M7 Q2 0701 TG FD Mathematics 7 PDF

Mathematics 7 Unit 7: Sets and the Set of Real Numbers Lesson 1: Introduction to Sets Table of Contents Learning Competency 1 Learning Targets...

Mathematics 7 Unit 7: Sets and the Set of Real Numbers Lesson 1: Introduction to Sets Table of Contents Learning Competency 1 Learning Targets 1 Prerequisite Skills 2 Lesson Proper 3 Introduction to the Lesson 3 Discussion 5 Define and Discover 5 Develop and Demonstrate 8 Alternative Digital Output 9 Practice and Feedback 9 Individual Practice 9 Group Activity 11 Performance Assessment 14 Study Guide and Worksheet Answer Key 14 Synthesis 24 Possible Answers to the Essential Questions 25 References 26  Mathematics Unit 7: Sets and the Set of Real Numbers 1 Introduction to Sets Learning Competency At the end of this lesson, the learners should be able to describe sets and their subsets, the union of sets, and the intersection of sets. Learning Targets At the end of the lesson, the learners should be able to do the following: Define a set. Describe the different kinds of a set. Illustrate real-life examples of sets. Essential Questions Instruct learners to reflect on the question for a minute; call on a few to give responses. Consolidate the responses and prepare the learners for the warm-up activity. If you will be using technology, refer to the slide presentation. If you will not be using technology, write the essential questions on the board. 1. Why is it important to understand the concept of subsets? 2. How does the concept of cardinality help in comparing sets? 7.1. Introduction to Sets 1  Mathematics Unit 7: Sets and the Set of Real Numbers Prerequisite Skills Skills classifying objects listing and describing counting numbers, integers, fractions, and decimal numbers listing odd and even numbers, prime numbers, and composite numbers 7.1. Introduction to Sets 2  Mathematics Unit 7: Sets and the Set of Real Numbers Lesson Proper A. Introduction to the Lesson Activity 1: Pick and Group! Suggested Time Frame: 10 mins 21st Century Skills Icons Legend Information, Media, and Technology Skills Communication Skills Teacher’s Notes Learning and Innovation Skills Life and Career Skills Materials pictures of objects Instructions 1. Before the lesson, prepare 10 pictures of any objects. You may decide what kind of objects you would like to have. 2. Show the pictures to the class. 3. Ask the learners to group the objects according to a common characteristic. 4. Ask some learners to share their answers in class. Guide Questions 1. How did you classify the given objects? Possible answer: “I classified the objects according to a common characteristic like color or shape.” 2. Why do you think it is important to classify objects? Possible answer: “It is important so that we can have a better way of 7.1. Introduction to Sets 3  Mathematics Unit 7: Sets and the Set of Real Numbers arranging the given objects. Activity 2: Treasure Hunt Suggested Time Frame: 10 mins Materials pen and paper Instructions 1. Ask the learners to form groups with 3–4 members. 2. Ask each group to look for 5 objects around the room that have common characteristics. 3. List down the objects and their common characteristics on a piece of paper. 4. Ask each group to look for other objects that have common characteristics. 