Unit 1: Further on Sets PDF

Unit 1: Further on Sets UNIT FURTHER ON SETS 1 Unit Outcomes Explain facts about sets. Describe sets in different ways. Define operations on sets. Demonstrate set operations using Venn di...

Unit 1: Further on Sets UNIT FURTHER ON SETS 1 Unit Outcomes Explain facts about sets. Describe sets in different ways. Define operations on sets. Demonstrate set operations using Venn diagram. Apply rules and principles of set theory for practical situations. Unit Contents 1.1 Sets and Elements 1.2 Set Description 1.3 The Notion of Sets 1.4 Operations on Sets 1.5 Application Summary Review Exercise 1  Unit 1: Further on Sets subset symmetric difference set description Venn diagram empty set intersection absolute complement union proper subset complement set Introduction In Grade 7 you have learnt basic definition and operations involving sets. The concept of a set serves as a fundamental part of the present day mathematics. Today this concept is being used in almost every branch of mathematics. We use sets to define the concepts of relations and functions. In this unit, you will discuss some further definitions, operations and applications involving sets. Activity 1.1 1. Define a set in your own words. 2. Which of the following are well defined sets and which are not? Justify your answer. a. Collection of students in your class. b. Collection of beautiful girls in your class. c. Collection of consonants of the English alphabet. d. Collection of hardworking teachers in a school. 1.1 Sets and Elements A set is a collection of well-defined objects or elements. When we say a set is well- defined, we mean that if an object is given, we are able to determine whether the object is in the set or not. 2  Unit 1: Further on Sets Note i) Sets are usually denoted by capital letters like 𝐴, 𝐵, 𝐶, 𝑋, 𝑌, 𝑍, etc. ii) The elements of a set are represented by small letters like 𝑎, 𝑏, 𝑐, 𝑥, 𝑦, 𝑧, etc. If 𝑎 is an element of set 𝐴, we say “𝑎 belongs to 𝐴”. The Greek symbol ∈ (epsilon) is used to denote the phrase “belongs to”. Thus, we write 𝒂 ∈ 𝑨 if 𝑎 is a member of set 𝐴. If 𝑏 is not an element of set 𝐴, we write 𝒃 ∉ 𝑨 and read as “𝑏 does not belong to set 𝐴” or “𝑏 is not a member of set A”. Figure 1.1 Example 1 a. The set of students in your class is a well-defined set since the elements of the set are clearly known. b. The collection of kind students in your school. This is not a well-defined set because it is difficult to list members of the set. c. Consider 𝐺 as a set of vowel letters in English alphabet. Then 𝑎 ∈ 𝐺, 𝑜 ∈ 𝐺, 𝑖 ∈ 𝐺, but 𝑏 ∉ 𝐺. Example 2 Suppose that 𝐴 is the set of positive even numbers. Write the symbol ∈ or ∉ in the blank spaces. a. 4 _____ 𝐴 b. 5_____ 𝐴 c. −2_____ 