The Leading Maths - 9

Allied Publication Pvt. Ltd. Kathmandu, Nepal Phone : 01-5388827 The Leading Allied For Class 9 Approved by Curriculum Development Center, GON Sanothimi, Bhaktapur, Nepal Author Ashok Dangol M.Ed. Maths (TU) Mathematics

Publisher Allied Publication Pvt. Ltd Kathmandu, Nepal Ph.: 01-5388827, 5378629 Email : [email protected] Author Ashok Dangol Copyright Author Edition 1st 2080 ICT Supporter Ashok Dangol Language Editor Nariswor Gautam Graphics Sunil Shrestha Printing Quantity 5000 Special Thanks Dileep Maharjan Nabaraj Pathak Jit Bahadur Khanal Binod Kumar Yadav Ashim Maskey Santosh Gupta Sarala Devkota Lalbabu Prasad Yadav Laxmi Prasad Gautam Nabaraj Paudel Nishchal Lama Kamal Nepal Dilkumar Maharjan Sanjay Maharjan Gobinda Prasad Pokharel Rudra Prasad Pokharel Hari Maharjan

This Allied The Leading Mathematics - 9 is basically meant for making the teachers and taught active while teaching and learning mathematics. The contents and extent of the series are strictly contained and arranged in accordance with the new vision and mission of the latest New Curriculum 9-10 of Secondary Level of CDC, Nepal. This series is basically an outcome of my untiring effort and patience. The long and dedicated service in teaching and popularization of mathematics has been a great asset in preparing this series. It has been designed as a textbook for English medium private and government school students with a new approach. This book provides maximum benefit to both teachers and students because of the following unique features: Unique features of this book; Ö Arranged especially focusing on child psychology of teaching and learning mathematics which are based on the Areas of Secondary Curriculum 9-10. Ö Prepared with the firm belief that “Mathematics begins at Home, grows in the Surroundings and takes shape in School (HSS Way)”, and sincere attempts have been made to make the learner, the teacher and reader feel “Mathematics is Fun, Mathematics is Easy and Mathematics is Everywhere (FEE Concept)”. Ö Written the focus of students’ activities and easily perform teaching-learning activities for teachers. Ö Well arranged the four colours of the whole book supports to find easily Themes, Chapters, Lessons, Examples, Practices, and other topics from Content. Ö Every Theme begins with its estimated teaching hours (Theory + Practical), competency, learning outcomes, revision contents for pre-knowledge. Ö Highlighted the important terms, notes and key points. Ö Included sufficiently all types of Classwork Examples and Home Assessments from simple to complex with suitable figures and reasons. Ö Included Project Works and Their Formats as self-practice to the students at home for some days' activities to memories long time about the entire chapters. Ö Included Confidence Level Tests as self-evaluations for Success and Competent by themselves. Ö Available Individual Practical Evaluation Sheets at the end. It is very much hoped that with all the above features, this book will be found really fruitful by teachers and students alike. Thank Allied Publication Pvt. Ltd., Kathmandu, Nepal for taking responsibility for publishing this book, Nariswor Gautam for language editing, and Sunil Shrestha for an attractive design. I would like to extend my sincere gratitude to the persons whose ideas or creations are directly or indirectly incorporated into the text. I would like to extend my thanks to the teachers and the students who helped me to verify the answers and to check the manuscript of this book. Also, many thanks to the schools that applied this book and suggested it to me. Finally, I heartily welcome criticisms, feedbacks and suggestions from readers so that it may appear with revise from in the coming edition and will be gratefully and thankfully acknowledged and honored. Author Preface to First Edition

Units Chapters Lessons Pages I SETS 1. Sets 1.1 Union and Intersection of Sets 12 Classwork Examples 17 Practice - 1.1 20 1.2 Difference and Complement of Sets 24 Classwork Examples 28 Practice - 1.2 30 1.3 Cardinality of Sets 34 Classwork Examples 40 Practice - 1.3 43 Confidence Level Test - I 47 Additional Practice - I 48 II ARITHMETIC 2. Tax 2.1 Income TAX 53 Classwork Examples 57 Practice - 2.1 64 2.2 Value Added TAX (VAT) 68 Classwork Examples 74 Practice - 2.2 78 3. Commission and Dividend 3.1 Commission 82 Classwork Examples 83 Practice - 3.1 88 3.2 Dividend and Bonus 91 Classwork Examples 94 Practice - 3.2 98 4. Home Arithmetic 4.1 Household Expenses for Use of Electricity 103 Classwork Examples 105 Practice - 4.1 109 4.2 Household Expenses for Use of Water 114 Classwork Examples 116 Practice - 4.2 119 4.3 Household Expenses for Use of Telephone 124 Classwork Examples 126 Practice - 4.3 129 4.4 Calculation of Taxi Fare 133 Classwork Examples 134 Practice - 4.4 136 Confidence Level Test - II 140 Additional Practice - II 141 CONTENTS

III MENSURATION 5. Area 5.1 Area of Scalene Triangle 149 Classwork Examples 152 Practice - 5.1 155 5.2 Surface Area of Room and Cost Estimation 160 Classwork Examples 163 Practice - 5.2 166 6. Prism and Cylinder 6.1 Surface Area and Volume of Prism 171 Classwork Examples 173 Practice - 6.1 176 6.2 Surface Area and Volume of Cylinder 179 Classwork Examples 180 Practice - 6.2 186 7. Sphere 7.1 Surface Area and Volume of Sphere 189 Classwork Examples 192 Practice - 7.1 195 Additional Practice - III 199 8.1 Introduction to Sequence and Its General Terms 202 Classwork Examples 205 Practice - 8.1 208 8.2 Introduction to Series 210 Classwork Examples 211 8. Sequence and Series Practice - 8.2 213 8.3 General Term of Arithmetic Sequence 215 Classwork Examples 217 Practice - 8.3 222 8.4 General Term of Geometric Sequence 224 Classwork Examples 226 Practice - 8.4 230 9.1 Factorization in the Form of (a ± b)3 and a3 ± b3 232 Classwork Examples 235 9. Factorization Practice - 9.1 236 9.2 Factorization in Form of (a4 + a2 b2 + b) 239 Classwork Examples 239 Practice - 9.2 242 IV ALGEBRA 10.1 Highest Common Factor (HCF) 244 Classwork Examples 245 10. HCF and LCM Practice - 10.1 247 10.2 Lowest Common Multiple (LCM) 249 Classwork Examples 250 Practice - 10.2 251

11.1 Solving Simultaneous Equations by Substitution Method 253 Classwork Examples 254 Practice - 11.1 257 11.2 Solving Simultaneous Equations by Elimination Method 258 11. Linear Equations Classwork Examples 258 Practice - 11.2 260 11.3 Verbal Problems on Simultaneous Equations in Two Variables 261 Classwork Examples 262 Practice - 11.1 265 12.1 Simplification of Indices By Using Their Laws 268 Classwork Examples 270 12. Indices Practice - 12.1 272 Confidence Level Test - IV 275 Additional Practice - IV 276 IV GEOMETRY 13. Construction 13.1 Angles of Triangle 279 Classwork Examples 284 Practice - 13.1 285 13.2 Properties of Isosceles Triangle 286 Classwork Examples 293 Practice - 13.2 295 13.3 Sides and Angles of a Triangle 298 Classwork Examples 300 Practice - 13.3 301 13.4 Similar Triangles 302 Classwork Examples 307 Practice - 13.4 308 14. Parallelogram 14.1 Properties of Parallelogram 311 Classwork Examples 315 Practice - 14.1 318 14.2 Application of Theorems of Parallelogram 321 Practice - 12.2 322 15. Construction 15.1 Construction of Scalene Quadrilaterals 324 Practice - 15.1 327 15.2 Construction of Trapeziums 328 Practice - 15.2 331 15.3 Construction of Rhombus 332 Practice - 15.3 334 16.1 Circle 335 Classwork Examples 342 16. Circle Practice - 16.1 344 Confidence Level Test - V 347 Additional Practice - V 348

VI STATISTICS AND PROBABILITY 17. Classification and Representation of Data 17.1 Collection of Data and Frequency Table 351 Practice - 17.1 354 17.2 Histogram 357 Classwork Examples 257 Practice - 17.2 358 17.3 Frequency Polygon 360 Classwork Examples 361 Practice - 17.3 361 17.4 Cumulative Frequency Curve (Ogive) 363 Classwork Examples 363 Practice - 17.4 364 18. Measurement of Central Tendency 18.1 Arithmetic Mean 367 Classwork Examples 368 Practice - 18.1 369 18.2 Median 371 Classwork Examples 371 Practice - 18.2 373 18.3 Quartiles 374 Classwork Examples 374 Practice - 18.3 376 18.4 Quartiles 377 Classwork Examples 377 Practice - 18.4 379 19.1 Introduction to Probability 380 Classwork Examples 383 Practice - 19.1 384 19.2 Empirical and Classical Probabilities 385 19. Probability Classwork Examples 386 Practice - 19.2 389 Confidence Level Test - VI 391 Additional Practice - VI 392 VII TRIGONOMETRY 20. Trigonometry 20.1 Trigonometric Ratios 396 Classwork Examples 398 Practice - 20.1 402 20.2 Trigonometric Ratios of Standard Angles 404 Classwork Examples 405 Practice - 20.2 408 Confidence Level Test - VII 409 Additional Practice - VII 410 Appendix Extra Materials I . Practice Question Set 411 II. Classroom Individual Form 414 III. Project and Practical Work Evaluation Sheet 415 IV. Internal Evaluation Form 416

Sets PB Allied The Leading Mathematics-9 Sets 9 Specification Grid Unit Areas Total working hour Knowledge Understanding Application Higher ability Total number of items Total number of questions Total Marks No. of items Marks No. of items Marks No. of item Marks No. of item Marks I Sets 12 1 1 1 1 1 3 1 1 4 1 6 Operating on sets and solve the behaviour problems on the cardinality of sets. Estimated Working Hours : Competency Learning Outcomes At the end of this unit, the students will be able to: operate on sets and represent in the Venn diagram. find the cardinality of the set. 1. Sets 1.1 Union and Intersection of Sets 1.2 Difference and Complement of Sets 1.3 Cardinality of Sets Chapters / Lessons SETS UNIT 12 (Th. + Pr.) HSS Way: Mathematics begins at Home, grows in the Surroundings and takes shape in School. I

Sets WARM-UP Introduction to Sets We all have a group of some specific objects and arrangement of goods at our homes, office and other fields in our life. For this, in mathematics, the term 'set' is used which means the collection of well-defined objectsthat refersthe characteristics/properties of the objects. Mathematically, a set is a collection of well-defined objects. First, the word set was introduced by a German mathematician Georg Cantor (1845 - 1918) in 1904. He developed set theory, which has become a fundamental theory in mathematics. Set Notation and Set Membership For convenience, we use a single capital or upper case letter such as A, B, C, ... ..., X, Y or Z to denote sets and the small or lower case letters such as a, b, c, ... ..., x, y, z or digits or numbers to denote the elements or members of a set. To indicate set membership, the Greek letter ∈ (epsilon) is used to denote the words is in or belongs to or is an element of. The non-membership is indicated by the symbol ∉ (does ot belong). In particular, if A = {a, b, c}, we can write “a ∈ A” and 3 ∉ A. Types of Sets SN Name Definition Notation / Examples 1. Null Set set having no element is called the empty set or null set or void set. φ (phi) or { } Female prime minister of Nepal 2. Singleton Set A set that has only on element, is known as singleton set or unit set. Largest country of the world, Highest peak of the world 3. Finite Set A set that has countable number of elements is said to be finite. Letters of the English alphabet, digits, {a, b, c, ...., z} 4. Infinite Set Aset that is not finite is called infinite set. Points in a line, {A1, A2, A3, ......} Stars in sky, cells in human body, 5. Universal Set A set which has elements of all the related sets is called a universal set. U Educational materials in a students' bag Venn Diagram Sets can best be visualized by means of what is known as “Venn diagram”. It was first used by John Venn (1834 – 1923) in 1881 in Symbolic Logic. In a Venn diagram, the elements of a set are represented by the points in a plane region. A rectangle is used to represent the universal set U. Other sets in the universal set U are represented by an oval or a circle inside the rectangle. We may shade them, if necessary, as shown below: In Fig. (i), A and B are in the set U. In Fig. (ii), U = { 1, 2, 3, 4, 5} and A = {3, 4}. G. Cantor John Venn A B U Fig. (i) A 3 1 5 4 2 U Fig. (ii) 10 Allied The Leading Mathematics-9 Sets 11