5. Ask some groups to share their answers in class. Guide Questions 1. What objects did you choose? Possible answer: Answers may vary. Learners may share what objects they selected. 2. What strategies did you use to select the objects? Possible answer: “We thought of a characteristic and looked for objects that have those characteristics.” Activity 3: Animal Watching Suggested Time Frame: 10 mins Materials laptop or tablet with an internet connection 7.1. Introduction to Sets 4  Mathematics Unit 7: Sets and the Set of Real Numbers Instructions 1. Ask the learners to form groups with 3–4 members. Each group should have at least one laptop or tablet with an internet connection. 2. Ask the learners to search for five pictures of animals through the internet. They may decide what kind of animals they want to share. 3. Ask the learners to think of a characteristic common to the animals that they searched. 4. Ask some groups to share their answers in class. Guide Questions 1. What kind of animals did you select? Possible answer: Answers may vary. Learners may share what kind of animals they selected. 2. How did you classify the given animals? Possible answer: “We thought of some common characteristics like color or size.” B. Discussion Define and Discover In this lesson, the following key concepts will be discussed: set – a collection of well-defined and distinct objects Sets are denoted by uppercase letters of the alphabet, such as 𝐴, 𝐵, and 𝐶. Example: Set 𝐴 is the set of counting numbers from 1 to 5. elements – the objects in a set Object 𝑎 is an element of set 𝐴 if and only if 𝑎 belongs to set 𝐴. In symbols, 𝑎 ∈ 𝐴. Consequently, object 𝑏 is not an element of set 𝐴 if and only if 𝑏 does 7.1. Introduction to Sets 5  Mathematics Unit 7: Sets and the Set of Real Numbers not belong to set 𝐴. In symbols, 𝑏 ∉ 𝐴. Example: Set 𝐴 is the set of counting numbers from 1 to 5. The elements of set 𝐴 are 1, 2, 3, 4, and 5. Since 3 is in set 𝐴, we say that 3 ∈ 𝐴. Also, since 6 is not in 𝐴, we say that 6 ∉ 𝐴. Representing a set A set can be represented in three ways: descriptive form, roster form, and set-builder notation. In descriptive form, the set is described in words. In roster form, all elements of the set are listed and enclosed in braces. In set-builder notation, a common property of all the elements of a set is written. Example: Set 𝐴 is the set of counting numbers from 1 to 5. This is in descriptive form. In roster form, set 𝐴 can be written as 𝐴 = {1, 2, 3, 4, 5}. In set-builder notation, set 𝐴 can be written as 𝐴 = {𝑥 | 1 ≤ 𝑥 ≤ 5} or 𝐴 = {𝑥 | 𝑥 is a counting number from 1 to 5}. Subset of a set A set 𝐴 is a subset of set 𝐵 if all the elements in 𝐴 are in 𝐵. In symbols, 𝐴 ⊆ 𝐵. A set 𝐴 is a proper subset of set 𝐵 if 𝐴 is a subset of 𝐵 but there is at least one element in 𝐵 that is not an element of 𝐴. In symbols, 𝐴 ⊂ 𝐵. Example: Consider 𝐴 = {1, 2, 3} and 𝐵 = {1, 2}. We can say that 𝐵 is a subset of 𝐴 since all the elements in 𝐵 are also in 𝐴. Thus, 𝐵 ⊆ 𝐴. 7.1. Introduction to Sets 6  Mathematics Unit 7: Sets and the Set of Real Numbers Moreover, 𝐵 is a proper subset of 𝐴 since 𝐵 is a subset of 𝐴 but there are elements in 𝐴 that are not in 𝐵, such as 3. Thus, 𝐵 ⊂ 𝐴. Number of subsets 𝑛 A set with 𝑛 elements has 2 subsets. Example: 3 Consider 𝐴 = {1, 2, 3}. Since set 𝐴 has 3 elements, it follows that 𝐴 has 2 = 8 elements. The subsets are as follows: {} {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3} finite set – a set with a limited number of elements. Example: Set 𝐴 = {1, 2, 3, 4, 5} is a finite set since this set contains 5 elements. infinite set – a set with an infinite number of elements. Example: Set 𝐵 is the set of multiples of 2. Thus, 𝐵 = {2, 4, 6, 8, 10,...