FOCUS-ON TEXTBOOK MATHS 8 Singapore Maths Approach Progressive Practices 21st Century Learning Skills Formative & Summative Assessments Digital Resources Enhancements
FOCUS-ON MATHS TEXTBOOK 8 First Published 2023 6001 Beach Road, #14-01 Golden Mile Tower, Singapore 199589. E-mail: [email protected] ISBN 978-981-17293-2-4 Distributed by PT. Penerbitan Pelangi Indonesia Ruko the Prominence, Block 38G No. 36, Jl. Jalur Sutera, Alam Sutera, Tangerang, 15143, Indonesia. Tel: [021]29779388 Fax: [021]30030507 Email: [email protected] Printed in Malaysia by Herald Printers Sdn. Bhd. Lot 508, Jalan Perusahaan 3, Bandar Baru Sungai Buloh, 47000 Selangor Darul Ehsan. 2023 All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, photocopying, mechanical, recording or otherwise, without the prior permission of
II FOCUS-ON MATHS is an exciting new series that has been developed to match the latest Indonesian Mathematics syllabuses (Phase D) for Grade 7 to Grade 9. The topic coverage in each grade is arranged to address all the learning achievements (Capaian Pembelajaran) as prescribed by the Indonesian Ministry of Education. The series adopts the Singapore Maths method which is a world-class maths teaching approach. FOCUS-ON MATHS is a complete mathematics programme that comprises the Textbook, Workbook, Teacher’s Guide and online resources. It is a comprehensive, task-based and learner-centred programme designed to cultivate students’ interest in the learning of Mathematics, equip them with an in-depth understanding of Mathematical concepts and help them to achieve their fullest potential in mathematics. The content is structured to develop 21st Century Skills and Higher Order Thinking Skills in students. Students are challenged with problem-solving tasks and problems in real-word contexts. This enables them to think independently and build foundations for many of the advanced applications of mathematics that are relevant to today’s world. A collection of STEM activities is also integrated in this series to foster inquiring minds, logical reasoning and collaboration skills in students. FOCUS-ON MATHS aims to develop a greater awareness of the nature of mathematics and bring greater connections between different topics in mathematics, as well as between mathematics and other subjects, creating a deeper and more robust understanding of mathematics. PREFACE 21st Century Learning Skills Indonesian Maths Syllabuses s Singapore Maths Method FOCUS-ON MATHS Formative and Summative Assesment Digital Resources Enrichment Project - Based Learning STEM Activities
III KEY FEATURES • Introduces the chapter in a real-world setting, allow students to realise the relevance and utility of mathematics in our daily lives. • Poses mind-stimulating questions for students to think creatively, logically and analytically. • Presents the important mathematical terms in the chapter. • Outlines the learning outcomes that will be covered in the upcoming lesson. • Provide an overview of the chapters in the book which sets the learning pace. • Recounts the history and development of mathematics and the contributions of great mathematicians. APPLICATION OF THIS CHAPTER CONCEPT MAP LEARNING OUTCOMES KEY TERMS MATHS HISTORY CREATIVE THINKING
IV • Provides students opportunities to work in small groups to explore mathematical concepts. • Challenges students with questions that promote critical and reflective thinking. • Prepares various types of activities aimed at involving students individually, in pairs or in groups inside or outside the classroom. • A quick assessment of students’ prior knowledge of the concepts learnt previously. • Demonstrates the steps in solving mathematical problems and techniques in answering questions. LET’S EXPLORE FLASHBACK WORKED EXAMPLE CRITICAL THINKING INTERACTIVE ZONE
V • Direct students to websites with extra learning materials or resources to enhance the learning experience. LET’S INVESTIGATE • Encourages students to explore the learning contents by themselves and involves them in active discussions during the lesson. SCAN ME
VI • Remind students of key information or support that is useful to tackle an exercise, or simply useful to know. • Summarise important terms, definitions and mathematical properties to be used as helpful reinforcements. • Provides extra mathematical, scientific or historical facts and information related to the topic. MATHS INFO MATHS TIPS
VII • Let students explore, model and solve non-routine prolems in real-world contexts to devlop students’ problem-solving skills with prompts to guide students using Polya’s 4-step approach. PROBLEM SOLVING
VIII • Provides opportunities to students to explore how certain mathematical concepts can be connected or related to other concepts. • Presents different methods of working in a practical and easy-to-follow way. ALTERNATIVE METHOD MATHS LINK
IX CALCULATOR CORNER • Shows the use of scientific calculators in calculations • Helps pupils practice answering questions that promote higher order thinking skill. HOT CHALLENGE
X • Provides questions with different difficulty levels that allow students to practise methods that have just been introduced. These range from simple ‘recall and drill’ activities to applications and problem-solving tasks. SUMMARY • Sums up and highlights important concepts and formulae for quick revision. PRACTICE • Prepares various types of activities aimed at involving students individually, in pairs or in groups inside or outside the classroom. TEAM WORK
XI MASTERY PRACTICE • Provides a further exercise to test students’ mastery of the concepts and skills learnt in each chapter. Students can download the free GeoGebra Classic software program to open the related files in . https://www.geogebra.org/download
XII CONTENTS CHAPTER 1 Sets XIV 1.1 Sets 3 1.2 Properties of Sets 9 1.3 Operations of Sets 18 CHAPTER 2 Algebraic Expressions 36 2.1 Simplifying Algebraic Expressions 39 2.2 Determining the Values of Algebraic Expressions 48 2.3 Applying Algebraic Expressions 50 CHAPTER 3 Linear Equations in Two Variables 60 3.1 Linear Equations in Two Variables 63 3.2 Simultaneous Linear Equations in Two Variables 67 CHAPTER 4 Linear Functions 80 4.1 Relations 83 4.2 Functions 87 4.3 Linear Functions 92 4.4 Graphing Linear Functions 97 4.5 Equations of Straight Lines 109 4.6 Solving Problems Involving Linear Functions 120
XIII CHAPTER 6 Triangles and Quadrilaterals 168 6.1 Triangles and Quadrilaterals 171 6.2 Geometric Properties and Types of Triangles 173 6.3 Perimeter and Area of Triangles 176 6.4 Geometric Properties and Types of Quadrilaterals 179 6.5 Perimeter and Area of Quadrilaterals 183 6.6 Solving Problems Involving Perimeter and Area 189 6.7 Perimeter and Area of Irregular Geometrical Plane 194 CHAPTER 7 Statistics 202 7.1 Mean, Mode and Median 205 7.2 Measures of Dispersion 214 7.3 Measures of Central Tendency 217 CHAPTER 5 Polygons and Congruency 128 5.1 Polygons 131 5.2 Interior Angles and Exterior Angles of Triangles 138 5.3 Interior Angles and Exterior Angles of Polygons 142 5.4 Congruence 151
1SETS Applications of this chapter The objects distributed across the world can be grouped under several groups based on their common characteristics. For example, buildings and human beings are categorised under two different groups where the building is categorised as a non-living thing whereas a human is categorised as a living thing. The photograph above show many groups of objects with certain characteristics. Can you identify the characteristics of each group of objects? XIV
