 1.5 1. Let A = {1, 3, 5, 6, 7}, B = {4, 6, 7, 9}, then

verify the following: n (A B) = n(A) + n(B) n(A B) Proof: A = {1, 3, 5, 6, 7} and B = {4, 6, 7, 9} A B = {1, 3, 4, 5, 6, 7, 9} and A B = {6, 7} n(A) = 5, n(B) = 4, n(A B) = 7 and

n(A B) = 2 L.H.S. = n(A B) = 7 ....(i) R.H.S. = n(A) + n(B) n(A B) = 5 + 4 2 = 7 ....(ii) L.H.S. = R.H.S. ....[From (i) and (ii)] n(A B) = n(A) + n(B) n(A B) 2. Let A and B be two sets such that n(A) = 5,

n(A B) = 9, n(A B) = 2. Find n(B). Solution: Given, n(A) = 5, n(A B) = 9, n(A B) = 2 n(B) = ? By using identity, n(A B) = n(A) + n(B) n(A B) 9 = 5 + n(B) 2 9 – 5 + 2 = n(B) n(B) = 6 3. In a school hostel, there are 100 students,

out of which 60 drink tea, 50 drink coffee and 30 drink both tea and coffee. Find the number of students who do not drink tea or coffee.

Solution: Let U be the universal set of students in hostel, T be the set of students who drink tea and C be the set of students who drink coffee. n(U) = 100, n(T) = 60, n(C) = 50, n(T C) = 30 By using the identity, n(T C) = n(T) + n(C) n(T C) = 60 + 50 30 = 110 30

n(T C) = 80 There are 80 students who either drink tea or

coffee or both. But there are 100 students in the hostel.

Number of students who neither drink tea nor coffee = n(U) n(T C) = 100 80 = 20

Students who do not drink tea or coffee is 20. 4. 110 children choose their favourite colour

from blue and pink. Every student has to choose at least one of the colours. 60 children choose blue colour, while 70 children choose pink colour. How many children choose both the colours as their favourite colour?

Solution: Let the number of children who choose blue

colour be n(B) and number of children who choose pink colour be n(P).

n(B) = 60 and n(P) = 70 Number of children who choose their

favourite colour from blue or pink. n(B P) = 110 By using the identity, n(B P) = n(B) + n(P) n(B P) 110 = 60 + 70 n(B P) n(B P) = 60 + 70 110 n(B P) = 20 The number of students who choose both

the colours as their favourite colours is 20. 5. Observe the figure and verify the following

equation: n(A B C) = n(A) + n(B) + n(C) n(A B)

n(B C) n(C A) + n(A B C) Proof: L.H.S. = n(A B C) A B C = {1, 2, 3, 4, 5, 6, 7, 8, 9} n(A B C) = 9 …. (i) Now, A = {1, 2, 3, 4, 5} n(A) = 5 B = {2, 3, 6, 7, 8} n(B) = 5 C = {3, 4, 6, 9} n(C) = 4 A B = {2, 3} n(A B) = 2

.1

.5

.2

.3 .4

.9

.6

.7

.8 BA

C

Page 13: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Std. IX : Algebra

10

B C = {3, 6} n(B C) = 2

C A = {3, 4} n(C A) = 2

A B C = {3} n(A B C) = 1

R.H.S.

= n(A) + n(B) + n(C) n(A B) n(B C)

n(C A) + n(A B C)

= 5 + 5 + 4 2 2 2 + 1

= 9 …. (ii)

L.H.S. = R.H.S. ....[From (i) and (ii)]

n(A B C) = n(A) + n(B) + n(C)

n(A B) n(B C) n(C A)

+ n(A B C) Problem Set - 1 1. Which of the following collections are sets? i. The collection of rich people in your

district. ii. The collection of natural numbers less

than 50. iii. The collection of most talented

persons of India. iv. The collection of first ten prime integers. v. The collection of all days in a week

starting with the letter ‘T’. vi. The collection of some months in a year. vii. The collection of all books in your

school library. viii. The collection of smart boys in your

class. ix. The collection of multiples of 7. x. The collection of students in your

class who got a lot of marks in the first unit test.