}. This is an infinite set since there are an infinite number of elements. equal sets – sets with the same elements Example: Consider 𝐴 = {1, 2, 3} and 𝐵 = {1, 2, 3}. The two sets are equal since they have the same elements. 7.1. Introduction to Sets 7  Mathematics Unit 7: Sets and the Set of Real Numbers empty set or null set – set with no elements Example: Set 𝐴 has no elements. Thus, 𝐴 = { } or 𝐴 = ∅. universal set – set containing all elements under discussion; denoted by 𝑈 Example: Let us consider all counting numbers from 1 to 10. Thus, 𝑈 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. cardinality of a set – the number of elements in a set Example: Consider 𝐴 = {1, 2, 3, 4, 5}. The cardinality of set 𝐴 is |𝐴| = 5. Develop and Demonstrate Example 1 Set 𝐴 consists of all positive integers less than 7 while set 𝐵 consists of all prime numbers less than 9. Represent set 𝐴 in roster form and set 𝐵 in set-builder notation. Solution In roster form, 𝐴 = {1, 2, 3, 4, 5, 6}. In set-builder notation, 𝐵 = {𝑥 | 𝑥 is a prime number less than 9}. Example 2 Set 𝐴 consists of all positive multiples of 3 that are less than 26 and set 𝐵 consists of all positive multiples of 8 that are less than 89. Find |𝐴| and |𝐵|. Solution Set 𝐴 consists of all positive multiples of 3 less than 26. Thus, 𝐴 = {3, 6, 9, 12, 15, 18, 21, 24}. Therefore, |𝐴| = 8. 7.1. Introduction to Sets 8  Mathematics Unit 7: Sets and the Set of Real Numbers Set 𝐵 consists of all positive multiples of 8 less than 89. Thus, 𝐵 = {8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88}. Therefore, |𝐵| = 11. Example 3 Set 𝑅 consists of all even numbers between 13 and 19. How many distinct subsets does set 𝑅 have? List all the subsets of set 𝑅. Solution Set 𝑅 consists of all even numbers between 13 and 19. Thus, 𝑅 = {14, 16, 18}. 3 Since 𝑅 has 3 elements, it follows that 𝑛 = 3. Thus, 𝑅 has 2 = 8 subsets. The subsets are listed below: {} {14} {16} {18} {14, 16} {14, 18} {16, 18} {14, 16, 18} Alternative Digital Output The calculator below can be used to create the set-builder notation of a set given a range of numbers with specific parameters. The calculator can be accessed here: https://www.omnicalculator.com/math/set-builder C. Practice and Feedback Individual Practice 1. Ask the learners to individually answer the following problems using a pen and a piece of paper. 2. Give them enough time to answer the items. 3. Call a random learner to show his or her work on the board afterward. 4. Let the learner share how he or she came up with the solution. 