1 Learning Outcomes • Explain what a set is. • Define a set using set notation and description. • Represent sets using Venn diagrams. • Recognise universal sets and empty sets. • Identify the cardinality of sets. • List and state the number of elements in a set. • Understand and use the concept of subset, universal set and complement of a set. • Understand intersection between sets. • Understand the union of sets. • Perform operations on sets. Operations on Sets Venn Diagram Sets Type of Sets Unary Binary Intersection of Sets Universal Sets Empty Set Union of Sets Subsets Difference of Sets Power Sets Complement of Sets The Elements of Sets Representing Sets Concept of Sets Word description Listing the elements Set Notation • Set • Set notation • Universal set • Empty set • Venn Diagram • Cardinality • Element • Subset • Complement • Power set • Equality • Intersection • Disjoint • Union • Difference Key Terms Maths History Georg Cantor was a Russian Mathematician who has discovered the theory of set that then become the fundamental theory in mathematics. Set theory is one of the greatest achievements in modern mathematics. Set theory has served unique role by systemising mathematical arguments. Concept Map
2 CHAPTER 1 Sets Flashback 1. List all the vowels in the alphabet. 2. List all the positive numbers less than 10. 3. List the colours in the rainbow. 4. List the things that we could wear. 5. List all the colours on our flag. 6. List the first seven prime numbers. In our daily life, we can classify the recyclable materials into several categories, such as paper, plastic, glass, aluminium, fabric and so on. But not all materials can be recycled. Items that cannot be recycled should be deposited correctly into trash cans or waste bin to ensure that the environment is preserved and sustained. (a) Can you help to sort the waste material? Draw lines to put them into the correct bins. (b) How can these categories of materials be represented mathematically? 1 How can the classification of solid waste be carried out effectively? Critical Thinking PAPER GLASS PLASTIC METAL
3 Sets CHAPTER 1 1.1 Sets A Representing sets (I) Introduction to sets From the results of Activity 1, the objects which have common characteristics will be classified into the same group. Each of these group is known as set. A set is a collection of objects which have a common property. The objects in a set are called the elements or members of the set. Sets are often symbolised by capital letters and the elements of sets are usually represented by small letters. A set can be defined by using description and set notation, { }. The table below shows three methods to describe a set. Method Example Word description A = Set of first five prime numbers Listing the elements A = {2, 3, 5, 7, 11} Set notation A = {x : 2 < x < 11, x is a prime number} Objective: To classify objects. Instruction: Perform the activity in groups 1. Observe each object given below. 2. Classify the objects according to specific groups. 3. What are the common characteristics of each group? 1
4 CHAPTER 1 Sets EXAMPLE 1 Mango Bus Banana Train Cat Pineapple Ship Snake Cow Classify all the objects in the diagram above into fruits, vehicles and animals. Solution: Fruits: Pineapple, mango, banana Vehicles: Bus, ship, train Animals: Cat, cow, snake EXAMPLE 2 A set A consists of the numbers 0, 2, 4, 6, 8, 10, 12 and 14. Describe the set A by using (a) description, (b) listing, (c) set builder notation. Solution: (a) A is the set of even numbers which are less than 15. (b) A = {0, 2, 4, 6, 8, 10, 12, 14} (c) A = {x : x is an even number and x , 15} Set notation { } is used when we describe a set using listing and set builder notation. (II) Identifying the elements of a set The symbol is used to denote ‘is an element of’ or ‘is a member of’. For example, p is an element of, or is a member of the set {p, q, r}, so we write p {p, q, r} and read it as ‘p is an element of the set {p, q, r}’ or ‘p is a member of the set {p, q, r }’. The symbol is used to denote ‘is not an element of’ or ‘is not a member of’. For example, s is not an element of, or is not a member of the set {p, q, r }, so we write s {p, q, r} and read as ‘s is not an element of the set {p, q, r }’ or ‘s is not a member of the set {p, q, r }’. EXAMPLE 3 If A = {odd numbers less than 12} and B = {factors of 18}, fill in each of the following boxes with the symbol or . (a) 7 A (b) 5 B (c) 13 A (d) 18 B
5 Sets CHAPTER 1 Solution: A = {odd numbers less than 12} B = {factors of 18} = {1, 3, 5, 7, 9, 11} = {1, 2, 3, 6, 9, 18} (a) 7 A (b) 5 B (c) 13 A (d) 18 B B Universal sets and empty sets (I) Universal sets The universal set, x, is a set containing all elements under a particular discussion. When sets within a universal set have no common elements, they are called disjoint sets. EXAMPLE 4 The diagram shows the relationship between the universal set x, set P and set Q respectively. Set P and set Q are disjoint sets. List the elements of (a) x, (b) P, (c) Q. Solution: (a) x = {a, b, c, d, e, f } (b) P = {a, d } (c) Q = {c, e, f } (II) Empty sets The empty set or null set is the set which has no elements. It is denoted by the symbol f or { }. {0} is not an empty set. It is a set which has one element, that is, zero. {f} is not the symbol for empty set. If a set has only one element and that element is 0, it does not mean that the set is an empty set. For example, Given that set A = {0}. This means set A contains only one element, which is 0. Set A is not an empty set. For a set A contained in the universal set , n(A) < n(). d a f c e b P Q ξ EXAMPLE 5 Determine whether each of the following set is an empty set. (a) K is a province in Indonesia which its name starts with the letter Q. (b) L = {prime numbers between 1 to 10} (c) M = {x : x is a factor of 24 and x . 25} (d) N = {x : x is a polygon that has less than 6 sides} • When listing a set which contains too many elements, an ellipsis sign is used to represent the sequence of elements described in the set. For example, A = {x : x is a multiple of 5 and x < 1000}. Therefore, set A can be written as A = {5, 10, 15, ..., 1000}.
6 CHAPTER 1 Sets Solution: (a) There is no province in Indonesia which the name starts with letter Q. Therefore, K = f. (b) L = {2, 3, 5, 7}. Therefore, L ≠ f. (c) The factor of a number cannot be greater than that number. Therefore, M = f. (d) N = {triangle, quadrilateral, pentagon}. Therefore, N ≠ f. EXAMPLE 6 Determine whether each of the following sets is an empty set. (a) P = {odd numbers which are divisible by 2} (b) Q = {months that start with the letter Z} (c) R = {prime numbers less than 10} Solution: (a) P = f (b) Q = f (c) R ≠ f {0} is not an empty set. It has one element, which is zero. R = {2, 3, 5, 7} • When describing a set using listing, the same elements need not to be repeated. For example, set P = {x : x are all the letters in the word “SUCCESS”}. Therefore, the listing for the set P is P = {S, U, C, E}. C Venn diagram Venn diagrams are used to give a pictorial view of the relationships of sets and subsets within the universal set. Usually, the universal set is represented by a rectangle and the other sets are represented by closed geometrical shapes such as ovals, circles, squares and so on. In a Venn diagram, each element or member of the set is represented by a dot. ξ P b c d f g a e Each dot represents an element. Universal set is represented by a rectangle. All the elements of P are written in the circle. Set P is represented by a circle. The letter that represents a set is written outside and closer to the circle. Elements of complement of set P are written randomly outside the circle.