Solution: From the given collections (ii), (iv), (v), (vii) and (ix) are sets. Remaining collections are not considered as sets as they have relative terms and their meaning may vary from person to person. 2. Write the following sets in roster form: i. A = {x|x I, x W} ii. B = {x|x is two digit number such that

the product of its digits is a multiple of ten}

iii. C = {x|x is a prime divisor of 120} iv. D = {x|x I and x2 < 10}

v. E = 2

n,2 n 4,n N

n 1

x x

Solution:

i. A = {… , 3, 2, 1} ii. B = {25, 45, 52, 54, 56, 58, 65, 85} iii. C = {2, 3, 5}

iv. D = {3, 2, 1, 0, 1, 2, 3}

v. Since, 2 n 4

n = 2, 3, 4

for n = 2, 2

n

n 1 =

2

2

(2) 1=

2

3

for n = 3, 2

n

n 1=

2

3

(3) 1=

3

8

for n = 4, 2

n

n 1 =

2

4

(4) 1 =

4

15

E = 2 3 4

, ,3 8 15

3. Write the following sets in the set builder

form: i. F = {I, N, D, A}

ii. G = {1, 1} iii. H = {3, 9, 27, 81, 243} iv. J = {15, 24, 33, 42, 51, 60}

v. K =

1 2 3 4 5, , , ,

2 5 10 17 26

Solution: i. F = {x|x is a letter in the word ‘INDIA’} ii. G = {y| square of y is 1} or G = {y|y is a square root of 1}

iii. H = {a|a = 3n, n N, n 5} iv. J = {b|b is a two digit number whose sum of

digits is 6}

v. When n = 1, c = 2

1 1

2(1) 1

When n = 2, c = 2

2 2

5(2) 1

When n = 3, c = 2

3 3

10(3) 1

When n = 4, c = 2

4 4

17(4) 1

When n = 5, c = 2

5 5

26(5) 1

K = 2

nc c = , n N, n 5

n +1

Page 14: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Chapter 01: Sets

11

4. Classify the following sets as ‘singleton’ or ‘empty’:

i. A = {x|x is a negative natural number} ii. B = {y|y is an odd prime number < 4} iii. C = {z|z is a natural number, 5 < z < 7} iv. D = {d|d N, d2 0} Solution: i. Each natural number is positive. A = { } It is an empty set. ii. B = {3} It is a singleton set. iii. C = {6} It is a singleton set. iv. There is no natural number whose square is

less than or equal to zero. D = { } It is an empty set. 5. Classify the following sets as ‘finite’ or

‘infinite’: i. A = {x|x is a multiple of 3} ii. B = {y|y is a factor of 13} iii. C = {…, 3, 2, 1, 0} iv. D = {x|x = 2n, n N} Solution: i. A = {3, 6, 9, 12, …} It is an infinite set. ii. B = {1, 13} It is a finite set. iii. C is an infinite set. iv. D = {20, 21, 22, 23, 24,...} = {2, 4, 8, 16, 32, …} D is an infinite set. 6. State which of the following sets are equal. i. N = {1, 2, 3, 4, …} ii. W = {0, 1, 2, 3, …} iii. A = {x|x = 2n, n W} iv. B = W {0} Solution: i. N ={1, 2, 3, 4, …} ii. W = {0, 1, 2, 3, …} iii. A = {20, 21, 22, 23, …} = {1, 2, 4, 8, …} iv. B = W {0} = {1, 2, 3, 4, …} Here, set N and set B are subset of each other. In set N and set B, the elements are the same. N = B

7. Let A = {7, 5, 2} and B = 3 125, 4, 49 .

Are the sets A and B equal? Justify your answer.

Solution: A = {7, 5, 2},

B = 3 125, 4, 49

B = {5, 2, 2, 7, 7} Here, A is a subset of B, but B is not a subset

of A. Elements of set A and set B are not equal. A B 8. If A = {1, 2, 3, 4}, B = {2, 4, 6, 8},

C = {3, 4, 5, 6} and U = {x|x N, x < 10}. Verify the following properties:

i. A (B C) = (A B) C ii. A (B C) = (A B) (A C) iii. A (C) = (A B) (A C) iv. (A B) = A B v. (A B) = A B vi. (A) = A Solution: Roster form of set U is as follows: 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. B C = {2, 3, 4, 5, 6, 8} A B = {1, 2, 3, 4, 6, 8} L.H.S. = A (B C) = {1, 2, 3, 4, 5, 6, 8} …. (i) R.H.S. = (A B) C = {1, 2, 3, 4, 5, 6, 8} …. (ii) From (i) and (ii), we get L.H.S. = R.H.S. A (B C) = (A B) C ii. B C = {4, 6} A B = {1, 2, 3, 4, 6, 8} A C = {1, 2, 3, 4, 5, 6} L.H.S. = A (B C) = {1, 2, 3, 4, 6} …. (i) R.H.S. = (A B) (A C) = {1, 2, 3, 4, 6} …. (ii) From (i) and (ii), we get L.H.S. = R.H.S. A (B C) = (A B) (A C) iii. B C = {2, 3, 4, 5, 6, 8} L.H.S. = A (B C) = {2, 3, 4} …. (i)