𝐴 d. 0____ 𝐴 Solution: The positive even numbers include 2, 4, 6, 8, …. Therefore, a. 4 ∈ 𝐴 b. 5 ∉ 𝐴, c. −2 ∉ 𝐴, d. 0 ∉ 𝐴. 3  Unit 1: Further on Sets Exercise 1.1 1. Which of the following is a well-defined set? Justify your answer. a. A collection of all boys in your class. b. A collection of efficient doctors in Black Lion Hospital. c. A collection of all natural numbers less than 100. d. The collection of songs by Artist Tilahun Gessese. 2. Let 𝐴 = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces: a. 5____𝐴 b. 8 ____ 𝐴 c. 0____𝐴 d. 4____ 𝐴 e. 7____𝐴 1.2 Set Description Sets can be described in the following ways. i) Verbal method (Statement form) In this method, the well-defined description of the elements of the set is written in an ordinary English language statement form (in words). Example 1 a. The set of whole numbers greater than 1 and less than 20. b. The set of students in this mathematics class. ii) Listing Methods a) Complete listing method (Roster Method) In this method, all elements of the set are completely listed. The elements are separated by commas and are enclosed within set braces, { }. Example 2 a. The set of all even positive integers less than 7 is described in complete listing method as {2, 4, 6}. b. The set of all vowel letters in the English alphabet is described in complete listing method as {𝑎, 𝑒, 𝑖, 𝑜, 𝑢 }. 4  Unit 1: Further on Sets b) Partial listing method We use this method, if listing of all elements of a set is difficult or impossible but the elements can be indicated clearly by listing a few of them that fully describe the set. Example 3 Use partial listing method to describe the following sets. a. The set of natural numbers less than 100. b. The set of whole numbers. Solution: a. The set of natural numbers less than 100 are 1, 2, 3 , … , 99. So, naming the set as 𝐴, we can express 𝐴 by partial listing method as 𝐴 = {1, 2, 3, … , 99}. The three dots after element 3 and the comma above indicate that the elements in the set continue in that manner up to 99. b. Naming the set of whole numbers by 𝕎 , we can describe it as 𝕎 = {0, 1, 2, 3, … }. So far, you have learnt three methods of describing a set. However, there are sets which cannot be described by these three methods. Here, below is another method of describing a set. iii) Set builder method (Method of defining property) The set-builder method is described by a property that its member must satisfy the common property. This is the method of writing the condition to be satisfied by a set or property of a set. In set brace, write the representative of the elements of a set, for example 𝑥, and then write the condition that 𝑥 should satisfy after the vertical line (|) or colon (:) Figure 1.2 