5. Provide the learner with feedback on the accuracy of his or her answer and 7.1. Introduction to Sets 9  Mathematics Unit 7: Sets and the Set of Real Numbers solution. In cases where there are misconceptions, guide the learners in the right direction to find the correct answer. Let’s Try It Problem 1 Set 𝐴 consists of odd integers from 40 to 54, and set 𝐵 consists of even integers from 5 to 29. Represent set 𝐴 in roster form and set 𝐵 in set-builder notation. Solution In roster form, 𝐴 = {41, 43, 45, 47, 49, 51, 53}. In set-builder notation, 𝐵 = {𝑥 | 𝑥 is an even integer from 5 to 29}. Let’s Try It Problem 2 Set 𝐴 consists of positive even multiples of 5 that are less than 60. Set 𝐵 consists of odd multiples of 7 from 1 to 60. Find |𝐴| and |𝐵|. Solution Set 𝐴 consists of positive even multiples of 5 that are less than 60. Thus, 𝐴 = {10, 20, 30 , 40, 50}. Therefore, |𝐴| = 5. Set 𝐵 consists of odd multiples of 7 from 1 to 60. Thus, 𝐵 = {7, 21, 35, 49}. Therefore, |𝐵| = 4. 7.1. Introduction to Sets 10  Mathematics Unit 7: Sets and the Set of Real Numbers Let’s Try It Problem 3 Set 𝑀 consists of all composite numbers from 8 to 13. How many subsets does 𝑀 have? List all the subsets of 𝑀. Solution 𝑛 A set with 𝑛 elements has 2 subsets. If 𝑀 consists of all composite numbers from 8 to 13, then 𝑀 = {8, 9, 10, 12}. It follows that 𝑛 = 4. 4 The number of distinct subsets of 𝑀 is 2 or 16. The subsets are as follows. {} {8} {9} {10} {12} {8, 9} {8, 10} {8, 12} {9, 10} {9, 12} {10, 12} {8, 9, 10} {8, 9, 12} {8, 10, 12} {9, 10, 12} {8, 9, 10, 12} Group Activity 1. Ask the learners to form a minimum of two groups to a maximum of five groups. 2. Each group will answer problem items 4 and 5. These questions are meant to test learners’ higher-order thinking skills by having them work collaboratively with their peers. 3. Give learners enough time to analyze the problem and work on their solution. 7.1. Introduction to Sets 11  Mathematics Unit 7: Sets and the Set of Real Numbers 4. Ask each group to assign a representative to show and discuss its solution on the board. 5. Provide the group with feedback on the accuracy of its answer and solution. In cases where there are misconceptions, give the group members the opportunity to work together and re-analyze the problem. Guide them in the right direction to find the correct answer. Let’s Try It Problem 4 Six learners in a class—Jack, Karlo, Leo, Mario, Nathan, and Owen—share their favorite sport in class. Jack: I play basketball and volleyball. Karlo: I play volleyball. Leo: I play football but not basketball. Mario: I play volleyball and football. Nathan: I play all three sports. Owen: I play basketball and football, but not volleyball. Set 𝐴 is the set of all learners who play basketball. Set 𝐵 is the set of all learners who play volleyball. Set 𝐶 is the set of all learners who play football. Determine the universal set of the problem and write all sets in roster form. Solution The learners who play basketball are Jack, Nathan, and Owen. Thus, we can write set 𝐴 as 𝐴 = {Jack, Nathan, Owen}. The learners who play volleyball are Jack, Karlo, Mario, and Nathan. Thus, we can write set 𝐵 as 𝐵 = {Jack, Karlo, Mario, Nathan}. 7.1. Introduction to Sets 12  Mathematics Unit 7: Sets and the Set of Real Numbers The learners who play football are Leo, Mario, Nathan, and Owen. Thus, we can write set 𝐶 as 𝐶 = {Leo, Mario, Nathan, Owen}. The universal set is the set of all learners in the problem. Thus, 𝑈 = {Jack, Karlo, Leo, Mario, Nathan, Owen}. Let’s Try It Problem 5 Consider the numbers from 1 to 30. Represent the following sets in roster form. Which of these sets have the same cardinalities? a. Set 𝐴 is the set of prime numbers. b. Set 𝐵 is the set of positive even multiples