Page 15: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Std. IX : Algebra

12

A B = {2, 4} A C = {3, 4} R.H.S. = (A B) (A C) = {2, 3, 4} .… (ii) From (i) and (ii), we get L.H.S. = R.H.S. A (C) = (A B) (A C) iv. A B = {1, 2, 3, 4, 6, 8} (A B) = {5, 7, 9} A = {1, 2, 3, 4} A = {5, 6, 7, 8, 9} B = {2, 4, 6, 8} B = {1, 3, 5, 7, 9} L.H.S. = (A B) = {5, 7, 9} …. (i) R.H.S. = A B = {5, 7, 9} …. (ii) From (i) and (ii), we get L.H.S. = R.H.S. (A B) = A B v. A B = {2, 4} (A B) = {1, 3, 5, 6, 7, 8, 9} A = {5, 6, 7, 8, 9} B = {1, 3, 5, 7, 9} L.H.S. = (A B) = {1, 3, 5, 6, 7, 8, 9} …. (i) R.H.S. = A B = {1, 3, 5, 6, 7, 8, 9} …. (ii) From (i) and (ii), we get L.H.S. = R.H.S. (A B) = A B vi. A = {5, 6, 7, 8, 9} L.H.S. = (A) = {1, 2, 3, 4} …. (i) R.H.S. = A = {1, 2, 3, 4} …. (ii) From (i) and (ii), we get L.H.S. = R.H.S. (A) = A 9. For each of the following sets, state with

reasons, whether it is a null set or not: i. A = {x|x I, x2 is not positive} ii. B = {b|b N, 2b + 1 is even} iii. C = {c|c N, c is odd and c2 is even} Solution: i. Whether a number is positive or negative its

square is always a positive. Square of an integer cannot be negative,

except zero, whose square is neither positive nor negative.

A = {0} Set A is not a null set. ii. Here, 2b is an even number and 1 is an odd

number. Since, addition of even and odd number

always give odd number, value of 2b + 1, for any value of b N is an odd number. Set B is a null set. iii. Square of an odd number is always an odd

number. c2 cannot be even. Set C is a null set. 10. Give an example of the set which can be

written in set builder form but cannot be written in roster form.

Solution: Consider the set of rational numbers ‘Q’. In set builder form, it is

Q = a

a I, b Iand b 0b

But same set Q cannot be written in roster form. 11. Write down all possible subsets of each of

the following sets: i. ii. A = {1} iii. B = {1, 2} iv. C = {a, b, c, d} Solution: i. Subset of a null set is only one i.e. ii. Subsets of set A are empty set { } and set A

itself. i.e. and {1} iii. All possible subsets of set B are , {1}, {2}, {1, 2}. iv. All possible subsets of set C are {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d},

{b, c}, {b, d}, {c, d}, {a, b, c}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c, d}

12. Write proper subsets of the following sets: i. A = {a, b} ii. B = {a, b, c} Solution: i. Proper subsets of A are {a} and {b}. ii. Proper subsets of B are {a}, {b}, {c}, {a, b},

{b, c}, {c, a}.

Page 16: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Chapter 01: Sets

13

13. Write the sets A and B such that A is finite, B is finite, A and B are disjoint sets.

Solution: Let B = {1, 3, 5, 7}, A = {2, 4, 6, 8} be the

finite sets. But their intersection is a null set, i.e. A B = A and B are disjoint sets. 14. Let A = {a, b, c, d}, B = {a, b, c},

C = {b, d, e}, then find the sets D and E satisfying the following conditions:

i. D A, D B ii. C E, B E = Solution: i. Since, D A and D B D = {a, b}/{a, c}/{a, d}/{a, b, d}/{b, c, d}

/{a, c, d} ii. Since, C E and B E = E must not contain any element of set B. C is

superset of set E. E = {d}/{e}/{d, e} 15. Let U = {x|x N, x < 10}, A = {a|a is even, a U}, B = {b|b is a factor of 6, b U}. Verify that: n(A) + n(B) = n(A B) + n(A B). Solution: Roster form of set U and set A is as follows: U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {2, 4, 6, 8} ….(Even numbers within

the universal set) n(A) = 4 Roster form of set B is as follows:

B = {1, 2, 3, 6} ….( b U)

n(B) = 4 A B = {1, 2, 3, 4, 6, 8} n(A B) = 6 A B = {2, 6} n(A B) = 2 L.H.S. = n(A) + n(B) = 4 + 4 = 8 ....(i) R.H.S. = n(A B) + n(A B) = 6 + 2 = 8 ....(ii) From (i) and (ii), we get L.H.S. = R.H.S. n(A) + n(B) = n(A B) + n(A B)

16. In a group of students, 50 students passed in English, 60 students passed in Mathematics and 40 students passed in both. Find the number of students who passed either in English or in Mathematics.

Solution: Let the number of students who passed in

English be denoted by set E. n(E) = 50 Let the number of students who passed in

Mathematics be denoted by set M. n(M) = 60 The number of students who passed in English

and Mathematics = n(M E) = 40 By using identity, n(E M) = n(E) + n(M) n(E M) = 50 + 60 – 40 = 110 40 = 70 The number of students who passed either

in English or in Mathematics is 70. 17. A T.V. survey says 136 students watch only

programme P1, 107 watch only programme P2, 27 watch only programme P3. 25 students watch P1 and P2 but not P3. 37 watch P2 and P3 but not P1. 53 students watch P1 and P3 but not P2. 40 students watch all three programmes and 80 students do not watch any programme. Find, with the help of Venn diagram.

i. Number of P1 viewers. ii. Number of P2 or P3 viewers. iii. Total number of viewers surveyed. Solution: Given data represented by Venn diagram is as

follows: i. From the Venn diagram, Number of P1 viewers = n(P1) = 136 + 25 + 53 + 40 = 254 Number of P1 viewers is 254.

P2 P1

P3

27

136

53

25 107

37 40

80

U

Page 17: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Std. IX : Algebra

14

ii. Number of P2 or P3 viewers = n(P2 P3) = 107 + 25 + 40 + 37 + 53 + 27 = 289 Number of P2 or P3 viewers is 289. iii. Total number of viewers surveyed = Number of only P1 viewers + Number of

P2 or P3 viewers + 80 = 136 + 289 + 80 = 505 Total number of viewers surveyed is 505. 18. Show that, it is impossible to have sets A

and B such that set A has 32 elements, set B has 42 elements, A B has 12 elements and A B has 64 elements.

Solution: Set A has 32 elements. n(A) = 32 Set B has 42 elements. n(B) = 42 Given that n(A B) = 12 and n(A B) = 64 By using the identity, n(A B) = n(A) + n(B) n(A B), L.H.S. = n(A B) = 64 ....(i) R.H.S. = n(A) + n(B) n(A B) = 32 + 42 12 = 62 ....(ii) From (i) and (ii), we get L.H.S. R.H.S. Sets A and B are impossible. 19. Let the universal set U be a set of all

students of your school. A is the set of boys, B is the set of girls and C is the set of students participating in sports. Describe the following sets in words and represent them by a Venn diagram:

i. B C ii. A (B C) Solution: i. B C represents the set of girls participating

in sports. ii. A (B C) represents the set of all boys or

set of girls that participate in sports.

20. Represent sets A, B, C such that A B, A C = and B C by Venn diagram and shade the portion representing A (B C).

Solution: i. A B set A is proper subset of set B. i.e. set A is inside set B. ii. A C = set A and set C do not intersect. iii. B C set B and set C intersect. 21. Let A, B, C be sets such that A B ,

B C and A C . Do you claim that A B C ? Justify your answer.

Solution: There are two possibilities: i. If A = {a, b} B = {b, c} C = {c, a} then, it is observed that A B , B C , C A , but, A B C = ii. If A = {a, b} B = {a, c} C = {a, b, c} then, it is observed that A B , B C , C A , but, A B C = {a} We cannot claim that A B C . 22. With the help of suitable example, verify

the following statements: If A B, B C, then A C. Solution: Let A = {x, y, z}, B = {a, x, y}, C = {y, w} Since, each element of set A does not exist in

set B. A B Each element of set B does not exist in set C. B C Each element of set A does not exist in set C. A C

A

B C

A (B C)

B C

A C B

B C

A C B

A (B C)