5  Unit 1: Further on Sets Note The set of natural numbers, whole numbers, and integers are denoted by ℕ, 𝕎, and ℤ, respectively. They are defined as ℕ = {1, 2, 3,... } 𝕎 = {0, 1, 2, 3,... }, ℤ = {.... −3, −2, −1, 0 , 1, 2, 3,... }. Example 4 Describe the following sets using set builder method. i) Set 𝐴 = {1, 2, 3 … 10} can be described in set builder method as: 𝐴 = {𝑥 | 𝑥 ∈ ℕ and 𝑥 < 11}. We read this as “𝐴 is the set of all elements of natural numbers less than 11.” ii) Let set 𝐵 = {0, 2, 4, …. }. This can be described in set builder method as: 𝐵 = {𝑥 | 𝑥 ∈ ℤ and 𝑥 is a non-negative even integer} or 𝐵 = {2𝑥 | 𝑥 = 0, 1, 2, 3, … } 𝑜𝑟 𝐵 = {2𝑥 | 𝑥 ∈ 𝕎}. Exercise 1.2 1. Describe each of the following sets using a verbal method. a. 𝐴 = { 5, 6, 7, 8, 9} b. 𝑀 = {2, 3, 5, 7, 11, 13} c. 𝐺 = {8, 9, 10, …. } d. 𝐸 = {1, 3, 5, … , 99} 2. Describe each of the following sets using complete and partial listing method (if possible): a. The set of positive even natural numbers below or equal to 10. b. The set of positive even natural numbers below or equal to 30. c. The set of non-negative integers. d. The set of even natural numbers. e. The set of natural numbers less than 100 and divisible by 5. f. The set of integers divisible by 3. 3. List the elements of the following sets: 6  Unit 1: Further on Sets a. 𝐴 = {3𝑥 | 𝑥 ∈ 𝕎} b. 𝐵 = {𝑥 | 𝑥 ∈ ℕ and 5 < 𝑥 < 10} 4. Write the following sets using set builder method. a. 𝐴 = {1, 3, 5 …. } b. 𝐵 = {2, 4, 6, 8} c. 𝐶 = {1, 4, 9, 16, 25} d. 𝐷 = {4, 6, 8, 10, … , 52 } e. 𝐸 = {−10,... , −3, −2, −1, 0, 1, 2, … , 5} f. 𝐹 = {1, 4, 9, …. } 1.3 The Notion of Sets Empty set, Finite set and Infinite set Empty Set A set which does not contain any element is called an empty set, void set or null set. The empty set is denoted mathematically by the symbol { } or Ø. Example 1 Let set 𝐴 = {𝑥 | 1 < 𝑥 < 2, 𝑥 ∈ ℕ}. Then, 𝐴 is an empty set, because there is no natural number between numbers 1 and 2. Finite set and Infinite set Definition 1.1 A set which consists of a definite number of elements is called a finite set. A set which is not finite is called an infinite set. Example 2 Identify the following sets as finite set or infinite set. a. The set of natural numbers up to 10 b. The set of African countries c. The set of whole numbers Solution: a. Let 𝐴 be a set and 𝐴 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. 7  Unit 1: Further on Sets Thus, it is a finite set because it has definite (limited) number of elements. b. The set of African countries is a finite set. c. The set of whole numbers is an infinite set. Note The number of elements of set 𝐴 is denoted by 𝑛ሺ𝐴ሻ. For instance, in the above example 2𝑎, 𝑛ሺ𝐴ሻ = 10. Read 𝑛ሺ𝐴ሻ as number of elements of set 𝐴 Exercise 1.3 1. Identify empty set from the list below. a. 