7 Sets CHAPTER 1 EXAMPLE 7 Represent the following sets using a Venn diagram. (a) A = {2, 4, 6, 8} (b) B = {x : x is a positive integer less than 6} Solution: (a) 2 4 6 8 A (b) 1 3 2 4 5 B EXAMPLE 8 Given that = {x : x is an even number and 1 , x < 20}, A = {2, 6, 10, 16, 20} and B = {multiples of 6}. Represent the relation between each of the following pair of sets by using Venn diagrams. (a) Sets and A (b) Sets and B In a Venn diagram, the number without a dot represents the number of elements in a set. For example, Usually, in a Venn diagram, the universal set is represented by a rectangle. • 7 7 A = {7} n(A) = 7 7 is an element of set A. Set A contains 7 elements. Solution: = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} (a) A = {2, 6, 10, 16, 20} ξ A 2 10 6 16 20 18 14 12 8 4 (b) B = {6, 12, 18} ξ B 6 12 18 10 16 14 8 20 2 4
8 CHAPTER 1 Sets EXAMPLE 9 ξ M b s f p d w e The Venn diagram shows the universal set and set M. List the elements of set M. Solution: = {b, d, e, f, p, s, w} M = {b, s, f} Practice 1.1 Basic Intermediate Advanced 1 Define each of the following sets by using description. (a) {2, 3, 5, 7, 11} (b) {2, 4, 6, 8, 10, 12} (c) {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} (d) {a, e, i, o, u} B Define each of the following by using set notation. (a) A = The set of whole numbers greater than 20 but less than 27. (b) B = The set of months of year beginning with J. (c) C = The set of colours in the traffic lights. (d) D = The set of odd numbers between 30 and 40. C A = {factors of 12} B = {x : 10 , x , 20, x is a prime number} C = {vowels in the word MODERN} Fill in each of the following boxes with the symbol or . (a) 4 A (b) 13 B (c) i C (d) 8 A D Determine whether each of the following is an empty set. (a) P = {cats which have ten heads} (b) Q = {prime numbers between 17 and 20} (c) R = {cubes with 7 sides} (d) S = {even numbers between 1 5 and 4 5 } E ξ 1 5 6 8 7 2 3 4 P Q The above Venn diagram shows the relationship between the universal set, x, set P and set Q. List the elements of (a) x, (b) P, (c) Q. F Represent each of the following sets by using a Venn diagram. (a) P = {multiples of 3 between 15 and 30} (b) Q = {consonants in the word SPIDER} (c) R = {red, blue, green} (d) S = {x : 11 < x < 17, x is an even number}
9 Sets CHAPTER 1 1.2 Properties of Sets A Cardinality of sets Cardinality means the number that we obtained after counting. Hence, the cardinality of a set is the total number of elements in the set. It also refers to the size of a set. For example, Set A = {p, q, r, s} and set B = {p, q, r}. Set A has cardinality of four compared to set B which only has cardinality of three. The cardinality of a set A can be denoted by n(A). (I) The number of elements of a set The number of elements in set A is denoted by the notation n(A). For example, A = {1, 3, 5}, thus n(A) = 3. In a Venn diagram, the number of elements in a set is represented by a number without a dot beside it. For example, the Venn diagram below indicates that n(A) = 4, that is, set A contains four elements. A 4 Discuss with your classmates, If A is an empty set, what can you say about n(A)? INTERACTIVE ZONE EXAMPLE 10 State the number of elements in each of the following sets. (a) A = {rose, orchid, tulip} (b) B = {factors of 8} (c) C = {x : x is a multiple of 5 less than 40} Solution: (a) A = {rose, orchid, tulip} n(A) = 3 (b) B = {factors of 8} = {1, 2, 4, 8} n(B) = 4 (c) C = {x : x is a multiple of 5 less than 40} = {5, 10, 15, 20, 25, 30, 35} n(C) = 7 List all elements in a set so that we can determine the number of elements easily and clearly.
10 CHAPTER 1 Sets EXAMPLE 11 Determine the number of elements for each of the following sets. (a) P = {car, bus, van, bicycle} (b) Q = {colours in a rainbow} (c) R = {x : x is a two-digit number such that the sum of the two digits is equal to 9} (d) S = {vocals in the word ‘PEMBELAJARAN’} Solution: (a) P = {car, bus, van, bicycle}, n(P) = 4 (b) Q = {red, orange, yellow, green, blue, indigo, violet}, n(Q) = 7 (c) R = {18, 27, 36, 45, 54, 63, 72, 81, 90} n(R) = 9 (d) S = {E, E, A, A, A}, n(S) = 5 B Subsets From the results of Activity 2, it is found that set A is a subset of set B if every element of set A is an element of set B. The symbol is used to denote ‘is a subset of’. For example, set A is a subset of set B, so we write A , B. The symbol is used to denote ‘is not a subset of’. For example, set P is not a subset of set Q, so we write P Q. An empty set is a subset of any set. For example, f , A. A set is a subset of itself. For example, A , A. Objective: To identify the subsets of a set. Instruction: Perform the activity in groups of four. 1. Prepare number cards labelled as follows. 2. Mark the two boxes to represent the following sets respectively: A = {multiple of 2} B = {multiple of 4} 3. Put the number cards in the correct box. 4. What do you notice the values of the numbers in the two boxes? 5. What is the relationship between set A and set B? 2
11 Sets CHAPTER 1 EXAMPLE 12 Given that A = {odd numbers between 2 and 11}, B = {3, 5, 7} and C = {factors of 4}, determine whether each of the following is true or false. (a) B , A (b) C , A (c) C , B (d) B C A set with n elements contains 2n number of subsets. e.g. {x, y, z} = 2n = 23 = 8 ∴ There are 8 subsets altogether. Solution: A = {odd numbers between 2 and 11} = {3, 5, 7, 9} B = {3, 5, 7} C = {1, 2, 4} (a) B , A is true. (b) C , A is false. (c) C , B is false. (d) B C is true. If set A is the subset of set B, then the relationship between A and B can be illustrated by a Venn diagram. Notice that set A is within set B. We can also say that A is contained in B. Every element of B is an element of A. All the elements of C are not the elements of A. EXAMPLE 13 Draw a Venn diagram to show the relationship between the sets in each of the following pairs. (a) J = {x, y, z}, K = {x, y} (b) P = {factors of 6}, Q = {whole numbers less than 10}. Solution: (a) K , J (b) P = {1, 2, 3, 6} Q = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} P , Q J K x y z Q P 1 2 3 6 9 8 0 4 5 7 Critical Thinking If F , G and H , G, is F = H? Explain your answer with the help of a Venn diagram.