Sets Relation Between Sets SN Name Definition Notation / Examples 1. Subset A set A is said to be a subset of a set B if every element of A is also an element of B. A is a subset of B, denoted by A ⊆ B. B is a super set of A, denoted by B ⊇ A. If A = {2, 4, 6} and B = {2, 3, 4, 5, 6}, then A ⊆ B or B ⊇ A. 2. Proper Subset A is a proper subset of B if and only if every element of A is in B, but there is at least one element of B that is not in A. If A = {1, 2, 4} and B = {1, 2, 4, 5}, then A is proper subset of B (A ⊂ B). 3. Improper Subset A is a improper subset of B if and only if B contains every element of A. If A = {1, 2, 4, 5} and B = {5, 2, 4, 1}, then A is improper subset of B (A ⊆ B). Disjoint and Intersecting Sets SN Name Definition Notation / Examples 1. Disjoint Sets Two sets A and B are said to be disjoint or non-overlapping if they have no element as common. The sets B = {b, o, y} and G = {g, i, r, l} are disjoint sets. 2. Intersecting Sets Two sets A and B are said to be intersecting or overlapping if they have at least one element as common. The sets M = {m, a, r, s} and E = {e, a, r, t, h} meet or intersect. But, in fact or actually, they do not intersect. Equal and Equivalent Sets Equality of Sets: Two sets A and B are said to be equal or identical or same if they have the same elements. For example, the sets T = {t, o, p} and P = {p, o, t} are equal, T = P; but the sets Q = {top} and R = {pot} are unequal Q ≠ R. Equivalent Sets: Two setsAand B are said to be equivalent if they have equal number of elements, may not be the same elements. We denote it by A ~ B (~ is read ‘ tilde’). For example, the sets T = {t, o, p} and P = {b, o, y} are equivalent sets, T ~ P. Note : All equal sets are equivalent, but not vice versa. When equivalent sets have the same elements, they are only equal sets. For more examples; (i) The sets of bones in two men of different height are only equivalent, but not equal sets. (ii) The sets of gears of two new bikes of the same CC (Cubic Capacity, volume of the chamber of the bike's engine) are equal sets. They are also equivalent. B U A 4 2 1 5 B U A 4 2 1 5 B o g b i y r l G U M m r e s a t h E U 10 Allied The Leading Mathematics-9 Sets 11

Sets 1.1 Union and Intersection of Sets At the end of this topic, the students will be able to: ¾ find the union and intersection of the given sets and represent in the Venn diagram. Learning Objectives I Introduction In set theory, the method of combining the elements of two or more sets is known as set operation. There are three major types of operations performed on sets such as; * Union of sets (∪) * Intersection of sets (∩) * Difference of sets ( – ) In real life, the operations on sets are vastly used in various sectors. Some of the uses of sets are discussed below; The group of me and my parents is most common set. When me and my mother, and me and my father live together, it forms a small family. What do you say about the family of child and his parents ? Father eats Daal-bhattarkari-achaar, but his son eats Daal-bhat-milk only. What do you say about the meals of father and his son ? What are common items? Some players like to play foot ball, some like to play volley ball and some like both. Also, some do not like neither of these. How many players like to play only one game and neither of both games ? II Union of Sets Activity 1 Look at the types and colours of flowers Sara and Ali have. Answer the following questions on the basis of the adjoining bounces of flowers: a) Write the colours of flowers with Sara in set notation form, say a set S. Sara's Bounce of Flowers Ali's Bounce of Flowers CHAPTER 1 SETS 12 Allied The Leading Mathematics-9 Sets 13

Sets b) Write the colours of flowers with Ali in set notation form, say a set A. c) Which same colours of flower do they have ? d) Which colours of flower does Sara have, but does not Ali? e) Which colours of flower does Ali have, but does not Sara? f) Which colours of flower do not they have ? Cay you say ? g) Prepare a Venn diagram for the colours of flowers that Sara and Ali have, say S = {Pink, Red, Blue} and A = {White, Red, Blue}. h) What type of Venn diagram is formed? What would be its name? Hence, we conclude that the flowers of all four colours are rose. This set of rose flowers is called union of colours of rose flowers, see the shaded portion in the Venn diagram. In set theory, it is called the union of the sets S and A. It is denoted by S ∪ A and read as S union A or S cup A. For example, let us consider two non-empty sets N = {1, 2, 3, 4, 5} and P = {2, 3, 5, 7} and represent them in the Venn diagram alongside. Write a set that represents all the elements of the sets N and P. Say, A = {1, 2, 3, 4, 5, 7} This set A is called the union of the sets N and P. Then A = N ∪ P = {1, 2, 3, 4, 5, 7} = {1, 2, 3, 4, 5} ∪ {2, 3, 5, 7} Thus, the union of sets A and B is the set of all the non-repeated elements which are contain in the sets A or B or both. In set-builder form, A ∪ B = { x : x ∈ A or x ∈ B} Alert to Union of Sets 1. The union of overlapping sets is not exactly the total elements or sum of all elements of the given sets. In set, we can take the repeated elements only once? Special Cases on Union of Sets 1) Consider two sets M = {m, o, n, k, e, y} and S = {s, h, a, r, p}. Find the union of the sets M and S. Compare M ∪ S with M and S. What do you find? So, M ∪ S = Set of all elements of M and S = {m, o, n, k, e, y, s, h, a, r, p} Hence, the union of disjoint sets is the set of all elements of the given sets. i.e., M ∪ S = Set of all elements. 2) Consider two sets M = {m, o, n, k, e, y} and K = {k, e, y}. Find the union of the sets M and K. Compare M ∪ K with M and K. What do you find? Pink White Red Blue A U S A ∪ B 1 7 4 2 3 5 U N P s h a m o k n e y r p U M S m o k n e y U K M 12 Allied The Leading Mathematics-9 Sets 13

Sets So, M ∪ K = Set of all elements of M only = {m, o, n, k, e, y} = M. Hence, the union of set and its proper subset is the set of all elements of the given super set. or larger set. i.e., If K ⊂ M, then M ∪ K = M = Lager Set. Note : If A is non-empty set and φ, empty, then A ∪ φ = A. Properties of Union of Two Sets 1) Idempotent Property: For two non-empty sets A and B, A ∪ A = A. 2) Identity Property: For two non-empty sets A and B, A ∪ φ = A. 3) Commutative Property: For two non-empty sets A and B, A ∪ B = B ∪ A. 4) Associative Property: For three non-empty sets A, B and C, A ∪ (B ∪ C) = (A ∪ B) ∪ C. 5) Distributive Property: For three non-empty sets A, B and C, A ∪ (B ∪ C) = (A ∪ B) ∪ (A ∪ C). Activity 2 Look at the adjoining Venn diagram of three sets. Then, a) Tell the elements in the set A, B and C. b) Which elements belong to the set A ∪ B ? c) Which elements does the set B ∪ C contain ? d) Which are the elements of the set A ∪ C ? e) Tell the elements that belong to A or B or C. f) Shade the region that contains the elements in the question (e). g) What is the name of the shaded portion in the question (f) ? The union of the sets A, B and C is denoted by A ∪ B ∪ C. Thus, the union of sets A, B and C is the set of all the non-repeated elements which is contained in the sets A or B or C. In set-builder form, A ∪ B ∪ C = { x : x ∈ A or x ∈ B or x ∈ C} Special Cases on Union of Sets In the given three disjoint sets O, A and R. What are the elements of the set O ∪ A ∪ R ? O ∪ A ∪ R = {o, m , a, n, g, e, l, r, u, v, i} = Set of all elements of the given sets Hence, the union of disjoint sets is the set of all elements of the given sets. i.e., O ∪ A ∪ R = Elements of O + Elements of A + Elements of R U U o m O T t u s i l g e A a n 14 Allied The Leading Mathematics-9 Sets 15

Sets Similarly, discuss the union of three sets for the following Venn diagrams: A U B a C b c A U B a C b c d e f e A U B a C b d c A U B a C d b e c A U B a C b c d U O m A a l T u t s i e g n III Intersection of Sets From the activity 2, which same colours of flower do Sara and Ali have ? Yes, the set formed by the same colours of rose flowers is called intersection of rose flowers, see the shaded portion in the Venn diagram. In set theory, it is called the intersection of the sets S and A. It is denoted by S ∩ A and read as S intersection A or S cap A. For example, in the above Venn diagram, write a set that contains the elements on both N and P only. Say B = {2, 3, 5}. This set B is called the intersection of the sets N and P. Then A = N ∩ P = {2, 3, 5} = {1, 2, 3, 4, 5} ∩ {2, 3, 5, 7} Thus, the intersection of the sets A and B is the sets of common elements of both A and B. In set-builder form, A ∩ B = {x : x ∈ A and x ∈ B} Special Cases on Intersection of Sets 1) Consider two sets M = {m, o, n, k, e, y} and S = {s, h, a, r, p}. Find the intersection of the sets M and S. Compare M ∩ S with M and S. What do you find? So, M ∩ S = Set of common elements of M and S = { } = φ. Hence, the union of disjoint sets is the set of total elements or sum of all elements of the given sets. i.e., M ∩ S = { }. Pink Red White Blue A U S A ∩ B 1 7 4 2 3 5 U N P s h a m o k n e y r p U M S 14 Allied The Leading Mathematics-9 Sets 15

Sets 2) Consider two sets M = {m, o, n, k, e, y} and K = {k, e, y}. Find the union of the sets M and K. Compare M ∩ K with M and K. What do you find? So, M ∩ K = Set of common elements of M only = {k, e, y} = K. Hence, the union of set and its proper subset is the set of all elements of the given proper subset or smaller set. i.e., If K ⊂ M, then M ∩ K = K = Smaller set. Note : If A is non-empty set and φ, empty, then A ∩ φ = φ. Properties of Intersection of Two Sets 1) Idempotent Property: For two non-empty sets A and B, A ∩ A = A. 2) Identity Property: For two non-empty sets A and B, A ∩ φ = φ. 3) Commutative Property: For two non-empty sets A and B, A ∩ B = B ∩ A. 4) Associative Property: For three non-empty sets A, B and C, A ∩ (B ∩ C) = (A ∩ B) ∩ C. 5) Distributive Property: For three non-empty sets A, B and C, A∩ (B ∩ C) = (A∩ B) ∩ (A∩ C). (i) Distributive Property over Union: For three non-empty sets A, B and C, A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). (ii) Distributive Property over Intersection: For three non-empty sets A, B and C, A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). Points to be Remembered 1. The union of sets is the set of all the non-repeated elements which is contained in the given sets. In set-builder form for two non-empty sets A and B, A ∪ B = { x : x ∈ A or x ∈ B}. 2. The union of overlapping sets is the set of all the non-repeated elements of the first or the second or both the given sets. For two non-empty overlapping sets A and B, A ∪ B = Set of (All elements – Common elements). 3. The union of disjoint sets is the set of all elements of the given sets. For two non-empty disjoint sets A and B, A ∪ B = Set of all elements. 4. The union of super set and its subset is the set of all elements of the super set. For two sets A and B, if A ⊂ B, then A ∪ B = Super set = B. 5. If A ⊂ U and A is non-empty set and φ, empty, then A ∪ U = U and A ∪ φ = A. m o k n e y U K M 16 Allied The Leading Mathematics-9 Sets 17