𝐴 = {𝑥 | 𝑥 ∈ ℕ and 5 < 𝑥 < 6} b. 𝐵 = {0} c. C is the set of odd natural numbers divisible by 2. d. 𝐷 = { } 2. Sort the following sets as finite or infinite sets. a. The set of all integers b. The set of days in a week c. 𝐴 = {𝑥 ∶ 𝑥 is a multiple of 5} d. 𝐵 = {𝑥 ∶ 𝑥 ∈ 𝑍, 𝑥 < −1} e. 𝐷 = {𝑥 ∶ 𝑥 is a prime number} Equal Sets, Equivalent Sets, Universal Set, Subset and Proper Subset Equal Sets Definition 1.2 Two sets 𝐴 and 𝐵 are said to be equal if and only if they have exactly the same or identical elements. Mathematically, it is denoted as 𝐴 = 𝐵. 8  Unit 1: Further on Sets Example 1 Let 𝐴 = {1, 2, 3, 4} and 𝐵 = {4, 3, 2, 1}. Then, 𝐴 = 𝐵. Set 𝐴 and set 𝐵 are equal. Equivalent Sets Definition 1.3 Two sets 𝐴 and 𝐵 are said to be equivalent if there is a one-to-one correspondence between the two sets. This is written mathematically as 𝐴 ↔ 𝐵 (or 𝐴~𝐵). Note Observe that two finite sets 𝐴 and 𝐵 are equivalent, if and only if they have equal number of elements and we write mathematically this as 𝑛 ሺ𝐴ሻ = 𝑛 ሺ𝐵ሻ. Example 2 Consider two sets 𝐴 = {1, 2, 3, 4} and 𝐵 = {Red, Blue, Green, Black}. In set 𝐴 there are four elements and in set 𝐵 also there are four elements. Therefore, set 𝐴 and set 𝐵 are equivalent. Universal Set (∪) Definition 1.4 A universal set (usually denoted by U) is a set which has elements of all the related sets, without any repetition of elements. Example 3 Let set 𝐴 = {2, 4, 6,... } and 𝐵 = {1, 3, 5,... }. The universal set U consists of all natural numbers, such that 𝑈 = {1, 2, 3, 4,... }. Therefore, as we know all even and odd numbers are part of natural numbers. Hence, set U has all the elements of set A and set B. 9  Unit 1: Further on Sets Subset (⊆) Definition 1.5 Set 𝐴 is said to be a subset of set 𝐵 if every element of 𝐴 is also an element of 𝐵. Figure 1.3 shows this relationship. Mathematically, we write this as 𝑨 ⊆ 𝑩. If set 𝐴 is not a subset of set 𝐵, then it is written as 𝐴 ⊈ 𝐵. Figure 1.3 Example 4 Let 𝐴 = {1, 2, 3} and 𝐵 = {1, 2, 3, 4} be sets. Here, set 𝐴 is a subset of set 𝐵, or 𝐴 ⊆ 𝐵, since all members of set 𝐴 are found in set 𝐵. In the above set 𝐴, find all subsets of the set. How many subsets does set 𝐴 have? Solution: The subsets of set 𝐴 are {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}, { }. The number of subsets of set 𝐴 is 8. Note i) 𝐴ny set is a subset of itself. ii) Empty set is a subset of every set. iii) If set 𝐴 is finite with 𝑛 elements, then the number of subsets of set 𝐴 is 2𝑛. In the above Example 3.b, 𝑛ሺ𝐴ሻ = 3. Then, the number of subsets is 23 = 8. Proper Subset (⊂) Definition 1.6 If 𝐴 ⊆ 𝐵 and 𝐴 ≠ 𝐵, then 𝐴 is called the proper subset of set 𝐵 and it can be written as 𝐴 ⊂ 𝐵. 10  Unit 1: Further on Sets Example 5 Given that sets 𝐴 = {2, 5, 7} and 𝐵 = {2, 5, 7, 8}. Set 𝐴 is a proper subset of set 𝐵, that is, 𝐴 ⊂ 𝐵 since 𝐴 ⊆ 𝐵 and 𝐴 ≠ 𝐵. Observe also that 𝐵 ⊄ A. In the above set 𝐴, find all the proper subsets. How many proper subsets does set 𝐴 have? Solution: The proper subsets of set 𝐴 are {2}, {5}, {7}, {2, 5}, {5, 7}, {2, 7}, { }. There are seven subsets. Note i) For any set 𝐴, 𝐴 is not a proper subset of itself. ii) The number of proper subsets of set 𝐴 is 2𝑛 − 1. iii) Empty set is the proper subset of any other sets. iv) If set 𝐴 is subset of set 𝐵 ሺ𝐴 ⊆ 𝐵ሻ, conversely 𝐵 is super set of 𝐴 written as 𝐵 ⊃ 𝐴. Exercise 1.4 1. Identify equal sets, equivalent sets or which are neither equal nor equivalent. a. 𝐴 = {1, 2, 3} and 𝐵 = {4, 5} b. 𝐶 = {𝑞, 𝑠, 𝑚} and 𝐷 = {6, 9, 12} c. 𝐸 = {3, 7, 9, 11} and 𝐹 = {3, 9, 7, 11} d. 𝐺 = {𝐼, 𝐽, 𝐾, 𝐿} and 𝐻 = {𝐽, 𝐾, 𝐼, 𝐿} e. 𝐼 = {𝑥 | 𝑥 ∈ 𝕎, 𝑥 < 5} and 𝐽 = {𝑥 | 𝑥 ∈ ℕ, 𝑥 ≤ 5} f. 𝐾 = {𝑥 | 𝑥 is a multiple of 30} and 𝐿 = {𝑥 | 𝑥 is a factor of 10} 2. List all the subsets of set 𝐻 = {1, 3, 5}. How many subsets and how many proper subsets does it have? 3. Determine whether the following statements are true or false. a. {𝑎, 𝑏} ⊄ {𝑏, 𝑐, 𝑎} b. {𝑎, 𝑒} ⊆ {𝑥 | 𝑥 is a vowel in the English alphabet} c. {𝑎} ⊂ { 𝑎, 𝑏, 𝑐 } 11  Unit 1: Further on Sets 4. Express the relationship of the following sets, using the symbols ⊂, ⊃, or = a. 𝐴 = {1, 2, 5, 10} and 𝐵 = {1, 2, 4, 5, 10, 20} b. 𝐶 = {𝑥 |𝑥 is natural number less than 10} and 𝐷 = {1, 2, 4, 8} c. 𝐸 = {1, 2} and 𝐹 = {𝑥 | 0 < 𝑥 < 3, 𝑥 ∈ ℤ} 5. Consider sets 𝐴 = {2, 4, 6}, 𝐵 = {1, 3 7, 9, 11} and 𝐶 = {4, 8, 11}, then a. Find the universal set b. Relate sets 𝐴, 𝐵, 𝐶 and 𝑈 using subset. 1.4 Operations on Sets There are several ways to create new sets from sets that have already been defined. Such process of forming new set is called set operation. The three most important set operations namely Unionሺ∪ሻ, Intersectionሺ∩ሻ, Complementሺ′ሻ and Difference ሺ−ሻ are discussed below. Union and Intersection Activity 1.2 Let the universal set is the set of natural numbers ℕ which is less than 12, and sets 𝐴 = {1, 2, 3, 4, 5, 6} and 𝐵 = {1, 3, 5, 7, 9}. a. Can you write a set consisting of all natural numbers that are in 𝐴 or in 𝐵? b. Can you write a set consisting of all natural numbers that are in 𝐴 and in 𝐵? c. Can you write a set consisting of all natural numbers that are in 𝐴 and not in 𝐵? Figure 1.4 Venn diagrams A Venn diagram is a schematic or pictorial representation of the sets involved in the discussion. Usually sets are represented as interlocking circles, each of which is 12  Unit 1: Further on Sets enclosed in a rectangle, which represents the universal set. Figure 1.4 above is an example of Venn diagram. Definition 