12 CHAPTER 1 Sets Sam is a manager at a food processing company. He wants to classify his staff according to departments. The following are the sets A, B, C and D, that are defined by Sam. A = All staff in the processing and quality control departments B = Staff in the processing department C = Staff in the quality control department D = Staff in the packaging department (a) Based on the information given, draw a Venn diagram to represent the relationship between sets , A, B, C and D. (b) What is the relationship between (i) sets A and B? (ii) sets C and D? (c) If Sam is an element of set E, write two pairs of relationship to show the position of set E. Explain your answer. EXAMPLE 14 List all the subsets of each of the following sets. (a) {p, q, r} (b) {2, 4, 6, 8} Solution: (a) All the subsets are f, {p}, {q}, {r}, {p, q}, {p, r}, {q, r} and {p, q, r}. (b) All the subsets are f, {2}, {4}, {6}, {8}, {2, 4}, {2, 6}, {2, 8}, {4, 6}, {4, 8}, {6, 8}, {2, 4, 6}, {2, 4, 8}, {4, 6, 8}, {2, 6, 8} and {2, 4, 6, 8}. Number of subsets of set {p, q, r} = 2n = 23 = 8 There are 8 subsets altogether. C Complement of sets The complement of set A is the set of all elements in the universal set, ξ, which are not elements of set A, and is denoted by A. For example, ξ Aʹ A 4 3 1 2 ξ = {1, 2, 3, 4}. A = {1, 2}. A = {3, 4}. What is the complement of empty set? Critical Thinking
13 Sets CHAPTER 1 EXAMPLE 15 Given that the universal set, ξ = {a, b, c, d, e}, find the complement of each of the following sets. (a) P = {d, e} (b) Q = {vowels in the word ‘snake’} Solution: (a) P = {d, e}, thus P = {a, b, c} (b) Q = {a, e}, thus Q = {b, c, d} EXAMPLE 16 Given that x = {x : 1 < x < 12, x is an integer} and A = {x : x is a factor of 12}, find (a) the complement of A, (b) n(A9). Solution: x = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} A = {1, 2, 3, 4, 6, 12} (a) A9 = {5, 7, 8, 9, 10, 11} (b) n(A9) = 6 EXAMPLE 17 The Venn diagram shows the elements in the universal set x, set P and set Q. Find Q9. Solution: Q 9 = {a, b, e, m, n} EXAMPLE 18 The Venn diagram shows the universal set, ξ, set F and set G respectively. List the elements of F and G. Solution: F = {2, 3}, thus F = {4, 5, 6, 7, 8}. G = {6, 7}, thus G = {2, 3, 4, 5, 8}. A9 is read as ‘A prime’. a e h g k d c n m f b P Q ξ ξ F G 4 8 5 6 7 2 3
14 CHAPTER 1 Sets D Relationship between sets The relationship between set, subset, universal set and the complement of a set can be illustrated using Venn diagrams. EXAMPLE 19 Given that the universal set, ξ = {positive integers less than 10}, A = {first four odd numbers}, B = {x : x is a prime number between 2 and 6} and C = {even numbers less than 10}, (a) draw a Venn diagram to represent the sets ξ, A, B and C. (b) state the relationship between (i) ξ and A, (ii) C and A, (iii) B and A. Solution: ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {1, 3, 5, 7} B = {3, 5} C = {2, 4, 6, 8} (a) ξ A B C 9 2 4 6 8 3 5 7 1 (b) (i) A , ξ (ii) C , A (iii) B , A E Power sets A power set is all subsets of a set also includes the empty set. On the other words, the subsets of a set are the members of a power set. Power set is denoted by P(S). A set that has ‘n’ number of elements has 2n subsets in all. The cardinality of a power set is given by |P(A)| = 2n . |P(S)| = 2n , n = the total number of elements in the given set. For example, Set A = {1, 2) Set A has a total number of 2 elements. Therefore, there are 22 subsets in power set A.
15 Sets CHAPTER 1 The empty set { } is a subset of set A. Also, the subsets of set A are {1}, {2} and {1, 2}. Altogether, we can get the power set of A: P(A) = {{ }, {1}, {2}, {1, 2}} Number of subsets of set A = 22 = 4 A set with a finite number of elements has a finite power set. For example, if set P = {r, s, t, u}, the power sets are countable. The power set of an infinite set has an infinite numbers of subsets. For example, if set Y has infinite numbers of elements, this set will have infinite numbers of subsets. Thus, the power set exists for both finite set and infinite set. EXAMPLE 20 Given that set S = {1, 2, 3, 4}. Find the power set of set S. Solution: Set S = {1, 2, 3, 4}. Elements of set S = 4, thus P(S) = 2n = 24 = 16. Subsets of set S = { }, {1}, {2}, {3}, {4}, {1, 2}, {2, 3}, {3, 4}, {4, 1}, {1, 3}, {1, 4}, {2, 4}, {1, 2, 3}, {2, 3, 4}, {3, 4, 1}, {4, 1, 2} {1, 2, 3, 4}. Power set, P(S) = {{ }, {1}, {2}, {3}, {4}, {1, 2}, {2, 3}, {3, 4}, {4, 1}, {1, 3}, {1, 4}, {2, 4}, {1, 2, 3}, {2, 3, 4}, {3, 4, 1}, {4, 1, 2} {1, 2, 3, 4}}. F Equality of sets Two sets are equal if they have exactly the same elements. The order of the elements in each set does not matter. For example, {a, b, c} = {b, a, c}. If two or more sets contain the same elements, then they are considered as equal sets. For example, Given that A = {K, A, I, N} B = {I, K, A, N} C = {N, A, I, K} In conclusion, A = B = C, so sets A, B and C are equal sets. Each element in sets A, B and C are the same. Set A = {a, b, c} { }, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} Subset Power
16 CHAPTER 1 Sets EXAMPLE 21 Determine whether the sets in each of the following pairs are equal. (a) A = {8, 11, 15}, B = {15, 8, 11}. (b) P = {odd numbers between 10 and 18}. Q = {11, 13, 17}. Solution: (a) A = B (b) P = {11, 13, 15, 17}, Q = {11, 13, 17}. Thus, P ≠ Q. Therefore, A = B, A = C and B = C. Given that A = {K, A, I, N} B = {K, A, I, N, T} C = {N, I, K, A, H} In conclusion, A ≠ B ≠ C, so sets A, B and C are not equal sets. Element T is not an element of sets A and C. Element H is not an element of sets A and B. EXAMPLE 22 Determine whether the given set is equal set. M = {x : x is a factor of 15} and N = {15, 5, 3, 1} Solution: M = {1, 3, 5, 15} and N = {15, 5, 3, 1} Each element in set M is equal to each element in set N. Therefore, M = N. EXAMPLE 23 Given that V = {x : x is the common multiples of 2 and 3, 10 < x < 30} and W = {12, 18, k, 30}, where k is a constant. If V = W, determine the value of k. Solution: V = {12, 18, 24, 30} and W = {12, 18, k, 30} Given that V = W. Therefore, k = 24. The order of elements in each set is not important when we compare the equality of two or more sets. For example, P = {A, E, I} and Q = {I, E, A}. P and Q are equal sets although they have different order of elements. The sign ‘≠’ means not equal to. Determine whether each of the following statement is true. Give your reason. (a) If A = B, then n(A) = n(B). (b) If n(P) = n(Q), then P = Q.