Sets 6. The intersection of sets is the set of all common elements contained in both given sets. In set-builder form for two non-empty sets A and B, A ∩ B = { x : x ∈ A and x ∈ B}. 7. The intersection of overlapping sets is the set of common elements of both given sets. For two non-empty overlapping sets A and B, A ∩ B = Set of common elements. 8. The intersection of disjoint sets is the set of non-elements of the given sets. For two nonempty disjoint sets A and B, A ∩ B = Set of non-elements of A and B = Null Set (φ). 9. The intersection of super set and its subset is the set of all elements of the subset. For two sets A and B, if A ⊂ B, then A ∩ B = Subset = A. 10. If A ⊂ U and A is non-empty set and φ, empty, then A ∩ U = A and A ∩ φ = φ. Example-1 Study the items on the plates P and Q alongside and answer the following questions: (a) List the items of each plate in set notations. (b) List all the items in the plates P and Q. (c) List the items only in both plates in set notation. (d) List the items only in the plate P, but not in Q. (e) Show all the items of the plates P and Q in a Venn diagram. Solution: From the given items on the plates P and Q alongside, (a) The items of each plate in set notations; P = {Maize, Peas, Soybeans, Grams, Beans} and Q = {Wheat, Pulses, Rice, Beans, Maize, Soybeans} (b) All the items in the plates P and Q, P ∪ Q = {Maize, Soybeans, Beans, Peas, Grams, Wheat, Rice, Pulses} (c) The items only in both plates in set notation, P ∩ Q = {Maize, Soybeans, Beans} (d) The items only in the plate P, but not in Q = {Peas, Grams} (e) Representing all the items of the plates P and Q in a Venn diagram as alongside, Plate P Plate Q Peas Grams Pulses Rice Maize Wheat Beans Soybeans Q U P 16 Allied The Leading Mathematics-9 Sets 17

Sets Example-2 Find the union of the sets A = {x : x is the number between 10 and 17} and B = {x : 13 ≤ x < 20}. Represent it in Venn diagram. (a) List the elements of each set. (b) Find all the items in the sets A or B or both. (c) Show all the items in the same Venn diagram. (d) Why is A ∪ B equal to B ∪ A ? Give suitable reason. Solution: (a) Here, A = {x : x is the number between 10 and 17} = {11, 12, 13, 14, 15, 16} B = {x : 13 ≤ x < 20} = {13, 14, 15, 16, 17, 18, 19} (b) All the items in the sets A or B or both, A ∪ B = {11, 12, 13, 14, 15, 16} ∪ {13, 14, 15, 16, 17, 18, 19} = {11, 12, 13, 14, 15, 16, 17, 18, 19} (c) Showing all the items in the same Venn diagram, (d) A ∪ B is equal to B ∪ A because they are commutative. You Can Do ! If you find B ∪ A, what is your answer ? So, A∪ B =B ∪ A. Thisis commutative property of union of two sets. Example-3 If U = {x : x ≤ 12, x ∈ N}, and its subsets A = {Prime numbers} and B = {Odd numbers} then list the element of the set A ∩ B and show A ∩ B in the Venn diagram. (a) List the elements of the sets U, A and B. (b) Present the above information in a Venn diagram. (c) Find the set of all the items that belongs to the sets both A and B. (d) What is the relation between A ∪ B and A ∩ B ? Solution: (a) Given, U = {x : x ≤ 12, x ∈ N} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,12} A = {Prime numbers} = {2, 3, 5, 7, 11} B = {Odd numbers} = {1, 3, 5, 7, 9, 11} (b) Presenting the above information in a Venn diagram. (c) Set of all the items that belongs to the sets both A and B, A ∩ B = {2, 3, 5, 7, 11} ∩ {1, 3, 5, 7, 9, 11} = {3, 5, 7, 11} (d) The set A ∩ B is the subset of A ∪ B. You Can Do ! If you find B ∩ A, what is your answer? So, A∩ B = B ∩ A. Thisis commutative property of intersection of two sets. A ∪ B 11 13 17 18 19 12 14 15 16 A B U A ∩ B 3 4 8 6 A B 1 9 2 5 7 11 10 12 U 18 Allied The Leading Mathematics-9 Sets 19

Sets Example-4 Amrit, Bandhana and Chandra are playing cards. Identify the marks on the corner of the cards and answer the questions below: (The cord A represents 1, J represents 11, Q represents 12 and K represents 13.) Cards in Amrit's hand Cards in Bandhana's hand Cards in Chandra's hand (a) List the sets of the numbers of pips or marks on the corners of each card in their hand separately in set notation. Suppose, the cards in Amrit's hand A, Bandhana's hand, B and Chandra's hand, C. (b) Find B ∪ C, A ∩ B and A ∩ C. (c) Prove that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). (d) Present A ∩ (B ∪ C) in a Venn diagram. Solution: From the giving playing cards; (a) Set of the numbers of pips or marks on the corners of cards with Amrit, A = {1, 2, 3, 4, 5, 7, 9, 10, 13} Set of the numbers of pips or marks on the corners of cards with Bandhana, B = {1, 2, 5, 6, 7, 8, 9, 11, 12, 13} Set of the numbers of pips or marks on the corners of cards with Chandra, C = {1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 13} (b) B ∪ C = {1, 2, 5, 6, 7, 8, 9, 11, 12, 13} ∪ {1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 13} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} A ∩ B = {1, 2, 3, 4, 5, 7, 9, 10, 13} ∩ {1, 2, 5, 6, 7, 8, 9, 11, 12, 13} = {1, 2, 5, 7, 9, 13} A ∩ C = {1, 2, 3, 4, 5, 7, 9, 10, 13} ∩ {1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 13} ={1, 2, 3, 4, 7, 9, 10, 13} (c) LHS = A ∩ (B ∪ C) = {1, 2, 3, 4, 5, 7, 9, 10, 13} ∩ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} = {1, 2, 3, 4, 5, 7, 9, 10, 13} RHS = (A ∩ B) ∪ (A ∩ C) = {1, 2, 5, 7, 9, 13} ∪ 1, 2, 3, 4, 7, 9, 10, 13} = {1, 2, 3, 4, 5, 7, 9, 10, 13} Hence, A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). (d) Presenting A ∩ (B ∪ C) in a Venn diagram. A B C 18 Allied The Leading Mathematics-9 Sets 19

Sets PRACTICE 1.1 Read Think Understand Do Keeping Skill Sharp 1. (a) Define union of sets. (b) What is the intersection of two sets ? Define it. (c) Why the union of sets is different from their intersection ? Give reason. (d) Write the union of the sets M = { m, a, p} and N = {n, e, p, a, l}. (e) What is the intersection of the sets P = {2, 3, 5} and E = {2, 4}? (f) If the given sets are disjoint, what is their intersection ? 2. Circle ( ) the correct answer. (a) The union of two overlapping sets is a set of ............ . (i) all the elements of the given two sets. (ii) common elements of the given two sets. (iii) all the non-repeated elements of the given two sets. (iv) all the repeated elements of the given two sets. (b) The intersection of sets A and B is defined as (i) {x : x ∈ A and x ∉ B} (ii) {x : x ∈ A or x ∈ B or both} (iii) {x : x ∈ A and x ∈ B} (iv) {x : x ∈ A or x ∈ B} (c) Which is the intersection of the sets C = {c, a, m, e, l} and D = {d, o, g} ? (i) 0 (ii) ϕ (iii) {0} (iv) {c, a, m, e, l, d, o, g} (d) Which is the union of the sets M = {m, a, n, g, o} and G = {g, r, a, p, e, s} ? (i) {m, a, n, g, o, r, p, e, s} (ii) {a, g} (iii) {m, n, o} (iv) {r, p, e, s} (e) A is the subset of B, where B = {1, 3, 5, 7}. If A ∪ {1, 3, 5, 7} = {1, 3, 5, 7}, then the set A = .......... . (i) {1, 4} (ii) {1, 2, 3} (iii) {1, 5} (iv) {5, 6, 7} 20 Allied The Leading Mathematics-9 Sets 21

Sets (f) Which is true property for the sets A, B and C? (i) A ∩ (B ∪ C) = (A ∩ B) ∩ (A ∩ C) (ii) A ∩ (B ∪ C) = (A ∪ B) ∪ (A ∪ C) (iii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (iv) A ∪ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) Check Your Performance Answer the given questions for each problem. 3. Study the types of fruits in the shops A and B alongside and answer the following questions: (a) List the name of fruits of each shop in set notation. (b) Find the name of all the fruits in A or B or both shops. (c) List the items only in both shops in set notation. (d) List the items only in the shop A, but not in B. (e) List the items only in the shop B, but not in A. (f) Show all the items of the shops A and B in a Venn diagram. 4. Study the given Venn diagram and answer the following questions: (a) Write the elements of the sets A and B. (b) Find the set A ∪ B and shade it. (c) Find the set A ∩ B and shade it. (d) Write the relation between A ∪ B and A ∩ B. 5. It is given that the sets A = {Prime numbers less than 12} and B = {Multiples of 2 less than 16}. Answer the following questions: (a) List the above information in set notation. (b) Show the above sets in a Venn diagram. (c) What is the union of A and B ? Find it. (d) Find B ∪ A and shade in the same Venn diagram. (e) Why are A ∪ B and B ∪ A equal? Give suitable reason. 6. The sets E and F are defined as E is the set of the letters in the word 'education' and F is the letters in the word 'equation'}. (a) List the given facts in set notation. (b) Show the above sets in a Venn diagram. (c) Find the intersection of E and F ? Shade in the same Venn diagram. (d) In which condition does E ∩ F equal to E ? Shop A Shop B 8 4 1 6 A B 3 5 9 2 7 10 U 20 Allied The Leading Mathematics-9 Sets 21

Sets 7. Observe the adjoining Venn diagrams. (a) Write the name of the shaded portions in the given Venn diagrams. (b) What are the differences between the shaded portions in both Venn diagrams ? Write. 7. (a) P Q U A B U 7. (b) 8. Study the adjoining given Venn diagrams. (a) Write the name of the shaded portions in the given Venn diagrams. (b) Write the relation between the shaded sets in the Venn diagrams. 8. (b) A B C U 8. (a) A B C U 9. Given U = {x : x ≤ 15, x ∈ N}, P = {x : x is a square number, x ∈ U} and Q = {x : x is a composite number, x ∈ U}. (a) Write the above information in set notation. (b) Show the above sets in a Venn diagram. (c) Find P ∪ Q and shade in the same Venn diagram. (d) Find P ∩ Q and shade in the same Venn diagram. (e) What kinds of the sets are P ∪ Q and P ∩ Q equal? Give suitable reason. 10. If U = {x : 5 ≤ x < 18, x ∈ N}, A = {x : x is the number divisible by 3}, B = {x : x is the number divisible by 4} and C = {x : x is the number divisible by 5}, find the elements of the following sets and show them in the Venn diagram separately. (a) Write the above information in listing method. (b) Find A ∩ (B ∪ C) and shade in the same Venn diagram. (c) Find (A ∪ B) ∩ C and shade in the same Venn diagram. (d) Which part is common in the sets A ∩ (B ∪ C) and (A ∪ B) ∩ C ? 11. Let A = {a, s, h, o, k}, M = {m, e, d, n, a, t, h}, R = {r, u, p, a} are the given sets. (a) Verify the following relations: (i) A ∪ M = M ∪ A (ii) R ∩ A = R ∩ A (b) Why are they equal ? Give reasons. 22 Allied The Leading Mathematics-9 Sets 23