of 5. c. Set 𝐶 is the set of multiples of 3. d. Set 𝐷 is the set of multiples of 7. Solution Set 𝐴 is the set of prime numbers. Thus, 𝐴 = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29} and |𝐴| = 10. Set 𝐵 is the set of positive even multiples of 5. Thus, 𝐵 = {10, 20, 30} and |𝐵| = 3. Set 𝐶 is the set of multiples of 3. Thus, 𝐶 = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30} and |𝐶| = 10. Set 𝐷 is the set of multiples of 7. Thus, 𝐷 = {7, 14, 21, 28} and |𝐷| = 4. Thus, 𝐴 and 𝐶 have the same cardinality, which is 10. 7.1. Introduction to Sets 13  Mathematics Unit 7: Sets and the Set of Real Numbers Performance Assessment This performance assessment serves as a formative assessment, divided into three sets based on the student's level of learning. Click on the link provided on Q-Link lesson page to access each worksheet. Worksheet 1 (for beginners) Worksheet 2 (for average learners) Worksheet 3 (for advanced learners) Study Guide and Worksheet Answer Key Check Your Understanding A. Write the following sets in roster form and set-builder notation. 1. set of counting numbers from 1 to 15. Answer: Roster form: 𝐴 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} Set-builder notation: 𝐴 = {𝑥 | 𝑥 is a counting number from 1 to 15} 2. set of multiples of 3 from 1 to 20. Answer: Roster form: 𝐴 = {3, 6, 9, 12, 15, 18} Set-builder notation: 𝐴 = {𝑥 | 𝑥 is a multiple of 3 from 1 to 20} 3. set of numbers from 1 to 50 with 4 in its digit. Answer: Roster form: 𝐴 = {4, 14, 24, 34, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49} Set-builder notation: 𝐴 = {𝑥 | 𝑥 is a number from 1 to 50 with 4 in its digit} 7.1. Introduction to Sets 14  Mathematics Unit 7: Sets and the Set of Real Numbers B. Consider set 𝐴 as the set of letters in the word MATHEMATICS. 1. Write the set in roster form. Solution: The letters in the word in order are A, C, E, H, I, M, S, and T. Note that the elements should be distinct. Answer: 𝐴 = {𝐴, 𝐶, 𝐸, 𝐻, 𝐼, 𝑀, 𝑆, 𝑇} 2. Determine the cardinality of set 𝐴. Solution: Count the number of elements of set 𝐴. Answer: |𝐴| = 8 C. Consider set 𝐵 as the set of letters in the word SENSELESSNESS. 1. Write the set in roster form. Solution: The letters in the word in order are E, L, N, and S. Note that the elements should be distinct. Answer: 𝐵 = {𝐸, 𝐿, 𝑁, 𝑆} 2. Determine the number of subsets of set 𝐵. Solution: 𝑛 A set with 𝑛 elements has 2 subsets. Since 𝐵 has 4 elements, it follows that it 4 has 2 = 16 subsets. Answer: 16 subsets 7.1. Introduction to Sets 15  Mathematics Unit 7: Sets and the Set of Real Numbers 3. List down the different subsets of set 𝐵. Answer: The subsets are as follows. {} {𝐸} {𝐿} {𝑁} {𝑆} {𝐸, 𝐿} {𝐸, 𝑁] {𝐸, 𝑆} {𝐿, 𝑁} {𝐿, 𝑆} {𝑁, 𝑆} {𝐸, 𝐿, 𝑁} {𝐸, 𝐿, 𝑆} {𝐸, 𝑁, 𝑆} {𝐿, 𝑁, 𝑆} {𝐸, 𝐿, 𝑁, 𝑆} Worksheet 1 A. Instructions: Write the following sets in roster form. 1. Set 𝐴 is the set of all consonants in the word QUIPPER. Solution: The consonants in the word QUIPPER are P, Q, and R. Answer: 𝐴 = {𝑃, 𝑄, 𝑅} 2. Set 𝐵 is the set of all positive even counting numbers below 17. Solution: The positive even counting numbers below 17 are 2, 4, 6, 8, 10, 12, 14, and 16. Answer: 𝐵 = {2, 4, 6, 8, 10, 12, 14, 16} 3. Set 𝐶 is the set of prime numbers less than 20. Solution: The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. 