1.7 The union of two sets 𝐴 and 𝐵, which is denoted by 𝑨 ∪ 𝑩, is the set of all elements that are either in set 𝐴 or in set 𝐵 (or in both sets). We write this mathematically as 𝐴 ∪ 𝐵 = {𝑥 | 𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵}. Figure 1.5 Definition 1.8 The intersection of two sets 𝐴 and 𝐵, denoted by 𝐴 ∩ 𝐵, is the set of all elements that are both in set 𝐴 and in set 𝐵. We write this mathematically as 𝐴 ∩ 𝐵 ={𝑥 | 𝑥 ∈ 𝐴 and 𝑥 ∈ 𝐵} Figure 1.6 Note Two sets 𝐴 and 𝐵 are disjoint if 𝐴 ∩ 𝐵 = ∅ Figure 1.7 Example 1 Let 𝐴 = {0, 1, 3, 5, 7} and 𝐵 = {1, 2, 3, 4, 6, 7} be sets. Draw the Venn diagram and find 𝐴 ∪ 𝐵 and 𝐴 ∩ 𝐵. 13  Unit 1: Further on Sets Solution: Figure 1.8 shows Venn diagram of set 𝐴 and 𝐵. Thus, 𝐴 ∪ 𝐵 = {0, 1, 2, 3, 4, 5, 6, 7} and 𝐴 ∩ 𝐵 = {1, 3, 7}. Figure 1.8 Example 2 Let 𝐴 = {2, 4, 6, 8, 10, … } and 𝐵 = {3, 6, 9, 12, 15, … } be sets. Then, find 𝐴 ∪ 𝐵 and 𝐴 ∩ 𝐵. Solution: 𝐴 ∪ 𝐵 = {𝑥 | 𝑥 is a positive integer that is either even or a multiple of 3} = {2, 3, 6, 9, 12, 15,.... } 𝐴 ∩ 𝐵 = {𝑥 | 𝑥 is a positive integer that is both even and a multiple of 3} = {6, 12, 18, 24,.... } Note i) Law of ∅ and 𝑈: ∅ ∩ 𝐴 = ∅, 𝑈 ∩ 𝐴 = 𝐴. ii) Commutative law: 𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴. iii) Associative Law: ሺ𝐴 ∩ 𝐵ሻ ∩ 𝐶 = 𝐴 ∩ ሺ𝐵 ∩ 𝐶ሻ. Exercise 1.5 1. Let 𝐴 = {0, 2, 4, 6, 8} and 𝐵 = {0, 1, 2, 3, 5, 7, 9}. Draw the Venn Diagram and find 𝐴 ∪ 𝐵 and 𝐴 ∩ 𝐵. 2. Let 𝐴 be the set of positive odd integers less than 10 and 𝐵 is the set of positive multiples of 5 less than or equal to 20. Find a) 𝐴 ∪ 𝐵, b) 𝐴 ∩ 𝐵. 3. Let 𝐶 = {𝑥 | 𝑥 is a factors of 20}, 𝐷 = {𝑦 | 𝑦 is a factor of 12}. Find a) 𝐶 ∪ 𝐷, b) 𝐶 ∩ 𝐷. 14  Unit 1: Further on Sets Complement of sets Definition 1.9 Let 𝐴 be a subset of a universal set 𝑈. The absolute complement (or simply complement) of 𝐴, which is denoted by 𝑨′ , is defined as the set of all elements of 𝑈 that are not in 𝐴. We write this mathematically as 𝐴′ = {𝑥: 𝑥 ∈ 𝑈 and 𝑥 ∉ 𝐴}. Figure 1.9 Example 1 a. Let 𝑈 = {0, 1, 2, 3, 4} and 𝐴 = {3, 4}. Then, 𝐴′ = {0, 1, 2}. Example 2 Let 𝑈 = {1, 2, 3, … , 10} be a universal set, 𝐴 = {𝑥 | 𝑥 is a positive factor of 10 in 𝑈} and 𝐵 = {𝑥 | 𝑥 is an odd integer in 𝑈} be sets. a. Find 𝐴′ and 𝐵′. b. Find ሺ𝐴 ∪ 𝐵ሻ′ and 𝐴′ ∩ 𝐵′. What do you observe from the answers? Solution: a. 𝐴 = {1, 2, 5, 10}, 𝐵 = {1, 3, 5, 7, 9}, Thus, 𝐴′ = {3, 4, 6, 7, 8, 9}, 𝐵 ′ = {2, 4, 6, 8, 10}. b. First, we find 𝐴 ∪ 𝐵. Hence, 𝐴 ∪ 𝐵 = {1, 2, 3, 5, 7, 9,10} and ሺ𝐴 ∪ 𝐵ሻ′ = {4, 6, 8}. On the other hand, from 𝐴′ and 𝐵 ′ , we obtain 𝐴′ ∩ 𝐵′ = {4, 6, 8}. Hence, we immediately observe ሺ𝐴 ∪ 𝐵ሻ′ = 𝐴′ ∩ 𝐵′. In general, for any two sets 𝐴 and 𝐵 , ሺ𝐴 ∪ 𝐵ሻ′ = 𝐴′ ∩ 𝐵′. It is called the first statement of De Morgan’s law. 15  Unit 1: Further on Sets De Morgan’s Law For the complement set of 𝐴 ⋃ 𝐵 and 𝐴 ⋂ 𝐵, 1st statement: (𝐴 ⋃ 𝐵ሻ′ = 𝐴′ ⋂ 𝐵 ′ , 2nd statement: (𝐴 ⋂ 𝐵ሻ′ = 𝐴′ ⋃ 𝐵 ′. Figure 1.10 Figure 1.11 Exercise 1.6 1. If the universal set 𝑈 = {0, 1, 2, 3, 4, 5}, and 𝐴 = {4, 5}, then find 𝐴′. 