17 Sets CHAPTER 1 Practice 1.2 Basic Intermediate Advanced A List down all the elements in each of the following sets. (a) A = {x : 8 < x < 14, x is a factor of 143} (b) B = {first five letters in the English alphabet} (c) C = {colours on our flag} (d) D = {whole numbers between 2.5 and 71 4} B State the number of elements in each of the following sets. (a) J = {8, 13, 16, 21, 35} (b) K = {numbers on a dice} (c) A = {x : 20 , x , 30, x is an odd number which is divisible by 2} (d) A = {factors of 30} C Given that A = {2, 3, 4, 7, 8, 11, 13, 15, 17}, B = {4, 7, 8} and C = {odd numbers between 7 and 13}, state whether each of the following is true or false. (a) B , A (b) A , C (c) C , A (d) B C D List down all the subsets of each of the following sets. (a) P = {3, 4} (b) Q = {x, y, z} (c) R = {prime numbers between 10 and 14} E ξ Q P c f d a e g b The above Venn diagram shows the relationship between the universal set, ξ, set P and set Q. List down all the elements of P and Q . F Given that the universal set, ξ = {x : 3 < x < 12, x is an integer}, list down the elements of the complement of each of the following sets. (a) P = {4, 6, 8, 10} (b) Q = {multiples of 3} (c) R = {prime numbers} (d) S = {factors of 12} (e) T = {x : 4 , x , 11, x is an even number} G Find the number of elements in the power sets of the following. (a) An empty set, set B = { }. (b) A set with ‘k + 1’ elements. 8 Determine whether the sets in each of the following pairs are equal. (a) P = {vowels in the word SCIENCE} Q = {A, I, E} (b) M = {x : 7 < x < 18, x is an even number} N = {8, 10, 12, 14, 16, 18} (c) V = {multiples of 4 less than 20} W = {4, 8, 12, 16} (d) C = {even numbers between 13 and 20} D = {factors of 12} 9 Given that A = {x : 9 , x , 16, x is an odd number}, B = {11, 13, 2x + 1} and A = B, find the value of x. It is given that the universal set, ξ = {2, 5, 7, 8, 9, 11, 12}, A = {odd numbers between 4 and 10}, B = {multiple of 5} and C = {factors of 8}. Draw a Venn diagram to illustrate the relationship between (a) set ξ and set C, (b) sets ξ, A and B. It is given that the universal set, ξ = {first six letters of the English alphabet}, P = {vowels}, Q = {consonants in the word ‘face’} and R = {f}. (a) Determine the relationships between (i) R and ξ, (ii) R and Q, (iii) P and Q, (iv) P and R, (v) ξ, Q and R. (b) Draw a Venn diagram to represent the sets ξ, P, Q and R.
18 CHAPTER 1 Sets 1.3 Operations of Sets A Intersection of sets (I) Determining and describing the intersection of sets The intersection of two sets, A and B, is the set of elements which are common to both set A and set B. It is written as A > B and read as ‘A intersection B’. > is the symbol for intersection. For example, A = {1, 2, 3, 4, 5} B = {3, 4, 5, 6, 7) Therefore, A > B = {3, 4, 5}. The intersection of two sets can be represented using a Venn diagram. A B ξ The shaded region represents set A > B. The intersection of two sets obeys the commutative law, A > B = B > A, for any two sets A and B. Given that set A and set B, observe the following cases. (a) If A B and B A, then A > B , A and A > B , B. (b) If A , B, then A > B = A. (c) If B , A, then A > B = B. ξ A B A B ξ B A ξ A B ξ
19 Sets CHAPTER 1 EXAMPLE 24 Given that the universal set, x = {x : 12 < x < 20, x is an integer}, M = {12, 14, 16, 18}, and N = {x : x is a multiple of 4}. (a) List the elements of set N and set M > N. (b) Draw a Venn diagram to show the relationship between x, M and N. Solution: x = {12, 13, 14, 15, 16, 17, 18, 19, 20} M = {12, 14, 16, 18} (a) N = {12, 16, 20} M > N = {12 , 14, 16 , 18} > {12 , 16 , 20} = {12, 16} (b) 18 20 17 19 13 15 12 14 16 ξ M N EXAMPLE 25 Given that A = {8, 9, 10, 11}, B = {10, 11, 12, 13}, C = {6, 7, 8, 10} and D = {12, 13}, find the following sets. (a) A > B (b) C > D (c) B > D (d) B > C Solution: (a) A = {8, 9, 10 , 11 } B = {10 , 11 , 12, 13} Therefore, A > B = {10, 11}. (b) C = {6, 7, 8, 10} D = {12, 13} Therefore, C > D = f or { }. Find the common elements. 10 and 11 are the common elements.
20 CHAPTER 1 Sets EXAMPLE 26 It is given that the universal set, ξ = {a, b, c, d, e, f, g}, P = {a, b, c} and Q = {b, c, d}. (a) Find (i) P > Q, (ii) P > Q. (b) Draw a Venn diagram to represent (i) P > Q, (ii) P > Q. Solution: (a) (i) P = {a, b, c} Q = {b, c, d} Therefore, P > Q = {b, c}. (ii) P = {a, b, c} Q = {a, e, f, g} Therefore, P > Q = {a}. (b) (i) ξ P a Q P ∩ Q e f g d b c (ii) ξ P a Q P ∩ Qʹ e f g d b c (c) B = {10, 11, 12 , 13 } D = {12 , 13 } Therefore, B > D = {12, 13}. (d) B = {10 , 11, 12, 13} C = {6, 7, 8, 10 } Therefore, B > C = {10}. (II) Complement of the intersection of sets The complement of set A > B is a set where the elements of the set are all elements in the universal set which are not the elements of A > B. The complement of set A > B is written as (A > B)ʹ. The set (A > B)ʹ can be represented by using a Venn diagram. The shaded region represents set (A > B)ʹ. A B ξ
21 Sets CHAPTER 1 The complement of the intersection of sets A and B is the set which does not contain the elements of A > B and is denoted by (A > B). EXAMPLE 27 Given that the universal set, ξ = {3, 5, 7, 9, 11, 13, 15}, A = {5, 7, 9, 11} and B = {9, 11, 13}. (a) Find (A > B). (b) Draw a Venn diagram and shade the region that represents (A > B). Solution: (a) ξ = {3, 5, 7, 9, 11, 13, 15} A = {5, 7, 9, 11} B = {9, 11, 13} A > B = {9, 11} Thus, (A > B) = {3, 5, 7, 13, 15}. (b) ξ A B 9 11 5 13 3 15 7 (III) Shading of Venn diagram for intersection of two sets EXAMPLE 28 Shade the region representing A > B. Solution: Step 1 Shade set A with horizontal lines. Step Shade set B with vertical lines. Step The area of the lines overlapped represents A > B. A B A B A B A B
22 CHAPTER 1 Sets Solution: ξ A 30 – x x 15 – x B 4 Let the number of students who like both soccer and volleyball be x. Let ξ = {students in the class}, A = {students who like soccer}, B = {students who like volleyball}. (a) Given that n(ξ) = 40 Therefore, 30 – x + x + 15 – x + 4= 40 49 – x = 40 x = 49 – 40 = 9 Therefore, 9 students like both soccer and volleyball. (b) Number of students who like to play soccer only = 30 – x = 30 – 9 = 21 Therefore, 21 students like to play soccer only. B Disjoint of sets Disjoint sets are sets which do not have common elements. If A and B are disjoint sets, then A > B = φ and n(A > B) = 0. On a Venn diagram, A and B are represented by two non-intersecting circles. A B EXAMPLE 29 A class consists of 40 students. 30 of them like to play soccer, 15 like volleyball and 4 like neither. Find the number of students who like (a) both soccer and volleyball, (b) soccer only.