Sets 12. Let U = { a, b, c, ..., k}, A = {a, b, d, f}, B = {c, e, g. i} and C = {o, a, e, h, i} are the given sets. (a) Verify the following relations: (i) A ∪ (B ∪ C) = (A ∪ B) ∪ C) (ii) A ∩ (B ∩ C) = (A ∩ B) ∩ C) (b) Show the above relations in the Venn diagrams. (c) Why are they equal ? Give reasons. 13. A farmer cultivates tomato, cabbage, radish, carrot, brinjal, lady's finger and peas in the farm P; tomato, cabbage, radish, garlic, brinjal, onion, turnip and potato in the farm Q and cauliflower, choko, cabbage, radish, cucumber, brinjal, pumpkin, turnip and capsicum in the farm R. List all the vegetables in each farm by supposing the bigging small letter of each vegetable. (a) List the name of vegetables of each farm in set notation. (b) Prove the following relations: (i) P ∩ (Q ∪ R) = (P ∩ Q) ∪ (P ∩ R) (ii) P ∪ (Q ∩ R) = (P ∪ Q) ∩ (P ∪ R) (c) Why are they equal ? Give reasons by using Venn diagram. 4. (a) {2, 7, 8, 10}, {3, 5, 8, 9, 10} (b) {2, 3, 5, 7, 8, 9, 10} (c) {8, 10} 5. (a) {2, 3, 5, 7, 11}, {2, 4, 6, 8, 10, 12, 14} (c) {2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14} (d) {2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14} 6. (a) {e, d, u, c, a, t, i, o, n}, {e, q, u, a, t, i, o, n} (c) {e, u, a, t, i, on, n} 7. (a) P ∪ Q (b) A ∩ B 8. (a) (A ∩ B) ∪ (B ∩ C) ∪ (C ∩ A) (b) C ∪ (A ∩ B) 9. (a) {1, 2, 3, 4, ..., 13, 14, 15}, {1, 4, 9}, {4, 6, 8, 9, 10, 12, 14, 15} (c) {1, 4, 6, 8, 9, 10, 12, 14, 15} (d) {4, 9} 10. (a) {5, 6, 7, 8, ..., 15, 16, 17}, {6, 9, 12, 15}, {8, 12, 16}, {5, 10, 15} (b) {12, 15} (c) {15} Answers 22 Allied The Leading Mathematics-9 Sets 23

Sets 1.2 Difference and Complement of Sets At the end of this topic, the students will be able to: ¾ find the difference and complement of the given sets and represent in the Venn diagram. Learning Objectives I Difference of Sets Activity 3 A father and his son take simple meal without maasu and saag in the morning as shown in the adjoining pictures. What items of food are there in the father's plate? List them in set notation by supposing the set of items as F ? What items of food are there in the son's plate? List them in set notation by supposing the set of items as S ? Which items are eaten by them in common ? Which different items are eaten by father than eaten by his son? Which different items are eaten by son than eaten by father? The items tarkari and achar eaten by father are different from the items eaten by his son. A set of these different items tarkari and achar is called the difference of S from F. It is denoted by F – S, read as 'F difference S'. This new set F – S is a set of all the elements of F but not in S as shown in the adjoining Venn diagram. In set builder form, F – S = {x : x ∈ F and x ∉ S} The item dudh taken by son is different from the items eaten by his father. A set of the different item dudh is called the difference of F from S. It is denoted by S – F, 'S difference F'. This new set S – F is a set of all the elements of S but not in F as shown in the adjoining Venn diagram. In set builder form, S – F = {x : x ∈ S and x ∉ F} Definition: The difference of B from A is the set of all the elements of A not belonging to B. For more, consider two sets A = {1, 2, 3}, B = {3, 4, 5}. Then, the difference of B from A, A – B = {1, 2, 3} – {3, 4, 5} Son's Meal Father's Meal F F – S S S – F U F – S S – F Bhat Tarkari Achar Dudh F S U Daal Meat Saag 24 Allied The Leading Mathematics-9 Sets 25

Sets Now, take out the common elements of A and B from A, we get A – B = {1, 2}. i.e., A – B = A – (A ∩ B) Similarly, the difference of A from B, B – A = {3, 4, 5} – {1, 2, 3} Now, take out the common elements of A and B from B, we get B – A = {4, 5}. i.e., B – A = B – (A ∩ B) The differences A – B and B – A are shown in the Venn diagram alongside. Alert to Difference of Sets 1. The difference of B from A means the cancellation of common elements of A and B from A only, not from B. Note : If A and B are two non-empty and unequal sets, then A – B ≠ B – A. Special Cases on Difference of Sets 1) Consider two subsets M = {m, o, n, e, y} and P = {p, a, i, s} of the universal set U = {p, o, r, k, m, e, n, i, y, a, s}. Find the difference of P from M. Compare M – P with M and P. What do you find? So, M – P = Set of all the elements containing in M but not P = {m, o, n, e, y} – {p, a, i, s} = {m, o, n, e, y} Similarly, P – M = Set of all the elements containing in P, but not M = {p, a, i, s} – {m, o, n, e, y} = {p, a, i, s} Hence, the difference of two disjoint sets is the set of its elements only that from cancelled. i.e., M – P = M and P – M = P if M and P are disjoint sets. 2) Consider two sets M = {m, o, n, e, y} and N = {n, e, y}, where N ⊆ M. Find the difference of N from M. Compare M – N with M and N. What do you find? So, M – N = Set of all the elements containing in M, but not N = {m, o, n, e, y} – {n, e, y} = {m, o} = Shaded portion = Super set without subset A – B B – A 3 1 2 4 5 A B U p a i m o n e M – P y s M P U p a i m o n e y s M P U P – M m o n e y U N M – N M 24 Allied The Leading Mathematics-9 Sets 25

Sets Hence, the difference of subset from super set is the set of all the elements containing in super set but not in subset. Similarly, N – M = Set of all the elements containing in N but not M = {n, e, y} – {m, o, n, e, y} = { } = φ. Hence, the difference of super set from subset is null set. i.e., If N ⊆ M, then N – M = φ = Null Set. Note : If A is non-empty set and φ, empty, then A – A = φ. Activity 4 Look the adjoining Venn diagram of three sets. Then, a) Tell the elements in the set A, B and C. b) Which elements belong to the set A ∪ B and A ∩ B ? c) Which elements do the set B ∪ C and B ∩ C contain ? d) Which are the elements of the sets A ∪ C and A ∩ C ? e) Tell the elements that belong to C – (A ∪ B) and shade it. The set of these elements containing only C. In set-builder form, C – (A ∪ B) = { x : x ∈ C and x ∉ (A ∪ B)} f) Tell the elements that belong to (A ∩ B) – C and shade it. The set of these elements containing only (A ∩ B). In set-builder form, (A ∩ B) – C = { x : x ∈ (A ∩ B) and x ∉ C} Similarly, discuss the sets (B ∪ C) – A, B – (A ∩ C), etc. II Complement of a Set Activity 5 In the previous simple meals without maasu and saag of father and his son, discuss following questions. The set of food items in the plate of father, F = {Bhat, Daal, Tarkari, Achar}. The set of food items in the plate of son, S = {Bhat, Daal, Dudh}. m o n e y U N M N – M = φ A U B a C g b h c d f i e Son's Meal Father's Meal 26 Allied The Leading Mathematics-9 Sets 27

Sets What are the food items on the table within plates and out of plate? U = {Bhat, Daal, Tarkari, Achar, Dudh, Maaasu, Saag} What types of items of food do not contain in the plate of the father ? List them in set notation by supposing the set of items as A. A = {Dudh, Maasu, Saag} = {Bhat, Daal, Tarkari, Achar, Dudh, Maasu, Saag} – {Bhat, Daal, Tarkari, Achar} = U – F = Shaded Portion in the Venn diagram This difference of F from the universal set U is denoted by F' (read 'F prime') of Fc (read 'F complement') or F (read 'F bar') and is called complement of F. Therefore, F' of Fc or F = U – F Hence, F = {x : x ∈ U and x ∉ F} = U – F, F⊂ U. Similarly, what items of food do not contain in the plate of the son ? List them in set notation by supposing the set of items as B. B = {Tarkari, Achar, Maasu, Saag} = {Bhat, Daal, Dudh, Tarkari, Achar, Maasu, Saag} – {Bhat, Daal, Dudh} = U – S = Shaded Portion in the Venn diagram This difference of F from the universal set U is denoted by S' of Sc or S and called complement of S. Therefore, S' of Sc or S = U – S Hence, S = {x : x ∈ U and x ∉ S} = U – S, S⊂ U. Definition: The complement of a set A is the set of all the elements in the universal set that do not belong to A. It is the negation of the given set. For more, consider the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and its subset A = {1, 2, 3}. Then A = U – A = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3} = {4, 5, 6, 7, 8, 9}. Show A in a Venn diagram alongside, Note : If U is universal set and φ, empty, then U = φ, φ = U and (A) = A. F U F S U S 1 4 6 5 8 7 9 2 3 A U A 26 Allied The Leading Mathematics-9 Sets 27

Sets Example-1 If a universal set U = {Whole numbers up to 8} and its subsets A = {Even numbers} and B = {Natural numbers less than 6}, then answer the following questions: (a) List the above information in set notation. (b) What are the differences from A to B and from B to A? Find. (c) Shade A – B and B – A in the same Venn diagram. (d) Why are the sets A – B and B – A not equal ? Give reason. Solution: (a) Here, U = {Whole numbers up to 8} = {1, 2, 3, 4, 5, 6, 7, 8}, A = {Even numbers}= {2, 4, 6, 8} and B = {Natural numbers less than 6}= {1, 2, 3, 4, 5}. (b) Now, we know that Difference from A to B, A – B = {2, 4, 6, 8} – {1, 2, 3, 4, 5} = {6, 8}. Difference from B to A, B – A = {1, 2, 3, 4, 5} – {2, 4, 6, 8} = {1, 3, 5}. (c) Showing A – B and B – A in a Venn diagram as shaded portions aside, (d) The sets A – B and B – A are different because the difference of two sets is not commutative. Example-2 Let U = {e, q, u, a, t, i, o, n}, A = {n, o}, B = {t, e, a} and C = {a, t}. (a) Find, (i) A ∪ B and A ∩ B (ii) A – C and A – C (b) Show the above sets in the Venn diagrams. (c) Why are A ∪ B and A ∩ B equal, but A – C and A – C not equal ? Give reason. Solution: Here, U = {e, q, u, a, t, i, o, n}, A = {n, o}, B = {t, e, a} and C = {a, t} (a) (i) A ∪ B = {n, o} ∪ {t, e, a} = {n, o, t, e, a} A ∪ B = U – (A ∪ B) = {e, q, u, a, t, i, o, n} – {n, o, t, e, a} = {q, u, i} and A = U – A = {e, q, u, a, t, i, o, n} – {n, o} = {e, q, u, a, t, i} B = U – B = {e, q, u, a, t, i, o, n} – {t, e, a} = { q, u, i, o, n} ∴ A ∩ B = {e, q, u, a, t, i} ∩ { q, u, i, o, n} = {q, u, i}. (ii) A – C = {n, o} – {a, t} = {n, o} ∴ A – C = U – (A – C) = {e, q, u, a, t, i, o, n} – {n, o} = {e, q, u, a, t, i}. 7 A B U 1 3 5 2 4 6 8 A – B and B – A 28 Allied The Leading Mathematics-9 Sets 29

Sets and A = U – A = {e, q, u, a, t, i, o, n} – {n, o} = {e, q, u, a, t, i} C = U – C = {e, q, u, a, t, i, o, n} – {a, t} = {e, q, u, i, o, n} ∴ A – C ={e, q, u, a, t, i} – {e, q, u, i, o, n} = {a, t}. (b) Showing the above sets in the Venn diagrams, q u i A B U t e a n o A ∪ B q u i A B U t e a n o A ∩ B A C U t n a o A – C A C U t a n o A – C (c) A ∪ B and A ∩ B are equal because of complement on inverse operations(union and intersection) of the sets i.e., negation of conjunction and disjunction, but A – C and A – C are not equal because of complement on the same difference operation. Example-3 It is given that a universal set U = {x: x is a positive integer less than 11} and its subsets A = {x: x is a positive integer less than 6} and B = {x: x is a positive integer between 4 and 8}. Give the answers of the following questions: (a) List the given information in set notation. (b) Write the De Morgan's properties. (c) Present the above facts in a Venn diagram. (d) Prove that De Morgan's property: A ∩ B = A ∪ B (e) Is A ∪ B = A ∪ B ? Why ? Give suitable reason. Solution: (a) Here, the given information in set notation is, U = {x: x is a positive integer less than 11} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {x: x is a positive integer less or equal to 5} = {1, 2, 3, 4, 5} and B = {x: x is a positive integer between 4 and 8} = {5, 6, 7}. (b) The De Morgan's properties are A ∪ B = A ∩ B and A ∩ B = A ∪ B. (c) Presenting the above facts in a Venn diagram, (d) Now, we have A ∩ B = {1, 2, 3, 4, 5} ∩ {5, 6, 7} = {5} ∴ A ∩ B = U – (A ∩ B) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {5} = {1, 2, 3, 4, 6, 7, 8, 9, 10}. 10 8 9 A B U 6 7 5 2 1 4 3 28 Allied The Leading Mathematics-9 Sets 29