2. Let the universal set 𝑈 = {1, 2, 3, … , 20}, 𝐴 = {𝑥 |𝑥 is a positive factor of 20} and 𝐵 = {𝑥 | 𝑥 is an odd integer in 𝑈}. Find 𝐴′ , 𝐵 ′ , ሺ𝐴 ∪ 𝐵ሻ′ and 𝐴′ ∩ 𝐵 ′. 3. Let the universal set be 𝑈 = {𝑥 | 𝑥 ∈ ℕ, 𝑥 ≤ 10}. When 𝐴 = {2, 5, 9}, and 𝐵 = {1, 5, 6, 8}, find a) 𝐴′ ⋂ 𝐵′ and b) 𝐴′ ⋃ 𝐵′. Difference of sets Definition 1.10 The difference between two sets 𝐴 and 𝐵, which is denoted by 𝐴 − 𝐵 , is the of all elements in 𝐴 and not in 𝐵; this set is also called the relative complement of 𝐴 with respect to 𝐵. We write this mathematically as 𝐴 − 𝐵 = {𝑥 | 𝑥 ∈ 𝐴 and 𝑥 ∉ 𝐵}. Figure 1.12 Note The notation 𝐴 − 𝐵 can be also written as 𝐴\𝐵. 16  Unit 1: Further on Sets Example 1 If sets 𝐴 = {0, 1, 2, 3, 4} and 𝐵 = {3, 4}, then 𝐴 − 𝐵 or 𝐴\𝐵 = {0, 1, 2}. Example 2 Let 𝑈 be a universal set of the set of one-digit numbers, 𝐴 be the set of even numbers, 𝐵 be the set of prime numbers less than 10. Find the following: a. 𝐴 − 𝐵 or 𝐴\𝐵 b. 𝐵 − 𝐴 or 𝐵\𝐴 c. 𝐴∪𝐵 d. 𝑈 − ሺ𝐴 ∪ 𝐵ሻ or 𝑈\ሺ𝐴 ∪ 𝐵ሻ Solution: Here, 𝐴 = {0, 2, 4, 6, 8}, 𝐵 = {2, 3, 5, 7}. Then, we illustrate the sets using a Venn diagram as follows. From the Venn diagram we observe: a. 𝐴 − 𝐵 = {0, 4, 6, 8} b. 𝐵 − 𝐴 = {3 ,5, 7} c. 𝐴 ∪ 𝐵 = {0, 2, 3, 4, 5, 6, 7, 8} d. 𝑈 − ሺ𝐴 ∪ 𝐵ሻ = {1, 9} Figure 1.13 Example 3 For the same sets in Example 2, find the following. What can you say from Example 2 a. and b.? What about d. and Example 2, a.? a. 𝐴′ b. 𝑈 − 𝐴 c. 𝐵′ d. 𝐴 ∩ 𝐵′ Solution: a. 𝐴′ = {1, 3, 5, 7, 9} b. 𝑈 − 𝐴 = {1, 3, 5, 7, 9} c. 𝐵′ = {0, 1, 4, 6, 8, 9} d. 𝐴 ∩ 𝐵′ = {0, 4, 6, 8} From a. and b., we can say, 𝐴′ = 𝑈 − 𝐴. From d. and Example 2, a., we can say, 𝐴 − 𝐵 = 𝐴 ∩ 𝐵′. 17  Unit 1: Further on Sets Theorem 1.1 For any two sets 𝐴 and 𝐵, each of the following holds true. ሺ𝐴′ ሻ′ = 𝐴 𝐴′ = 𝑈 − 𝐴 𝐴 − 𝐵 = 𝐴 ∩ 𝐵′ 𝐴 ⊆ 𝐵 ⟺ 𝐵 ′ ⊆ 𝐴′ Exercise 1.7 From the given Venn diagram, find each of the following: a. 𝐴 − 𝐵 or 𝐴\𝐵 b. 𝐵 − 𝐴 or 𝐵\𝐴 c. 𝐴 ∪ 𝐵 d. 𝑈 − ሺ𝐴 ∪ 𝐵ሻ or 𝑈\ሺ𝐴 ∪ 𝐵ሻ Symmetric Difference of Two Sets Definition 1.11 Symmetric Difference For two sets A and B, the symmetric difference between these two sets is denoted by 𝐴∆𝐵 and is defined as: 𝐴∆𝐵 = ሺ𝐴\𝐵ሻ ∪ ሺ𝐵\𝐴ሻ , which is ሺ𝐴 − 𝐵ሻ ∪ ሺ𝐵 − 𝐴ሻ or = ሺ𝐴 ∪ 𝐵ሻ\ሺ𝐴 ∩ 𝐵ሻ In the Venn diagram, the shaded part represents 𝐴∆𝐵 Figure 1.14 Example 1 Consider sets 𝐴 = {1, 2, 4, 5, 8} and 𝐵 = {2, 3, 5, 7}. Then, find 𝐴∆𝐵. Solution: First, let us find 𝐴\𝐵 = {1, 4, 8} and 𝐵\𝐴 = {3, 7}. Hence, 𝐴∆𝐵 = ሺ𝐴\𝐵ሻ ∪ ሺ𝐵\𝐴ሻ Figure 1.15 18  Unit 1: Further on Sets = {1, 4, 8} ∪ {3, 7} = {1, 3, 4, 7, 