23 Sets CHAPTER 1 C Union of sets (I) Determining and describing the union of sets The union of two sets, A and B, is the set of elements which belong either to A or B or to both A and B. It is written as A < B and read as ‘A union B’. < is the symbol for union. For example, A = {1, 2, 3, 4, 5} B = {3, 4, 5, 6, 7} Therefore, A < B = {1, 2, 3, 4, 5, 6, 7} The union of two sets can be represented using a Venn diagram. The shaded region in the Venn diagram below represents set A < B. A B ξ A B ξ For the case A > B ≠ φ For the case A > B = φ The union of two sets obeys the commutative law, A < B = B < A for any two sets A and B. ξ A B The common elements are written only once. Given that set A and set B, observe the following cases. (a) If A B and B A, then A , A < B and B , A < B. A B ξ (b) If A , B, then A < B = B. (c) If B , A, then A < B = A. B A ξ A B ξ team work 1. Given that n(ξ) = 120, explain in words the actual meaning of (D < E)' and state the value of n [(D < E )]. 2. Why n (D < E ) = n(ξ) – n[(D < E )']. Work with your team members. Let ξ = {skilled workers in the electronic factory}, D = {workers who posses diploma}, E = {workers who posses degree}. A < B ξ D 59 – x 66 – x 20 x E
24 CHAPTER 1 Sets Solution: (a) ξ = {7, 8, 9, 10, 11, 12, 13, 14} P = {8, 9, 10, 11} Q = {10, 11, 12, 13} P < Q = {8, 9, 10, 11, 12, 13} Therefore, (P < Q) = {7, 14}. (b) n (P < Q) = 2 EXAMPLE 32 Given that the universal set, x = {x : 5 < x < 15, x is an integer}, A = {7, 9, 11, 13, 15} and B = {x : x is a factor of 12}. (a) List the elements of set B and set A < B. (b) Draw a Venn diagram to show the relationship between the universal set x, set A and set B. Solution: x = {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} A = {7, 9, 11, 13, 15} (a) B = {6, 12} A < B = {7, 9, 11, 13, 15} < {6, 12} = {6, 7, 9, 11, 12, 13, 15} EXAMPLE 30 Given that A = {2, 5, 8}, B = {3, 5, 11} and C = {7, 11, 13}, find (a) A < B, (b) B < C. Solution: (a) A = {2, 5, 8} B = {3, 5, 11} Therefore, A < B = {2, 3, 5, 8, 11}. (b) B = {3, 5, 11} C = {7, 11, 13} Therefore, B < C = {3, 5, 7, 11, 13}. The complement of the union of sets A and B is the set which does not contain the elements of A < B and is denoted by (A < B). EXAMPLE 31 It is given that the universal set, ξ = {7, 8, 9, 10, 11, 12, 13, 14}, P = {8, 9, 10, 11} and Q = {10, 11, 12, 13}. (a) Find the set (P < Q). (b) Find n(P < Q).
25 Sets CHAPTER 1 (II) Shading of Venn diagram for union of sets EXAMPLE 33 Shade the region representing A < B. Solution: Step 1 Shade set A with horizontal lines. Step Shade set B with vertical lines. Step All the area that are shaded represents A < B. A B A B A B A B D Solving problems involving the intersection and union of sets EXAMPLE 34 The Venn diagram below shows the days Ali performed two sport activities in a week. Set A represents cycling activity and set B represents running activity. A B ξ • Monday • Sunday • Friday • Thursday • Tuesday • Saturday • Wednesday (b) ξ 7 13 11 9 6 10 14 5 8 12 15 A B
26 CHAPTER 1 Sets (a) Therefore, (18 – x) + x + (22 – x) = 35 40 – x = 35 x = 5 n(P > Q) = 5 Therefore, there are 5 pupils who join both societies. (b) n(P > Q)ʹ = (18 – x) + (22 – x) n(P > Q)ʹ = 40 – 2x = 40 – 2(5) = 30 Therefore, there are 30 pupils who join only one society. (a) How many days in a week did Ali miss cycling activity? (b) State the days Ali performed both sport activities. Solution: (a) Aʹ = {Tuesday, Thursday, Saturday} n(Aʹ) = 3 Therefore, there are 3 days in a week Ali missed cycling activity. (b) A > B = {Wednesday, Friday} Therefore, Ali performed both sport activities on Wednesday and Friday. EXAMPLE 35 Class 8 Amanah has 35 pupils. 18 pupils are members of Mathematics Society and 22 pupils are members of English Language Society. Given that each pupil is the member of at least one of these two societies, find the number of pupils who are members of (a) both societies, (b) only one society. Solution: Let P = {members of Mathematics Society} Q = {members of English Language Society} x = number of pupils who join both societies. Therefore, n(P > Q) = x. 18 – x x P Q 22 – x
27 Sets CHAPTER 1 EXAMPLE 36 The Venn diagram shows the participation of ten pupils in two contests. Set A represents the pupils who participated in the drawing contest and set B represents the pupils who participated in the essay writing contest. (a) State the pupils who did not participate in both contests. (b) State the number of pupils who participated in one contest only. Solution: (a) (A < B)ʹ = {Firdaus, Subra, Anjung} Therefore, pupils who did not participate in both contests are Firdaus, Subra and Anjung. (b) A < B = {Susita, Hua An, Rajoo, Jefri, Devi, May Lin, Sabri} A > B = {Jefri, Devi} Let P = {pupils who participate in one contest only} A B ξ n(P)= n(A < B) – n(A > B) = 7 – 2 = 5 Therefore, there are 5 pupils who participated in one contest only. EXAMPLE 37 A food stall only sells Nasi lemak and Roti canai. A survey was carried out on 50 customers about their favourite food. The result of the survey showed that 32 customers like Nasi lemak, 28 customers like Roti canai and 18 customers like both types of food. (a) Represent the given information using a Venn diagram. (b) Hence, find the number of customers who (i) do not like both types of food. (ii) like one type of food only. A B ξ • Susita • Firdaus • Subra • Anjung • Hua An • Rajoo • May Lin •Jefr • Sabri i •Devi
28 CHAPTER 1 Sets E Difference of sets (I) Difference of two sets If A and B are two sets, then their difference is given by A – B or B – A. For example, A = {2, 3, 4} and B = {4, 5, 6}. A – B means elements of A which are not the elements of B. A – B = {2, 3} In general, B – A = {x : x B, and x A}. If A and B are disjoint sets, then A – B = A and B – A = B. (II) Find the difference of sets using Venn diagram The difference of two subsets A and B is a subset of U, denoted by A – B and is defined by A – B = {x : x A and x B}. Let A and B be two sets. The difference of A and B, written as A – B, is the set of all those elements of A which do not belongs to B. Thus A – B = {x : x A and x B} or A – B = {x A : x B}. Clearly, x A – B. x A and x B. Solution: Let P = {customers who like Nasi lemak} Q = {customers who like Roti canai} Therefore, n() = 50 n(P) = 32 n(Q) = 28 n(P > Q) = 18 (a) ξ P 14 18 10 8 Q (b) (i) n(P < Q)ʹ = 8 Therefore, 8 customers do not like both types of food. (ii) n(P < Q) – n(P > Q) = 42 – 18 = 24 Therefore, 24 customers like one type of food only.