B C

Page 18: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Chapter 01: Sets

15

23. If A and B are any two sets, then prove that i. (A B) = A B ii. (A B) = A B [Hint: Show (A B) A B and vice

versa] Solution: Let U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 3, 4, 5} B = {3, 4, 5, 6, 7} i. A B = {1, 2, 3, 4, 5, 6, 7} L.H.S. = (A B) = {8} …. (i) A = {6, 7, 8} B = {1, 2, 8} R.H.S. = A B = {8} …. (ii) From (i) and (ii), we get L.H.S. = R.H.S. (A B) = A B ii. Now, A B = {3, 4, 5} L.H.S. = (A B) = {1, 2, 6, 7, 8} …. (i) R.H.S. = A B = {1, 2, 6, 7, 8} …. (ii) From (i) and (ii), we get L.H.S. = R.H.S. (A B) = A B One-Mark Questions 1. Write the following set in set builder form. A = {2, 3, 5, 7, 11, 13, 17} Solution: The set builder r form of set B is: A = {x|x is a prime number, x < 18} 2. Write the following set in roster form. B = {x|x is a natural number and 4 x < 10Solution: The Roster form of set B is: B = {4, 5, 6, 7, 8, 9} 3. If A = {1, 2, 3, 4, 5, 6} and B = {1, 3, 5, 7}

then draw Venn diagram for A B. Solution:

4. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} is the universal set and C = {5, 6, 7, 8} then find C.

Solution: Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and C = {5, 6, 7, 8} C = {1, 2, 3, 4, 9} 5. If A = {9, 11, 13, 15} and B = {1, 3, 5, 7}

then find A B. Solution: Given, A = {9, 11, 13, 15} and B = {1, 3, 5, 7} A B = { } or 6. If A and B are two sets such that n(B) = 8,

n(A B) = 11, n(A B) = 6, find n(A). Solution: By using identify, n(A B) = n(A) + n(B) n(A B) 11 = n(A) + 8 6 n(A) = 11 2 n(A) = 9 7. State which of the following sets are equal:

A = {x|x W, x < 6} B = {1, 2, 3, 4, 5, 6} C = {0, 1, 2, 3, 4, 5}

Solution: Here, A = {0, 1, 2, 3, 4, 5} and C = {0, 1, 2, 3, 4, 5}

A = C 8. Classify the following sets as singleton or

empty. i. A = {x|x is a natural number, x 5

and x 7} ii. B = {x|x is an even prime number}

Solution: i. There is no common number for x < 5

and x > 7 A = {} Set A is empty set. ii B = {2} Set B is singleton set. 9. If A = {2, 3, 4, 5} and B = {1, 2, 5, 6}, then

find A B. Solution: Given, A = {2, 3, 4, 5} and B = {1, 2, 5, 6} A B = {1, 2, 3, 4, 5, 6}

A B

71

53

246

A B

U

Page 19: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Std. IX : Algebra

16

10. If A = {3}, write all possible subsets of set A. Solution: and {3} 11. If U = {1, 2, 3, 4} and X = {2, 4}, then find X. Solution: Given, U = {1, 2, 3, 4} and X = {2, 4} X = {1, 3} Additional Problems for Practice Based on Exercise 1.1  1. Write the following sets in the roster form: i. A = {x|x is a prime number which is a

divisor of 30} ii. B = {x|x is an even natural number} iii. C = {x|x is an integer and x2 < 5} iv. F = {x|x is a letter in the word ‘LITTLE’} v. E = {x|x W, x N} vi. D = {x|x is a square root of 81} 2. Write the following sets in the set builder form: i. A = {2, 4, 6, 8, 10, 12, 14} ii. B = {5, 10, 15, 20, ….} iii. C = {7, 72, 73, 74} iv. D = {51, 53, 55, 57, 59} v. E = {2, 3, 5, 7, 11, 13, 17, 19} Based on Exercise 1.2  1. State which of the following sets are singleton

or empty sets: i. A = {x|x 5 = 0} ii. B = {y|y is an even prime number greater

than 2} iii. D = {x|x N and 3x 1 = 0} iv. E = {x|x I, x is neither a positive nor a

negative number} v. C = {x|x N and x < 7 and x > 11} 2. Classify the following sets into finite or infinite: i. A = {x|x is a multiple of 1} ii. C = {x|x is a point on a line} iii. D = {1, 2, 3, 4, …., 100} iv. E = {x|x N and x is an odd number} Based on Exercise 1.3  1. Write the subset relations among the following

sets: P = set of all residents in Nagpur X = set of all residents in Vadodara Y = set of all residents in Maharashtra T = set of all residents in Gujarat