Sets A = U – A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {1, 2, 3, 4, 5} = {6, 7, 8, 9, 10}. B = U – B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {5, 6, 7} = {1, 2, 3, 4, 8, 9, 10}. ∴ A ∪ B = {6, 7, 8, 9, 10} ∪ {1, 2, 3, 4, 8, 9, 10} = {1, 2, 3, 4, 6, 7, 8, 9, 10}. Hence, A ∩ B = A ∪ B. (e) No, A ∪ B = A ∪ B is not because it does not hold complement on inverse operations. PRACTICE 1.2 Read Think Understand Do Keeping Skill Sharp 1. (a) Define difference of two sets. (b) What is the complement of a set ? Define it. (c) Write one difference between difference of two sets and complement of a set. (d) Write the set C – O if the sets C = {c, o, v, i, d} and O = {o, m, n, i, v, r, u, s}. (e) What is the complement of the set P = {2, 4, 6, 8, 10}, which is the subset of a universal set U = {1, 2, 3, .........., 10}? 2. Circle ( ) the correct answer. (a) The complement of a set is a set of ............ . (i) elements contained in the given set. (ii) elements contained in the given set but not in the universal set. (iii) elements not contained in the first set but in the second set. (iv) elements not contained in the given set, but in the universal set. (b) The complement of the set M is defined as ........... (i) {x : x ∈ A and x ∉ A} (ii) {x : x ∈ M or x ∈ U} (iii) {x : x ∈ M and x ∉ U} (iv) {x : x ∈ A and x ∈ B} (c) Which is the difference of the sets N = {3, 4, 5, 6} and P = {2, 3, 5} ? (i) {2, 3, 4, 5, 6} (ii) {3, 5} (iii) {2} (iv) {4, 6} 30 Allied The Leading Mathematics-9 Sets 31

Sets (d) Which is the complement of the sets M = {m, a, n, g, o} in the universal set U = {p, o, m, e, g, r, a, n, t} ? (i) {p, o, m, e, g, r, a, n, t} (ii) {p, e, r, t} (iii) {m, a, n, g, o} (iv) {r, p, e} (e) Which is true for three sets P, Q and R ? (i) A ∩ B = A ∪ B (ii) A ∩ B = A ∩ B (iii) A ∪ B = A ∪ B (iv) A ∩ B = A ∪ B Check Your Performance Answer the given questions for each problem. 3. Observe the utensils in the adjoining kitchen rooms of Maya and Yangri. Suppose the set of utensils of Maya by M and Yangri by Y. Then answer the questions asked below: (a) List the elements of M and Y in set notation. (b) Which utensils are more with Maya that of Yangri? Find it. (c) Which utensils are more with Maya that of Yangri? Find it. (d) Illustrate the above sets in Venn diagrams. (e) Identify the resulting sets in (b) and (c) equal or not? Why ? 4. Observe the adjoining Venn diagrams. (a) Write the name of the shaded portions in the given Venn diagrams and write them in set builder also. (b) Write one similar character on both Venn diagrams. 4. (a) P Q U M U 4. (b) 5. Study the adjoining Venn diagrams. (a) Write the name of the shaded portions in the given Venn diagrams. (b) Write the relation between the shaded sets in both Venn diagrams. 5. (a) A B C U A 5. (b) B C U 30 Allied The Leading Mathematics-9 Sets 31

Sets 6. Suppose, U = {English months of a year}, J = {Months beginning with 'J'}, S = {6th to 8th months}, W = {1st three months}. (a) Write the given information in set notation. (b) List the elements of each of the following sets. (i) J – W (ii) J – S (iii) W ∆ S (iv) J ∪ W (v) J ∩ S (vi) J ∪ S (vii) J – (S ∪ W) (viii) J ∩ S ∩ W – W (ix) (J – S) ∩ (J – W) (c) Show each of the above sets in the Venn diagrams. (d) Which sets are equal in (b) ? 7. Suppose, P = {The letters in the word of a lung disease caused by inhalation of very fine silicate or quartz dust 'pneumonoultramicroscopicsilicovolcanoconiosis'} and its subsets C = {The letters in the word of a heart artery disease 'coronary'}, H = {The letters in the word of a lever disease 'hemochromatosis'} and D = {The letters in the word of a brain disease 'Dementias'}. (a) Write the given information in set notation. (b) List the members of each of the following sets. (i) C ∩ (H ∪ D) (ii) C ∪ (H ∩ D) (iii) (C ∩ D) ∪ (H ∩ D) (iv) C ∪ H ∪ D (v) C – (H ∩ D) (vi) (C – H) ∪ (C – D) (c) Show each of the above sets in the Venn diagrams. (d) Which sets are equal in (b) ? Why ? Give reasons. 8. Suppose, U = {x: x is a whole number up to 15}, M = {x: x is a multiple of 3 from 1 to 15}, O = {x: x is an odd number from 1 to 15} and F = {x: x is a factor of 12}. (a) Write the given information in set notation. (b) List the elements containing each of the following. (i) at least one set. (ii) at most two sets. (iii) neither all sets. (iv) M, but not in O ∩ F. (v) M ∪ F, but not in M ∩ F. (vi) not in F, but not in M ∪ O. (c) Shade each of the above sets in the Venn diagrams. 9. Use the given Venn diagram and find the sets by listing the elements. (a) P ∪ (Q – R) (b) (P ∪ Q) ∩ R (c) P – (Q – R) (d) (Q – P) ∩ (Q – R) (e) P – Q ∩ R (f) R ∪ Q – P (g) P – (Q ∪ R) (h) (P ∪ R) – (P ∩ R) (i) (P ∪ Q) - R) 20 16 17 18 2 3 1 5 4 6 10 9 7 8 11 14 12 15 13 P Q 19 U R 32 Allied The Leading Mathematics-9 Sets 33

Sets 10. Suppose, U = {x/x ≤ 20 and x∈ N}, A = {x/x is a prime number}, B = {x/x is an odd number} and C = {x/x is an even number}. (a) Write the given information in set notation. (b) List the elements containing each of the following. i. A ii. A ∩ C iii. A ∪ B iv. U − A ∩ C v. A ∪ B ∪ C vi. B ∩ C – A (c) Shade each of the above sets in the Venn diagrams. 11. Suppose, U = { a, b, c, .............., k}, A = {a, b, d, e, g}, B = {c, e, g, i} and C = {o, a, e, h, i} are the given sets. Verify the following relations: (a) A – B = A ∩ B = B – A (b) A ∪ B = A ∩ B (c) (A – B) ∪ (B – A) = (A ∪ B) – (A ∩ B) (d) A ∪ (B – C) = (A ∪ B) – (A ∪ C) (e) A – (B ∩ C) = (A – B) ∪ (A – C) (f) A ∩ B ∩ C = A ∪ B ∪ C 4. (a) P – Q = {x : x ∈ P and x ∉ Q} (b) = U – M = {x : x ∈ U and x ∉ M} 5. (a) (B∩C)–A, A–(B∩C) (b) (B∩C)–A = (B– A)∩(C–A), A–(B∩C) = (A–B)∪(A–C) 8. (a) {1, 2, 3, 4, ..., 12, 13, 14, 15}, {3, 6, 9, 12, 15}, {1, 3, 5, 7, 9, 11, 13, 15}, {1, 2, 3, 4, 6, 12} (b) {1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 15}, {1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, {8, 10, 14}, {6, 9, 12, 15}, {1, 2, 4, 9, 15}, {8, 10, 14} 9. (a) {1, 2, 3, 4, 5, 6, 7, 9, 10, 13, 15} (b) {2, 3, 8, 11} (c) {2, 3, 6, 7, 9, 10} (d) {13, 15} (e) {7} (f) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14} (g) {6, 9, 10} (h) {1, 4, 5, 6, 8, 9, 10, 11, 12, 14} (i) {1, 4, 5, 6, 9, 10, 13, 15} 10. (a) {1, 2, 3, ..., 18, 19, 20}, {2, 3, 5, 7, 11, 13, 17, 19}, {1, 3, 5, 7, 9, 11, 13, 15, 17, 19} {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} (b) {1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20}, {1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} {4, 6, 8, 10, 12, 14, 16, 18, 20}, {2}, { } Answers Project Work List out the name of the utensils in your kitchen room and represent them in set notation by taking any two capital letters and list out any two or three utensils that are important but you and your friend's kitchen rooms do not contain. Then find their union, intersection, difference and complements. Also, make a report for it and present it in your classroom by constructing a Venn diagram. 32 Allied The Leading Mathematics-9 Sets 33

Sets 1.3 Cardinality of Sets At the end of this topic, the students will be able to: ¾ find the cardinality of the operations on sets. Learning Objectives I Introduction Activity 6 Open your geometric box and observe the materials containing in it. List the name of the materials in set notation, say M. What is the type of the set M ? Is it null, singleton, doubleton, finite, infinite, or other type ? The set M has fixed or countable number of materials that can be easily counted. So, the set M is a finite set. How many materials are there in the set M ? Suppose your geometric box has 9 materials. This number of materials 9 is called Cardinality of the set M. That is, M has cardinality of 9. It measures the size of the set. The cardinality of a set is denoted by vertical bars, like absolute value signs; for instance, for a set M its cardinality is denoted |M| or n(M) or #M. In our school level, we use n(M) for cardinality of M. The cardinality of set is equivalent to the cardinal number of the set M. We use the cardinal number for comparing the cardinality of two or more sets. The cardinality of any set is one-to-one correspondence with the set of natural numbers asfollows; N = { 1 2 3 4 ......... ......... .......... 25 26 } ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ A = { a b c d ......... ......... .......... y z } So, the cardinality of A or the cardinal number of A is 26. It is denoted by n(A) = 26. Remember: One-to-one correspondence is an early learning math-skill that involves the act of counting each object in a set once, and only once with one touch per object. More Interesting: When M is finite, |M| is simply the number of distinct elements in M. When any set A is infinite, i.e., A = {3, 6, 9, 12, 15, .............}, it has infinite number of elements that do not easily count them. The elements of this set A are one-to-one correspondence with the set of natural numbers N, N = {1, 2, 3, 4, 5, .........., n} How many bones are there in our body ? But, how many cells are there in our whole body ? Discuss. Geometric Box 34 Allied The Leading Mathematics-9 Sets 35