29 Sets CHAPTER 1 EXAMPLE 38 Given that A = {1, 2, 3} and B = {4, 5, 6}. Find the difference between the two sets (a) A and B, (b) B and A. Solution: The two sets are disjoint as they do not have any elements in common. (a) A – B = {1, 2, 3} = A (b) B – A = {4, 5, 6} = B EXAMPLE 39 Let A = {a, b, c, d, e, f } and B = {b, d, f, g}. Find the difference between the two sets (a) A – B, (b) B – A. Solution: (a) A – B = {a, c, e} Therefore, the elements a, c and e belong to A but not to B. (b) B – A = {g} Therefore, the element g belongs to B but not A. In the adjoining diagram the shaded part represents A – B. U A – B A B Similarly, the difference B – A is the set of all those elements of B that do not belong to A. Thus, B – A = {x : x A and x B} or A – B = {x B : x A}. In the adjoining diagram the shaded part represents B – A. U B – A A B In particular, A – B = φ if A , B and A – B = A if A > B = φ. The subset of A – B is also called the complement of B relative to A. The difference A – B can be expressed in terms of the complement as A – B = A > B. Properties of difference of sets: A – (B > C) = (A – B) < (A – C) A – (B < C) = (A – B) > (A – C) Difference of two sets A – B Difference of two sets B – A
CHAPTER 1 Sets 30 A Given that A = {4, 7, 8, 9}, B = {3, 7, 8, 9} and C = {6, 9, 10, 13}, find (a) A > B, (b) B > C, (c) A > C. B It is given that the universal set, ξ = {a, e, i, o, u}, M = {a, i, u} and N = {a, e, u}. (a) Find M > N. (b) Draw a Venn diagram to represent the sets ξ, M and N. Hence, shade the region representing M > N. C It is given that the universal set, ξ = {4, 5, 6, 7, 8, 9, 10, 11}, A = {5, 7, 8, 9, 11}, B = {5, 6, 8, 9, 10} and C = {4, 5, 8, 10, 11}. Find (a) (A > B)', (b) (B > C)'. D In each of the following Venn diagrams, identify the shaded subset. (a) A B ξ (b) ξ A B C E It is given that x = {x : 10 < x < 20, x is an integer}, X = {11, 13, 15, 17, 19} and Y = {x : x is a prime number less than 20}. (a) List the elements of set X < Y. (b) Show the relationship between the universal set, x, set X and set Y with a Venn diagram. F Given that the universal set, ξ = {2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {3, 4, 8, 10}, B = {4, 5, 7, 10} and C = {4, 8, 9}. Find (a) (A < B)', (b) (B < C)'. G Given that the universal set, ξ = {x : 15 < x < 25, x is an integer}, P = {x : x > 18} and Q = {x : x is a factor of 60}, find (P < Q)'. H A B ξ 3 5 8 2 The above Venn diagram shows the number of teachers in a school. Given that ξ = {all the teachers in the school}, A = {teachers who wear spectacles} and B = {teachers who wear black shoes}, find the number of teachers who (a) wear spectacles and black shoes, (b) wear spectacles or black shoes or both, (c) wear neither spectacles nor black shoes. I In a school of 400 students, 100 students like to play table tennis, 150 students like to play basketball and 75 students like to play both table tennis and basketball. Find the number of students who like neither table tennis nor basketball. J For the given Venn diagrams, shade the regions to represent each of the following. (a) A B A' > B Practice 1.3 Basic Intermediate Advanced
Sets CHAPTER 1 31 (b) A B A' > B' (c) A B A < B (d) A B A < B K A B ξ • Grape • Watermelon • Papaya • Kiwi • Mango • Guava • Banana • Orange • Apple The Venn diagram above shows the types of fruits sold by two stalls A and B. (a) Which fruits are sold by both stalls? (b) State the fruits sold by only one of the stalls. L In a survey about the types of vehicles owned by 40 families in a housing area, it was found that 25 families owned cars and 28 families owned motorcycles. Given that each family owned at least one type of vehicle. (a) Using K = {families that owned cars} and M = {families that owned motorcycles} represent the information above using a Venn diagram. (b) Find the number of families that (i) owned both types of vehicles, (ii) owned only one type of vehicle. M •Nasi lemak •Fried chicken •Curry noodles •Fried noodles •Burger •Fried rice •Tom yam •Noodle soup •Fried egg A B C ξ The Venn diagram above shows the types of food sold by three food stalls, A, B and C. (a) State the types of food sold by stall A or stall B. (b) State the types of food which are not sold by stall B or stall C. N A survey was carried out to study the favourite subjects of 50 pupils. The result of the survey showed that 25 pupils like Science, 29 pupils like History, 12 pupils like Mathematics and Science, 13 pupils like Mathematics and History, 4 pupils like Science only and 5 pupils like all three subjects. Given that there are no pupils who dislike all three subjects. (a) By using M = {pupils who like Mathematics}, S = {pupils who like Science} and J = {pupils who like History}, represent the given information using a Venn diagram. (b) Hence, find the number of pupils who (i) like Mathematics or Science, (ii) like History only, (iii) dislike History or Science. O It is given that P = {x : x is a natural number between 10 and 16}, Q = {y : y is a even number between 8 and 20} and R = {7, 9, 11, 14, 18, 20}. (a) Find the difference of two sets P and Q. (b) Find Q – R. (c) Find R – P. (d) Find Q – P.