8}. Or 𝐴∆𝐵 = ሺ𝐴 ∪ 𝐵ሻ\ሺ𝐴 ∩ 𝐵ሻ = {1, 3, 4, 7, 8}. Example 2 Given sets 𝐴 = {𝑑, 𝑒, 𝑓} and 𝐵 = {4, 5, 6}. Then, find 𝐴∆𝐵. Solution: First, we find 𝐴\𝐵 = {𝑑, 𝑒, 𝑓} and 𝐵\𝐴 = {4,5,6}. Hence, 𝐴∆𝐵 = ሺ𝐴\𝐵ሻ ∪ ሺ𝐵\𝐴ሻ = {𝑑, 𝑒, 𝑓} ∪ {4, 5, 6} = {𝑑, 𝑒, 𝑓, 4, 5, 6}. Figure 1.16 Exercise 1.8 1. Given 𝐴 = {0, 2, 3, 4, 5} and 𝐵 = {0, 1, 2, 3, 5, 7, 9}. Then, find 𝐴∆𝐵. 2. If 𝐴∆𝐵 = ∅, then what can be said about the two sets? 3. For any two sets 𝐴 and 𝐵, can we generalize 𝐴∆𝐵 = 𝐵∆𝐴 ? Justify your answer. Cartesian Product of Two Sets Definition 1.12 Cartesian Product of Two Sets The Cartesian product of two sets 𝐴 and 𝐵, denoted by 𝐴 × 𝐵, is the set of all ordered pairs ሺ𝑎, 𝑏ሻ where 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵. This also can be expressed as 𝐴 × 𝐵 = {ሺ𝑎, 𝑏ሻ: 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵}. Example 1 Let 𝐴 = {1, 2} and 𝐵 = {𝑎, 𝑏}. Then, find a) 𝐴 × 𝐵 b) 𝐵 × 𝐴 Solution: a. 𝐴 × 𝐵 = {ሺ1, 𝑎ሻ, ሺ1, 𝑏ሻ, ሺ2, 𝑎ሻ, ሺ2, 𝑏ሻ} b. 𝐵 × 𝐴 = {ሺ𝑎, 1ሻ, ሺ𝑎, 2ሻ, ሺ𝑏, 1ሻ, ሺ𝑏, 2ሻ} 19  Unit 1: Further on Sets Example 2 If 𝐴 × 𝐵 = {ሺ1, 𝑎ሻ, ሺ1, 𝑏ሻ, ሺ2, 𝑎ሻ, ሺ2, 𝑏ሻ, ሺ3, 𝑎ሻ, ሺ3, 𝑏ሻ}, then find sets 𝐴 and 𝐵. Solution: 𝐴 is the set of all first components of 𝐴 × 𝐵, that is, 𝐴 = {1, 2, 3}, and 𝐵 is the set of all second components of 𝐴 × 𝐵, that is, 𝐵 = {𝑎, 𝑏}. Exercise 1.9 1. Let 𝐴 = {1, 2, 3} and 𝐵 = {𝑒, 𝑓}. Then, find a) 𝐴 × 𝐵 b) 𝐵 × 𝐴. 2. If 𝐴 × 𝐵 = {ሺ7,6ሻ, ሺ7,4ሻ, ሺ5,4ሻ, ሺ5,6ሻ, ሺ1,4ሻ, ሺ1,6ሻ}, then find sets 𝐴 and 𝐵. 3. If 𝐴 = {𝑎, 𝑏, 𝑐} , 𝐵 = {1, 2, 3} and 𝐶 = {3, 4}, then find 𝐴 × ሺ𝐵 ∪ 𝐶ሻ. 4. If 𝐴 = {6, 9, 11}, then find 𝐴 × 𝐴. 5. If the number of elements of set 𝐴 is 6 and the number of elements of set 𝐵 is 4, then the number of elements of 𝐴 × 𝐵 is _____________________. 1.5 Application Number of Elements of union of two sets For the two subsets 𝐴 and 𝐵 of a universal set 𝑈, the following formula on the number of elements holds. That is 𝑛ሺ𝐴 ∪ 𝐵ሻ = 𝑛ሺ𝐴ሻ + 𝑛ሺ𝐵ሻ − 𝑛ሺ𝐴 ∩ 𝐵ሻ. Especially in the case 𝐴 ∩ 𝐵 = ∅, thus, Figure 1.17 𝑛ሺ𝐴 ∩ 𝐵ሻ = 0, and the following holds: 𝑛ሺ𝐴 ∪ 𝐵ሻ = 𝑛ሺ𝐴ሻ + 𝑛ሺ𝐵ሻ. If 𝐴 ∩ 𝐵 ≠ ∅, then 𝑛ሺ𝐴 ∪ 𝐵ሻ = 𝑛ሺ𝐴ሻ + 𝑛ሺ𝐵ሻ − 𝑛ሺ𝐴 ∩ 𝐵ሻ. Figure 1.18 20  Unit 1: Further on Sets Example Let 𝐴 and 𝐵 be two finite sets such that 𝑛ሺ𝐴ሻ = 20, 𝑛ሺ𝐵ሻ = 28, and 𝑛ሺ𝐴 ∪ 𝐵ሻ = 36, then find 𝑛ሺ𝐴 ∩ 𝐵ሻ. Solution: Using the formula 𝑛ሺ𝐴 ∪ 𝐵ሻ = 𝑛ሺ𝐴ሻ + 𝑛ሺ𝐵ሻ − 𝑛ሺ𝐴 ∩ 𝐵ሻ we have 36 = 20 + 28 − 𝑛ሺ𝐴 ∩ 𝐵ሻ. This gives 𝑛ሺ𝐴 ∩ 𝐵ሻ = ሺ20 + 28ሻ − 36 = 48 − 36 = 12. Exercise 1.10 1. Let 𝐴 and 𝐵 be two finite sets such that 𝑛ሺ𝐴ሻ = 34, 𝑛ሺ𝐵ሻ = 46 and 𝑛ሺ𝐴 ∪ 𝐵ሻ = 70. Then, find 𝑛ሺ𝐴 ∩ 𝐵ሻ. 2. There are 60 people attending a meeting. 42 of them drink tea and 27 drink coffee. If every person in the meeting drinks at least one of the two drinks, find the number of people who drink both tea and coffee. (Hint: Use a Venn diagram). 21