Sets Note: (a) If E = Set of the highest peak of the world = {Mt. Everest}, then n(E) = 1. So, the cardinality of singleton set is 1. (b) If C = Set of couple of human beings = {Husband, Wife}, n(C) = 2. So, the cardinality of doubleton set is 2. (c) If S = Set of seas of Nepal = { } = φ, then n(S) = 0. So, the cardinality of null set is 0. (d) If N = Set of natural numbers less than 3 = {1, 2} and W = Set of whole numbers less than 2 = {0, 1}, then n(N) = n(W) = 2. So, the cardinalities of the equivalent sets is equal. (e) If P = {p, o, t} and T = {t, o, p}, then n(P) = 3 and n(T) = 3. So, the cardinality of equal sets is equal. (f) If P = {g, o, d} and T = {d, o, g}, then P ∪ T = {g, o, d}, P ∩ T = {d, o, g} and P T = { } = φ. Also, n(P) = 3, n(T) = 3, n(P ∪ T) = 3, n(P ∩ T) = 3 and n(P − T) = n(φ) = 0. So, the cardinality of union and intersecting of equal sets is equal to themselves, and the cardinality of difference of equal sets is 0. (g) If S = {s, t, o, p} and T = {t, o, p}, then n(S) = 4 and n(T) = 3. Also, S ⊂ T and n(S) > n(T). So, the cardinality of super set is greater than its subset, (h) If P = {p, o, t} and S = {s, p, o, t}, then n(P) = 3 and n(S) = 4. Also, P ⊂ S and n(P) < n(S). So, the cardinality of subset is less than its super set. II Cardinality of Operations of Two Sets Activity 7 Observe and count the flowers. How many red and white rose flowers are there ? Write the cardinality of red flowers R and white flowers W. Similarly, how many total rose flowers are there ? Write the cardinality of rose flowers R ∪ W. Can you establish the cardinality of red flowers R, white flowers W and total flowers R ∪ W ? What is it? The cardinality of the union of the disjoint sets is equal to the sum of cardinalities of these sets. If A and B are two disjoint sets, then n(A ∪ B) = n(A) + n(B). 34 Allied The Leading Mathematics-9 Sets 35

Sets Activity 8 Rina threw a die seven times and appeared the following number of spots: 2, 1, 5, 2, 4, 1, 5 John threw a die seven times and appeared the following number of spots: 3, 5, 3, 5, 3, 1, 5 Write the numbers of spots obtained by Rina and John by supposing the sets R and J respectively. What is the set of intersection of number of spots obtained by Rina and John ? What is the set of union of number of spots obtained by Rina and John ? From the above activities, R = {1, 2, 4, 5}, J = {1, 3, 5}, R ∩ J = {1, 5}, R ∪ J = {1, 2, 3, 4, 5} Show the above sets in a Venn diagram. Write the cardinality of all the above sets. n(R) = 4, n(J) = 3, n(R ∩ J) = 2, n(R ∪ J) = 5 Can you establish the relation between n(R), n(J) and n(R ∩ J) with n(R ∪ J)? 5 = 4 + 3 – 2 or, n(R ∪ J) = n(R) + n(J) – n(R ∩ J) The cardinality of the union of the overlapping sets is equal to the cardinality of intersection of the sets from the sum of cardinalities of these sets. If A and B are two overlapping sets, then n(A ∪ B) = n(A) + n(B) – n(A ∩ B). For more, what is the set of number of spots obtained by Rina only, but not John ? What is the set of number of spots obtained by John only, but not Rina ? R – J = {2, 4}, J – R = {3} Write the cardinality of all the above sets. n(R – J) = 2, n(J – R) = 1 Can you establish the relation between n(R – J), n(J – R) and n(R ∩ J) with n(R ∪ J)? 5 = 2 + 1 + 2 or, n(R ∪ J) = n(R – J) + n(J – R) + n(R ∩ J) i.e., n(R ∪ J) = n(R ∩ J) + n( R ∩ J) + n(R ∩ J) Also, n(R) = no(R) + n(R ∩ J) and n(J) = no(J) + n(R ∩ J) In cardinality of set notation, n(R – J) or n(R ∩ J) represents the number of elements contained only in the set R. So, it is also denoted by no(R), read as 'n not R' or 'Number of only R'. 2 4 3 1 5 R J U 6 2 4 3 1 5 R J U 6 36 Allied The Leading Mathematics-9 Sets 37

Sets T Then, n(R ∪ J) = no(R) + no(J) + n(R ∩ J) 'OR' n(R) + no(J) 'OR' no(R) + n(J) Again, what is the set of number of spots not obtained by Rina and John ? What is the set of number of spots containing on the dice ? R ∪ J = {6}, U = {1, 2, 3, 4, 5, 6} and n( R ∪ J) = 1, n(U) = 6. Can you establish the relation between n(R), n(J), n(R ∩ J) and n( R ∪ J) with n(U)? 6 = 4 + 3 – 2 + 1 or, n(U) = n(R) + n(J) – n(R ∩ J) + n( R ∪ J) i.e., n(U) = n(R ∪ J) + n(R ∪ J) Similarly, n(U) = no(R) + no(J) + n(R ∩ J) + n(R ∪ J) Important Facts on Cardinal Numbers of Sets 1. If A ⊆ B, then n(A ∪ B) = n(B), n(A ∪ B) = n(A), n(A – B) = 0. 2. If A ⊆ U, then n(A ∪ U) = n(U), n(A ∪ U) = n(A), n(A – U) = 0, n(U) = 0 and n(φ) = n(U). 3. If A and B are two disjoint sets, then n(A – B) = n(A), n(B – A) = n(B) and n(A ∪ B) = 0. B A U III Cardinality of Operations of Three Sets Activity 9 Observe and discuss the three primary colours. What are the three primary colours ? => The distinct three primary colours are red (R), yellow (Y) and blue (B). They are disjoint sets. If we mix two primary colours, we get new colour, called a secondary colour. For example, mixing red and blue produces purple (P); yellow and red makes orange (O); blue and yellow mixed make green (G); red, yellow and blue mixed make brown (N) in equal quantities. The mixing colours is overlapping sets. The secondary colours obtained by only two primary colours are as the intersection of only two sets and that of three colours as the intersection of three sets. Then, we suppose, R B Y RYB Three Primary Colours Mixing RYB Three Primary Colours R C C Y G B O P N U no(R∩B) no(R) no(Y) no(B) Y n(Y) n(B) n(R) n(U) R B n(R∩Y∩B) no(R∩Y) no(R∩B) 36 Allied The Leading Mathematics-9 Sets 37

Sets n(P) = no(R ∩ B); n(O) = no(Y ∩ R), n(G) = no(B ∩ Y); and n(N) = n(R ∩ Y ∩ B). So, n(R ∪ Y ∪ B) = no(R) + no(Y) + no(B) + n(P) + n(O) + n(G) + n(N) i.e., n(R ∪ Y ∪ B) = no(R) + no(Y) + no(B) + no(R ∩ B) + no(Y ∩ R) + no(B ∩ Y) + n(R ∩ Y ∩ B) The number of elements in the indicated operations of three non-empty sets is the cardinality of the related operations. The cardinality of different parts of the operations of three intersection of non-empty sets A, B and C is shown alongside. In the Venn diagram, n(A) = No. of elements in A; n(B) = No. of elements in B; n(C) = No. of elements in C; n(A ∩ B) = No. of elements in both A and B; n(B ∩ C) = No. of elements in both B and C; n(A ∩ C) = No. of elements in both A and C; n(A ∩ B ∩ C) = No. of elements in A, B and C; no(A) = No.of elements in only A; no(B) = No.of elements in only B; no(C) = No.of elements in only C; no(A ∩ B) = No. of elements in only A and B; no(B ∩ C) = No. of elements in only B and C; no(A ∩ C) = No. of elements in only A and C; n(A ∪ B ∪ C) = No. of elements in A, B and C. Then, n(A) = no(A) + no(A ∩ B) + no(A ∩ C) + n(A ∩ B ∩ C) n(B) = no(B) + no(A ∩ B) + no(B ∩ C) + n(A ∩ B ∩ C) n(C) = no(C) + no(A ∩ C) + no(B ∩ C) + n(A ∩ B ∩ C) and n(A ∪ B ∪ C) = no(A) + no(B) + no(C) + no(A ∩ B) + no(B ∩ C) + no(A ∩ C) + n(A ∩ B ∩ C) Again, n(U) = n(A ∪ B ∪ C) + n(A ∪ B ∪ C) n(U) = no(A) + no(B) + no(C) + no(A∩B) + no(B∩C) + no(A∩C) + n(A∩B∩C) + n(A∪B∪C) no(A) no(B) no(C) n(ABC) no(AB) no(BC) no(AB) A BU C n(U) n(A) n(B) n(A ∪ B ∪ C) 38 Allied The Leading Mathematics-9 Sets 39

Sets Points to be Remembered 1. The number of distinct elements of a finite set is called Cardinality of the set. It measures the size of the set. For a set M its cardinality is denoted by n(M). 2. The cardinality of set is equivalent to the cardinal number of all sets that can be put in oneto-one correspondence with each other. 3. The cardinality of singleton set is 1 and the cardinality of doubleton set is 2. 4. The cardinality of null or empty or void set is 0. 5. The cardinal number of equivalent and equal sets is equal. 6. The cardinality of union and intersecting of equal sets is equal to each other, and the cardinality of difference of equal sets is 0. 7. The cardinality of super set is greater than the cardinality of its subset. 8. The cardinality of subset is less than the cardinality of its super set. 9. The cardinality of the union of the disjoint sets is equal to the sum of cardinalities of these sets. If A and B are two disjoint sets, then n(A ∪ B) = n(A) + n(B). 10. The cardinality of the union of the overlapping sets is equal to the difference of the cardinality of intersection of the sets from the sum of cardinalities of these sets. If A and B are two overlapping sets, then we have, n(A – B) = n(A ∩ B) = no(A) and n(B – A) = n(A ∩ B) = no(B). So, n(A) = no(A) + n(A ∩ B) and n(B) = no(B) + n(A ∩ B) n(A ∪ B) = n(A) + n(B) – n(A ∩ B) = no(A) + no(B) + n(A ∩ B) = n(A) + no(B) = no(A) + n(B) and n(U) = n(A ∪ B) + n(A ∪ B) n(U) = n(A) + n(B) – n(A ∩ B) + n(A ∪ B) n(U) = no(A) + no(B) + n(A ∩ B) + n(A ∪ B) n(U) = n(A) + no(B) + n(A ∪ B) n(U) = no(A) + n(B) + n(A ∪ B) 11. For intersecting three sets A, B and C, n(A) = no(A) + no(A ∩ B) + no(A ∩ C) + n(A ∩ B ∩ C) n(B) = no(B) + no(A ∩ B) + no(B ∩ C) + n(A ∩ B ∩ C) n(C) = no(C) + no(A ∩ C) + no(B ∩ C) + n(A ∩ B ∩ C) n(A ∪ B ∪ C) = no(A) + no(B) + no(C) + no(A ∩ B) + no(B ∩ C) + no(A ∩ C) + n(A ∩ B ∩ C) n(U) = n(A ∪ B ∪ C) + n(A ∪ B ∪ C) n(U) = no(A) + no(B) + no(C) + no(A∩B) + no(B∩C) + no(A∩C) + n(A∩B∩C) + n(A∪B∪C) A n(A) n(B) n( A ∩ B) n(U) B U A no(A) n(A) n(B) no(B) n( A ∩ B) n(A ∩ B) n(U) B U no(A) no(B) no(C) n(ABC) no(AB) no(BC) no(AB) A BU C n(U) n(A) n(B) n(A ∪ B ∪ C) n(C) 38 Allied The Leading Mathematics-9 Sets 39