CHAPTER 1 Sets 32 Section A 1. The diagram shows a Venn diagram that contains the elements of the universal set , set A and set B. • q • u • p • r • s • t B A List all the elements of set B9. A {r, s, t} C {p, q, u} B {q, u} D {p, q, r, u} 2. Given that H = {letter in the word ‘KEUSAHAWANAN’}. State n(H ). A 7 B 8 C 9 D 10 3. Set Q is two-digit number and the difference between these digits is 6. State set Q by listing. A {16, 26, 36, 46, 56, 66, 76, 86, 96} B {6, 17, 28, 39} C {60} D {17, 28, 39, 71, 82, 93} 1 Summary Summary Summary Operations on sets • The intersection of two sets, A and B, is the set of elements which are common to both set A and set B, and is denoted by A > B. • The union of two sets, A and B, is the set of elements which belong either to A or B or to both A and B, and is denoted by A < B. • The complement of the intersection of sets A and B is denoted by (A > B). • The complement of the union of sets A and B is denoted by (A < B). Sets Empty set • Set which does not contain any element. • Denoted by { } or f. • Also called null set. Complement of a set • The complement of set A is the set of all elements in the universal set, ξ, which are not elements of set A. • Denoted by A. Subsets • Set A is a subset of set B if every element of A is an element of B. • , denotes ‘is a subset of’. Equal sets • Sets that have exactly the same elements, A = B. Power set • A power set is all subsets of a set also includes the empty set. Universal set • The set which contains all the elements under discussion. • Denoted by ξ.
Sets CHAPTER 1 33 4. The diagram shows a Venn diagram of set X, set Y and set Z in a universal set. X Y Z Which of the following is the correct relationship? A Z , Y9 B X , Z C Y , Z D Y , X 5. Given that = {x: x is multiple of 3 and x , 30}. If set A = {x: x is a factor of 24}, determine n(A9). A 5 B 6 C 7 D 8 6. If A = {φ, {a}, }, find the P(A). A {{ }, {f}, {a}, {}} B {{ }, {f}, {{a}}, {}, {f, {a}}, {f, {}, {{a}, }, {f, {a}, }} C {{ }, {f}, {a}} D {{ }} 7. P Q ξ In the diagram, is the universal set. Given that n() = 20, n(P) = 8, n(Q) = 12 and n(P > Q) = 6. Therefore, n(Pʹ) = A 4 B 8 C 12 D 16 8. Given that universal set, = {C, O, M, P, U, T, E, R, S}, set P = {M, O, U, S, E } and set Q = {M, E, P, S }, Find n(P > Q). A 1 B 2 C 3 D 4 9. F B ξ The diagram shows a Venn diagram with the universal set, = {Grade 8 pupils}, set F = {pupils who take Physics} and set B = {pupils who take Biology}. Given that n(F) = 34, n(B) = 68, n(F > B) = 12 and the number of pupils who do not take any of the subject, Physics and Biology is 6. Find the number of pupils who do not take Physics. A 56 B 62 C 68 D 72 10. R S x + 2 x 2x + 1 The Venn diagram shows the number of participants of two contests. Given that the universal set, = R < S, set R = {participants of calligraphy writing contest} and set S = {participants of drawing contest}. Given that there are 17 persons participated in calligraphy writing contest only, find the total number of participants. A 59 B 61 C 63 D 65 11. In a survey about electric cooking appliances owned by 500 families, it was found that 36% of the families have microwave ovens, 24% have convection ovens, 12% have air fryers.
CHAPTER 1 Sets 34 It was also found that 8% of the families have microwave ovens and convection ovens, 5% have microwave ovens and air fryers, 4% have convection ovens and air fryers whereas 2% have all the three types of appliances. Find the number of families that have microwave ovens only. A 100 B 125 C 136 D 142 12. M G S 46 12 80 x 5x ξ The diagram shows a Venn diagram with set M = {members of Mathematics club}, set S = {members of History club} and set G = {members of Geography club}. Given that the number of members of History club is 1 4 of the number of members of Mathematics club. Calculate the number of members of Geography club. A 150 B 182 C 242 D 310 Section B 1. 2, 3, 5, 7, 11 The diagram shows five numbers. (a) Define the above numbers using (i) description, (ii) set notation. (b) Given that P = {prime numbers less than 10}, state whether each of the following is true or false. (i) 1 P (ii) 2 P 2. Given that set P = {3, 6, 9}, set Q = {prime numbers between 8 and 11} and set R = {first three multiples of three}, state whether each of the following is true or false. (a) set P = set R (b) set Q = {f} (c) 6 R (d) n(Q) = 0 3. (a) Given that A = {odd numbers between 10 and 16}, B = {11, 13, 2x + 1} and A = B, find the possible value of x. (b) Given that P = {prime numbers less than 10}. State whether each of the following is true or false. (i) 1 P (ii) 2 P 4. Given that the universal set, ξ = {x : 7 < x < 12, x is an integer}, A = {10, 11} and B = {x : x is a multiple of 5}, draw a Venn diagram to represent the sets ξ, A and B. 5. Given that the universal set, ξ = {x : 1 , x , 10, x is an integer}, P = {2, 4} and Q = {multiples of 3}. (a) Draw a Venn diagram to represent the sets ξ, P and Q. (b) List all the subsets of P. (c) List the elements of the complement of set Q. (d) Find n(P). 6. K 5 J ξ 4 2 7 8 10 The Venn diagram shows the relationship between the universal set, ξ, set J and set K. (a) List down all the subsets of K. (b) List all the elements of K. (c) State the number of subsets of J. (d) Find n(J). 7. Given that the universal set, ξ = {x : 20 < x < 30, x is an integer}, P = {x : x is a multiple of 4},
Sets CHAPTER 1 35 Q = {x : x is a number such that the sum of its digits is less than 7} and R = {x : x . 27}, (a) list all the elements of the sets P and Q, (b) find the set P < R, (c) find n(P > Q). 8. Answer the questions based on the Venn diagram below. Football 11 5 7 15 Badminton (a) How many students like both sports? (b) How many students do not like badminton? (c) How many students like only football? (d) How many students like football? (e) How many students do not like both sports? 9. (a) Given that the universal set, ξ = {x : 17 < x , 29, x is an integer} and set M = {x : x is a number such that the sum of its digits is an odd number}, find set M'. (b) 3 J K 7 The diagram shows a Venn diagram with the universal set, ξ = K < J. List all the subsets of set J. 10. (a) Q P The diagram shows a Venn diagram with the universal set, ξ = P < Q. Given that n(P) = 86, n(Q) = 62 and n(P > Q) = 24, find n(ξ). (b) Given that P = {a, b, c}, find the number of subsets of P. 11. (a) Given that K = {factors of 18} and L = {prime numbers less than 10}, list all the elements of K < L. (b) Given that the universal set, ξ = {x : 6 < x < 15, x is an integer}, P = {x : x > 10} and Q = {x : x is a multiple of 3}, find P > Q. 12. E F 3x +1 2x x + 5 The Venn diagram shows the number of students in photography club and chess club. Given that x = E < F, E = {photography club students} and F = {chess club students}. If the number of students who join one club is 54, find (a) the value of x, (b) the number of students who join both club. 13. If set A = {1, 2, 3, 4} and set B = {2, 3, 5, 7}. Find (a) A – B, (b) B – A. 14. If X = {11, 12, 13, 14, 15}, Y = {10, 12, 14, 16, 18} and Z = {7, 9, 11, 14, 18, 20}. Find (a) X – Y – Z, (b) Y – X – Z, (c) Z – X – Y.