2. Draw a Venn diagram showing subset relations of the following sets:

A = {2 , 8} B = {x|x = 2n, n 4 and n N} C = {x|x is an even natural number 20} 3. If A = {x, y}, write all possible subsets of A. 4. State true or false: i. is a subset of itself. ii. If A B and B A, then A = B. iii. The empty set is a subset of all sets. Based on Exercise 1.4  1. Find the union of each of the following pairs

of sets: i. A = {5, 15, 25}, B = {10, 20, 30} ii. H = {3, 6, 9, 12, 15} , F = {3, 4, 5, 6} iii. M = {x|x N and x is a divisor of 12} N = {x|x N and x is a prime divisor of

12} 2. Find the Intersection of the following pairs of

sets: i. A = {5, 6, 7}, B = {8, 9, 10} ii. M = {10, 20, 30, 40, 50}, N = {20, 40, 60} iii. N is a set of natural numbers and W is a

set of whole numbers. iv. P = {a, b, p, d, q}, R = {q, r, s, p} 3. If U = {x|x is a natural number less than 15} is

a universal set A = {1, 3, 4, 5, 9}, B = {3, 5, 7, 9, 12} Verify that (A B) = A B 4. U = {x|x I and 3 x 3}, A = {2, 0, 2}, B = {0, 1, 2, 3} Find i. A ii. B iii. (A B) iv. A B Based on Exercise 1.5  1. With the help of following figure, write the

following sets: i. A ii. B iii. U iv. A B v. A B

A B

7

105

1

49

U

311

6

28

12

Page 20: Algebra Std Ix Maharashtra Board

Publications Pvt. Ltd. Target Chapter 01: Sets

17

2. Let A and B be two sets such that n(A) = 17,

n(B) = 23, n(A B) = 38. Find n(A B) 3. 240 students in a school were interviewed and

their hobbies were noted. 150 students were interested in stamp collection. 80 took delight in reading books, 40 of them do not like either. What is the number of students who liked both stamp collection and reading books?

4. In a class of 50 students, 35 like Physics, 30

like Mathematics and 3 like neither. How many like both the subjects and how many like Physics only?

5. From the above diagram find:

i. A B ii. n (A B)

iii. (A B) iv. n (A B)

v. A B Answers to additional problems for practice Based on Exercise 1.1  1. i. A = {2, 3, 5} ii. B = {2, 4, 6, 8, ….}

iii. C = {2, 1, 0, 1, 2} iv. F = {L, I, T, E} v. E = {0}

vi. D = {9, 9} 2. i. A = {x|x = 2n, n N and n < 8}

ii. B = {x|x = 5n and n N}

iii. C = {x|x = 7n, 1 n 4}

iv. D = {x|x N, x is an odd integer and 50 < x < 60}

v. E = {x|x is a prime number and 1 < x < 20} Based on Exercise 1.2 1. Singleton sets are A, E Empty sets are B, D, C 2. Finite set is D. Infinite sets are A, C, E.

Based on Exercise 1.3 1. P Y, X T 2. A = {2, 8}, B = {2, 4, 8, 16} C = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}

3. , {x}, {y}, {x, y} 4. i. True ii. True iii. True Based on Exercise 1.4 1. i. A B = {5, 10, 15, 20, 25, 30}

ii. H F = {3, 4, 5, 6, 9, 12, 15}

iii. M N = {1, 2, 3, 4, 6, 12} 2. i. A B =

ii. M N = {20, 40}

iii. N W = {1, 2, 3, ….}

iv. P R = {p, q} 4. i. A = {3, 1, 1, 3}

ii. B = {3, 2, 1}

iii. (A B) = {3, 1}

iv. A B = {3, 1} Based on Exercise 1.5 1. i. A = {2, 3, 6, 7, 8, 11, 12}

ii. B = {1, 3, 4, 6, 9, 11, 12} iii. U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

iv. A B = {1, 2, 4, 5, 7, 8, 9, 10}

v. A B = {5, 10} 2. 2 3. 30 4. 18 students like both subjects and 17 students

like only Physics. 5. i. A B = {6, 12}

ii. n(A B) = 7

iii. (A B) = {3, 9, 15, 18, 24}

iv. n(A B) = 2

v. A B = {3, 9, 15, 18, 24}

A B

18126

3

159

U

24

A

B

2

C

8

10

12

20 1418

6

4 16