Sets Example-1 If A = Set of colours in Nepal's flag, B = Set of the capital of Nepal C = Set of fingers of a hand of normal man and D = Set of the provinces of Nepal (a) Write the elements of the given sets in set notation. (b) What is cardinality of a set ? Define it. (c) Find the cardinality of the sets A, B, C and D. (d) Write the relation between cardinality of a set with one to one correspondence. Solution: (a) A = Set of colours in Nepal's flag = {Red, White, Blue} B = Set of the capital of Nepal = {Kathmandu} C = Set of fingers of a hand of normal man = {Thumb, Index finger, Middle finger, Ring finger, Little finger}, D = Set of the provinces of Nepal = {Koshi, Madhesh, Bagmati, Gandaki, Lumbani, Karnali, Sudurpashchim} (b) The number of unique elements in a set is called its cardinality. It is the size of the set. It is denoted n(A), |A| or #A# that means the cardinal number of the finite set. (c) Cardinality of A, n(A) = 3, Cardinality of B, n(B) = 1, Cardinality of C, n(C) = 5, Cardinality of D, n(D) = 7. (d) Cardinality of the given set is one to one correspondence with each element of the given set to each element of the second set of natural numbers. Example-2 It is given that a universal set U = {1, 2, 3, ...., 10} and its subsets sets A = {2, 4, 6, 8} and B = {1, 2, 3, 4, 5}. (a) Write the cardinality of the sets U, A and B. (b) Find cardinal numbers of A ∪ B, A ∩ B, A – B and B – A. (c) Represent the above results in a Venn diagram. (d) Why is n(A ∩ B) not more than n(A ∪ B) ? Give suitable reason. Solution: (a) Here, U = {1, 2, 3, ...., 10}, A = {2, 4, 6, 8} and B = {1, 2, 3, 4, 5} Cardinality of the sets U, n(U) = 10, Cardinality of the sets A, n(A) = 4 Cardinality of the sets B, n(B) = 5. (b) Now, we know that A ∪ B = {2, 4, 6, 8} ∪ {1, 2, 3, 4, 5} = {1, 2, 3, 4, 5, 6, 8} ∴ n(A ∪ B) = 7 Madhesh Bagmati Koshi Gandaki Lumbini Karnali Sudurpashchim 40 Allied The Leading Mathematics-9 Sets 41

Sets A ∩ B = {2, 4, 6, 8} ∪ {1, 2, 3, 4, 5} = {2, 4} ∴ n(A ∪ B) = 2 A – B = {2, 4, 6, 8} – {1, 2, 3, 4, 5} = {6, 8} ∴ n(A – B) = 2 B – A = {1, 2, 3, 4, 5} – {2, 4, 6, 8} = {1, 3, 5} ∴ n(B – A) = 3 A ∪ B = U – (A ∪ B) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {1, 2, 3, 4, 5, 6, 8} = {7, 9, 10} ∴ n(A ∪ B ) = 3 (c) Representing the above results in a Venn diagram, (d) n(A ∩ B) is not more than n(A ∪ B) because A ∩ B is the subset of A ∪ B. Example-3 In the adjoining figure, some different colourful pencils are on the table. Suppose B represents pencils in the blue case B and G, in the glass. Also, you can suppose the first letters of the colours of pencils in small letters. (a) Write the elements of the given sets and their cardinalities in set notation. (b) Show the above results in a Venn diagram. (c) Establish the following relations: (i) n(B ∪ G) = n(B) + n(G) – n(B ∩ G) (ii) n(U) = n(B) + n(G) – n(B ∩ G) + n(B ∪ G ) (d) Why is n(B) not less than n(B ∩ G)? Solution: (a) From the given picture, the set of colourful pencils in blue case, B = {green, yellow, red, pink, black, orange, sky, white} = {g, y, r, b, p, o, s, w} ∴ n(B) = 8 Set of colourful pencils in the glass, G = {brown, sky, red, green, orange, yellow, pink} = {b1, s, r, g, o, y, p} ∴ n(B) = 7 (b) Showing the above results in a Venn diagram by supposing n(B ∩ G) = x and n(B ∪ G ) = y. (c) Now, we have B ∪ G = {g, y, r, b, p, o, s, w} ∪ {b1, s, r, g, o, y, p} = {g, y, r, p, o, s, b, w, b1} ∴ n(B ∪ G) = 9 2 3 3 2 A B U 40 Allied The Leading Mathematics-9 Sets 41

Sets B ∩ G = {g, y, r, b, p, o, s, w} ∪ {b1, s, r, g, o, y, p} = {g, y, r, p, o, s} ∴ n(B ∩ G) = x = 6 (i) Now, 9 = 8 + 7 – 6 or, n(B ∪ G) = n(B) + N(G) – n(B ∩ G). Proved. (ii) Set of colourful pencils out of case and glass, B ∪ G = {blue, crimson} = {b2, c} ∴ n(B ∪ G) = y = 2 U = {g, y, r, p, o, s, b, w, b1, b2, c} ∴ n(U) = 11 Now, 11 = 8 + 7 – 6 + 2 or, n(U) = n(B) + N(G) – n(B ∩ G) + n(B ∪ G). Proved. (d) n(B) is not less than n(B ∩ G) because B is super set of (B ∩ G). Example-4 Suppose, A = {2, 4, 6} and B = {2, 3, 5, 7}. (a) Write the cardinalities of the given sets A, B and A ∩ B. (b) Illustrate the above information in a Venn diagram with their cardinalities. (c) Find n(A∪ Β) by using the relation n(A ∪ B) = n(A) + n(B) – n(A ∩ B). (d) Establish the relation between n(A ∪ B) with no(A), no(B) and n(A ∩ B). Solution: (a) Here, A = {2, 4, 6} ∴ n(A) = 3 B = {2, 3, 5, 7} ∴ n(B) = 4 and A ∩ B = {2} ∴ n(A ∩ B) = 1 (b) Illustrating the above information in a Venn diagram with their cardinalities. (c) Now, we have n(A ∪ B) = n(A) + n(B) – n(A ∩ B) = 3 + 4 – 1 = 7 – 1 = 6 (d) A – B = {2, 4, 6} – {2, 3, 5, 7} = {4, 6} ∴ n(A – B) = no(A) = 2 B – A = {2, 3, 5, 7} – {2, 4, 6} = {3, 5, 7} ∴ n(B – A) = no(B) = 3 ∴ 6 = 2 + 3 + 1 or, n(A ∪ B) = no(A) + no(B) + n(A ∩ B). n(A) = 3 4 2 6 3 5 7 A B U n(B) = 4 n(A∩B) = 1 42 Allied The Leading Mathematics-9 Sets 43

Sets PRACTICE 1.3 Read Think Understand Do Keeping Skill Sharp 1. (a) Define cardinality of a set. (b) What is the cardinality of a null set ? (c) Write the two examples of singleton set. (d) What is the cardinality of E = {6, 7, 8, 9, ........... 23} ? (e) If P = {u, n, i, o} and Q = {i, n, t, e, r, s, c, t, i, o, n} the cardinality of n(P ∩ Q) = ....... 2. Circle ( ) the correct answer. (a) What is the cardinality of singleton set? (i) 0 (ii) 1 (iii) 2 (iv) 3 (b) What is the cardinality of equivalent sets? (i) equal (ii) equivalent (iii) unequal (iv) non-equivalent (c) Which is the union of the sets N = {1, 3, 5, 7} and P = {2, 4, 8} ? (i) 5 (ii) 6 (iii) 7 (iv) 8 (d) What is the cardinality of the intersection of the sets M = {m, a, n, g, o} and P = {p, o, m, e, g, r, a, n, t} ? (i) 9 (ii) 8 (iii) 14 (iv) 13 (e) If P and Q are overlapping sets in the universal set U, then ............. (i) n(U) = n(C) + n(P) + n(C ∪ P) (ii) n(U) = n(C) + n(P) + n(C ∩ P) + n(C ∪ P) (iii) n(U) = n(C) + n(P) – n(C ∩ P) + n(C ∪ P) (iv) n(U) = no(C) + no(P) – n(C ∩ P) + n(C ∪ P) Check Your Performance Answer the given questions for each problem. 3. Think about the following sets and answer the given questions: C = Set of colours in rainbow, H = {Hands of a normal human being}, L = Set of the longest river of the world, Y = {Name of your parents}, P = {Pages of your Allied Maths}, D = Set of the districts of Nepal, N = {National parks of Nepal}, S = Set of the capitals of SAARC countries, A = {Sum of angles of first 5 regular polygons}, M = {World heritage sites in Nepal}, M = Set of the height of mountains more than 8000 m in Nepal} [If you do not know the name of different things mentioned above, you can use google search.] 42 Allied The Leading Mathematics-9 Sets 43

Sets (a) Write the elements of the given sets in set notation. (b) Define cardinality of a set. (c) Find the cardinality of the above mentioned sets. (d) Which set has the greatest cardinality and that of the least cardinality ? 4. Answer the given questions for the following numbers (a) and (b) both. (a) It is given that a universal set U = {1, 2, 3, ...., 12} and its subsets A = {1, 3, 9, 12} and B = {2, 3, 5, 7, 11}. (b) If A = {Even Prime numbers} and B = {Whole numbers less than 6}, then find cardinal numbers of A ∪ B, A ∩ B, A – B and B – A. Also, represents in a Venn diagram. (i) Write the cardinality of the sets U, A and B. (ii) Find the cardinal numbers of A ∪ B, A ∩ B, A – B and B – A. (iii) Represent the above results in a Venn diagram. (iv) Why is n(A ∩ B) not more than n(A ∪ B) ? Given suitable reason. 5. If M = Set of multiples of 5 less than 40 and E = Set of factors of 24, then; (a) Write the elements of the given sets and their cardinalities in set notation. (b) Show the above information in a Venn diagram. (c) Find n(M ∪ E), n(M ∩ E), n(M – E) and n(E – M). (d) Establish the relation n(M ∪ E) = n(M) + n(E) – n(m ∩ E). 6. Observe the given Venn diagram and answer the following questions; (a) Write the elements of the sets mentioned in the adjoining Venn diagram in roster and set-builder forms. (b) Find the cardinalities of U, M, N, M ∩ N and M ∪ N. (c) Show the above cardinalities in another Venn diagram. (c) Find n(M) and n(M ∩ N). (d) Establish the relation between n(U), n(M), n(N), n(M ∩ N) and n(M ∪ N). (e) Prove that: n(U) = no(M) + n(N) + n(M ∪ N). 7. Observe the given Venn diagram and answer the following questions; (a) Write the elements of the sets mentioned in the given Venn diagram in roster form. (b) Find the following cardinalities: (i) n(U), n(P), n(Q), n(R) (ii) n(P ∪ Q), n(Q ∩ R) (iii) no(P ∩ R), no(P ∩ Q) (v) no(P), no(R) (iv) n(P ∩ Q ∩ R), n(P ∪ Q ∪ R) (vi) n(P ∪ (Q ∩ R)), n(Q ∩ (P ∪ R)) (vii) n(P ∪ R), n(P ∩ Q) (viii) n(P ∪ Q ∪ R), n(P ∩ Q ∩ R) (c) Prove that: (i) n(P ∪ Q ∪ R) = n(P ∪ Q) + no(R) (ii) n(U) = n(P ∪ Q ∪ R) + n(P ∪ Q ∪ R) 1 2 3 6 9 10 5 7 4 8 M N U c h s o e k p f m j n a d q r P R Q U 44 Allied The Leading Mathematics-9 Sets 45

Sets 8. Observe the given Venn diagram of the cardinality and answer the following questions; (a) Find the cardinalities of n(U), n(A), n(B) and n(C). (b) Prove that: (i) n(A ∪ B ∪ C) = n(A ∪ B) + no(C) (ii) n(A ∪ B ∪ C) = n(A) + no(B) + no(C) + no(B ∩ C) (iii) n(A) = no(A) + no(A ∩ B) + no(A ∩ C) + n(A ∩ B ∩ C) (iv) n(A) = no(B) + no(C) + no(B ∪ C) + n(A ∪ B ∪ C) (v) n(U) = n(A ∪ B ∪ C) + n(A ∪ B ∪ C) (vi) n(A∪ B ∪ C) = n(A) + n(B) + n(C) – n(A∩ B) – n(A∩ C) – n(B∪C) + n(A∩B∩C) (vii) n(U) = no(A) + n(B ∪ C) + n(A ∪ B ∪ C) 9. Observe the different balls of Saru and Karina in the adjoining pictures. Name their balls in set notation by supposing S for the set of Saru's balls and K for the set of Karina's balls. (a) Write the elements of the given sets and their cardinalities in set notation. (b) Show the above information in a Venn diagram. (c) Find n(S), n(K), n(S ∪ K) and n(S ∩ K). (d) Is n(S ∪ K) = n(K)? Why? 10. If F = Set of factors of 18 and O = Set of odd numbers from 2 to 15, then; (a) Write the elements of the given sets in set notation. (b) Find n(F), n(O), n(F ∩ O), n(F – O) and n(O – F). (c) Show the above cardinalities in a Venn diagram. (d) Prove that: n(F ∪ O) = no(F) + no(O) – n(F ∩ O) (e) Why is no(E) not more than n(E) ? Give suitable reason. 11. Answer the given questions for the following sets in the numbers (a) and (b): (a) Suppose, P = {x: x is a prime number less than 15} and F = {x: x is a factor of 30} then, (b) Suppose, P = {Square numbers up to 100} and F = {Multiples of 9 less than 100} then, (i) Write the elements of the given sets in set notation. (ii) Write the relation between n(P ∪ F), n(P), n(F), and n(P ∩ F). Find in (P ∪ F). (iii) Write relation n(P ∪ F), no(P), no(F) and n(P ∩ F). Find n(P ∪ F) (iv) Show the above cardinalities in a Venn diagram. (v) Why is no(E) not more than n(E) ? Give suitable reason. 12. If n(A ∪ B) = 25, n(A) = 12 and n(B) = 17 then, (a) Define the cardinality of the union of A and B. (b) Illustrate the above information in a Venn diagram. (c) Find n(A ∩ B) and n(A ∆ B). (d) Write the relation between n(A ∪ B), n(A ∩ B) and n(A ∆ B). 8 2 3 5 6 7 4 4 A C B U Saru's Balls Karina's Balls 44 Allied The Leading Mathematics-9 Sets 45

Sets 13. If n(U) = 42, n(A) = 15, n(B) = 25 and n(A ∪ B) = 9 then, (a) Define the cardinality of the intersection of A and B. (b) Present the above information in a Venn diagram. (c) Find n(A ∩ B) and n(A ∆ B). (d) Write the relation between n(A ∪ B), n(A ∩ B) and n(A ∆ B). 14. It is given that n(U) = 45, n(A) = 25, n(B) = 28 and n(A ∩ B) = 13. (a) What is the meaning of n(A∪B) ? (b) Illustrate the above information in a Venn diagram. (c) Find n(A∪B), n(at most one set) and n(only one set). (d) Write the relation between n(U), n(A ∩ B), n(A ∆ B) and n(A∪B). 15. In a school there are 500 students. 60% of the students like mathematics and 50% of the students like science. (a) Represent this information in a Venn diagram. (b) Find the number of students who like mathematics only. (c) Find the number of students who like science only. (d) Find the number of students who like both mathematics and science. 16. In a group of 90 students 54 take English, 51 take Newari and 9 take neither. How many students take both subjects. Draw Venn-diagram to help you answer the questions. 17. In a group of 74 people, if the ratio of children who like Coke only and Pepsi only is 3:2 and the number of people, who like both drinks is 17 and who do not like both is 7, find the number of people who like Coke and represent the fact in a Venn diagram. 4. (a) (i) 12, 4, 5 (ii) 8, 1, 3, 4 (b) 3, 1, 0, 2 5. (a) {5, 10, 15, 20, 25, 30, 35}, {2, 4, 6, ..., 34}, 7, 17 (b) 21, 3, 4, 14 6. (a) {1, 2, 3}, {2, 4, 6, 8} (b) 10, 3, 4, 1, 4 (d) 7, 9 7. (a) {c, e, h, k, o, s}, {a, h, n, p, q, r, s}, {a, e, h, j, n, o}, {a, c, d, e, f, g, h, j, k, l, m, n, o, p, q, r, s} (b) (i) 15, 6, 7, 6 (ii) 11, 3 (iii) 2, 1 (iv) 1, 12 (v) 2, 1 (vi) 8, 4 (vii) 6, 13 (viii) 3, 14 9. (c) 6, 8, 8, 6 10. (a) {1, 2, 3, 6, 9, 18}, {3, 5, 7, 9, 11, 13, 15} (b) 6, 7, 2, 4, 5 (d) 11, 4, 5, 2 11. (a) (i) {2, 3, 5, 7, 11, 13}, {1, 2, 3, 5, 6, 10, 15, 30} (ii) 6, 8, 3 (iii) 3, 5, 11 (b) (i) {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}, {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99} (ii) 18 (iii) 18 12. (c) 4, 21 13. (c) 7 14. (c) 5, 32, 27 15.(b) 250 (c) 200 (d) 50 16. 24 17. 47 Answers Project Work List out the name of the you and your friend's family members and represent them in set notation by taking any two capital letters for sets and every beginning small letter of the family members for elements. Then find the number of all the family members of you and your friend by computing union of sets and using formula. Also, make a report for it and present in your classroom by constructing a Venn diagram. 46 Allied The Leading Mathematics-9 Sets 47

Sets CONFIDENCE LEVEL TEST - I Unit I : Sets Class: 9, The Leading Maths Time: 45 mins. FM: 24 Attempt all questions. 1. Let U = {Natural numbers up to 15} and its subsets A = {x: x is a factor of 12} and B = {y: y is a multiple of 3}. (a) List out the elements of U, A and B. [1] (b) Illustrate the above information in a Venn diagram. [1] (c) Prove that: A ∆ B = (A ∪ B) – (A ∩ B). [3] (d) Write the relation between A ∆ B and A∪B. [1] 2. Find the union of the sets A = {x : x is a number from 5 to 20} and B = {x : x is a prime number}. (a) List the elements of each set. [1] (b) Find all the items in the sets A or B or both. [3] (c) Show all the items in the same Venn diagram. [1] (d) Why is A ∪ B equal to B ∪ A ? Give suitable reason. [1] 3. The relation of sets U, A, B and C are shown below in Venn diagram. (a) Write the cardinality n(A ∪ B ∪ C). [1] (b) Find the value of C – B. [1] (c) Prove that: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) [3] (d) When n(A) = n(B), Is A = B?. Justify with reason. [1] 4. P and Q are the subsets of a universal set U where U = {x: x < 12, x ∈ W}, P = {y: y = 2n, n > 8} and Q = {z: z = n + 2, n < 3}. Answer the following questions. (a) List the elements of U, P and Q. [1] (b) Show the given information in Venn Diagram. [1] (c) Establish the relation n(P ∪ Q) = n(P) + n(Q) – n(P ∩ Q) [3] (d) Write the relation between P – Q and Q – P. [1] Best of Luck 7 5 2 0 4 1 A C B U 46 Allied The Leading Mathematics-9 Sets 47

Sets Additional Practice – I 1. Sets A = {multiples of 3}, B = {factors of 6} and C = {odd numbers} are the subsets of a universal set U = {natural numbers from 1 to 10}. (a) Write the cardinality of U. (b) Show the above information in a Venn-diagram. (c) Prove that: A – (B ∪ C) = (A ∪ B) – (B ∪ C) (d) What type of sets are A ∪ B ∪ C and B ∪ C? Write with reason. 2. Sets U, A and B are given below. U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 3, 5, 7} and B = {2, 3, 4, 8} (a) Write the cardinality of A. (b) Represent the above information in the Venn diagram. (c) Prove that: (A ∩ B) = A ∪ B (d) Is n (A – B) and n (B – A) always equal? Justify it. 3. Sets U, E and F are given below. U = {natural number less than 15}, E = {even number less than 12} and F = {factors of 12}. (a) Define cardinality of the set. (b) Find the value of E ∩ F. (c) Prove that: n(E ∩ F) = 9 (d) Are n(E ∩ F) = n(E ∪ F)? Justify. 4. The relation of sets U, A, B and C are shown below in Venn diagram. (a) Write the cardinality A ∩ B ∩ C. (b) Find the value of A – B. (c) Prove that: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (d) Is n(A ∪ B ∪ C ) + n(A ∪ B ∪ C) = n(U)? Justify with reason. 5. The relation of sets U, A and B are shown below in Venn diagram. (a) Write the cardinality of Set A. (b) Find the value of A ∪ B. (c) Prove that: n(A ∩ B) = n(A ∪ B) (d) In what condition n(A ∪ B) = n(U)? Justify with reason. 6. Two sets O and E are as given below in the description method. O = Set of odd numbers less than 11, E = Set of even numbers less than 11 (a) Write the cardinality of set O. (b) Show the above information in a Venn diagram. (c) Prove that: O – E ≠ E – O but n (O – E) = n (E – O) (d) Here, O – E = O? Is it always true? Why it so? 7 3 5 2 6 4 1 A C B U 7 3 5 2 6 1 A B U 48 Allied The Leading Mathematics-9 Sets PB

Arithmetic Arithmetic Solving the arithmetic problems related to daily life. Estimated Working Hours : Competency Learning Outcomes At the end of this unit, the students will be able to: state the concept of tax, commission and dividend. solve the problems on tax, commission and dividend. solve the problems related to home arithmetic. Lessons for Unit 2. Tax 2.1 Income Tax 2.2 Value Added Tax 3. Commission and Dividend 3.1 Commission 3.2 Bonus and Dividend 4 Home Arithmetic 4.1 Use of Electricity 4.2 Use of Water 4.3 Use of Telephone 4.4 Use of Taxi meter Chapters / Lessons ARITHMETIC UNIT 28 (Th. + Pr.) HSS Way: Mathematics begins at Home, grows in the Surroundings and takes shape in School. II Specification Grid Unit Areas Total working hour Knowledge Understanding Application Higher ability Total number of items Total number of questions Total Marks No. of items Marks No. of items Marks No. of item Marks No. of item Marks II Arithmetic 28 2 2 2 3 3 5 2 3 9 3 13 PB Allied The Leading Mathematics-9 Tax 49

Arithmetic Arithmetic WARM-UP Number System: A number system or system of numeration is the representation of numbers by using symbols or digits in a consistent way. In digital electronics, a mathematical value or a number is used to measure or count different objects and to perform arithmetic calculations. Classification of Decimals (Real Numbers) Name Definition Set of Numbers Operations Natural Numbers Numbers used for counting objects starting from 1, 2, 3, …. N = {1, 2, 3, ....} Addition and multiplication Whole Numbers Numbers used for counting objects with nothing starting from 0, W = {0, 1, 2, 3, ...} Addition, multiplication and subtraction for equal numbers Integers Numbers used for counting objects with nothing and due thing/s Z = {..., – 3, – 2, – 1, 0, 1, 2, 3, .....} Addition, multiplication and subtraction Fractions Numbers used for representing a part or some parts from whole parts F = {x: x = p/q; p, q ∈ W, q ≠ 0} Addition, multiplication, and division Decimal Numbers Numbers with a whole and a fractional part or fraction with denominator 10 or multiple of 10 D = {x:x = r. p q; p, q, r ∈ Z, q ≠ 0 or q = 10} Addition, multiplication, subtraction and division Terminating Decimal Decimal numbers with finite number of digit/s after the decimal point whose denominator as factor or multiple of 10 T ={x: x = p q; p, q ∈ Z, q ≠ 0 or q = factor or multiple of 10} Addition, multiplication, subtraction and division Recurring Decimal Decimal numbers with repeating digits in specific pattern or at regular intervals after the decimal point C ={x:x = p/q;p,q,r ∈ Z, q ≠ 0 or q ≠ factor or multiple of 10} Addition, multiplication and subtraction Rational Numbers Numbers used for dividing or quotient of integers; fractions, terminating and recurring decimal numbers Q ={x:x = p q; p, q ∈ Z, q ≠ 0} Addition, multiplication, subtraction and division Irrational Numbers Numbers that cannot be expressed as quotient of two integers; nonterminating and non-recurring decimal numbers. Q ={x:x ≠ p q; p, q ∈ Z, q ≠ 0}, different order of surd and exponent of numbers Addition, subtraction and multiplication Real Numbers Union of rational and irrational numbers in the number system. R ={x:x = Q ∪ Q} Addition, multiplication, subtraction and division Scientific Notation A form of presenting very large or small numbers in a simpler form of decimal number with exponent of 10. 120000 = 1.2 × 105 0.00025 = 2.5×10– 4 Addition, multiplication, subtraction and division 50 Allied The Leading Mathematics-9 Tax 51