7.1. Introduction to Sets 16  Mathematics Unit 7: Sets and the Set of Real Numbers Answer: 𝐶 = {2, 3, 5, 7, 11, 13, 17, 19} B. Instructions: Write the following sets in set-builder notation. 1. 𝐴 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢} Solution: The letters 𝑎, 𝑒, 𝑖, 𝑜, and 𝑢 are vowels of the alphabet. Answer: 𝐴 = {𝑥 | 𝑥 is a vowel} 2. 𝐵 = {1, 2, 3, 4, 5, 6, 7, 8} Solution: The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are counting numbers less than 9. Answer: 𝐵 = {𝑥 | 𝑥 is a counting number less than 9} 3. 𝐶 = {2, 4, 6, 8, 10} Solution: The numbers 2, 4, 6, 8, and 10 are even numbers less than 11. Answer: 𝐶 = {𝑥 | 𝑥 is an even number less than 11} C. Instructions: Determine the number of subsets of each of the following sets and list down all of its subsets. 1. 𝐴 = {0, 1} Solution: 2 Since 𝐴 has 2 elements, it follows that it has 2 = 4 subsets 7.1. Introduction to Sets 17  Mathematics Unit 7: Sets and the Set of Real Numbers Answer: 𝐴 has 4 subsets. The subsets are as follows: {} {0} {1} {0, 1} 2. 𝐶 = {𝑥, 𝑦, 𝑧} Solution: 3 Since 𝐶 has 3 elements, it follows that it has 2 = 8 subsets. Answer: 𝐶 has 8 subsets. The subsets are as follows: {} {𝑥} {𝑦} {𝑧} {𝑥, 𝑦} {𝑥, 𝑧} {𝑦, 𝑧} {𝑥, 𝑦, 𝑧} D. Instructions: Answer the problem below. Set 𝐴 consists of all prime numbers less than 10, and set 𝐵 consists of the first ten counting numbers. Is set 𝐴 a subset of set 𝐵? Explain your answer. Solution: Set 𝐴 in roster form is 𝐴 = {2, 3, 5, 7} and set 𝐵 in roster form is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Note that all elements of 𝐴 are also in 𝐵. This means that 𝐴 ⊆ 𝐵 Answer: 𝐴 is a subset of 𝐵 7.1. Introduction to Sets 18  Mathematics Unit 7: Sets and the Set of Real Numbers Worksheet 2 A. Instructions: Write the following sets in roster form. 1. Set 𝐴 is the set of all consonants in the word PHILIPPINES. Solution: The consonants in the word PHILIPPINES are H, L, N, P, S. Answer: 𝐴 = {𝐻, 𝐿, 𝑁, 𝑃 𝑆} 2. Set 𝐵 is the set of all positive odd counting numbers below 17. Solution: The positive odd counting numbers below 17 are 1, 3, 5, 7, 9, 11, 13, and 15. Answer: 𝐵 = {1, 3, 5, 7, 9, 11, 13, 15} 3. Set 𝐶 is the set of prime numbers less than 30. Solution: The prime numbers less than 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Answer: 𝐶 = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29} B. Instructions: Write the following sets in set-builder form. 1. 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒} Solution: The letters 𝑎, 𝑏, 𝑐, 𝑑, and 𝑒 are the first five letters of the alphabet. Answer: 𝐴 = {𝑥 | 𝑥 is one of the first five letters of the alphabet} 7.1. Introduction to Sets 19  Mathematics Unit 7: Sets and the Set of Real Numbers 2. 𝐵 = {2, 3, 5, 7} Solution: The numbers 2, 3, 5, and 7 are prime numbers less than 8. Answer: 𝐵 = {𝑥 | 𝑥 is a prime number less than 8} 3. 𝐶 = {3, 6, 9, 12, 15, 18} Solution: The numbers 3, 6, 9, 12, 15, and 18 are multiples of 3 that are less than 19. Answer: 𝐶 = {𝑥 | 𝑥 is a multiple of 3 that is less than 19} C. Instructions: Determine the number of subsets of each of the following sets and list down all of its subsets. 1. 𝐴 = {𝑎, 𝑏, 𝑐} Solution: 3 Since 𝐴 has 3 elements, it follows that it has 2 = 8 subsets. Answer: 𝐴 has 8 subsets. The subsets are as follows: {} {𝑎} {𝑏} {𝑐} {𝑎, 𝑏} {𝑎, 𝑐} {𝑏, 𝑐} {𝑎, 𝑏, 𝑐} 2. 𝐶 = {0, 1, 2, 3} Solution: 4 Since 𝐶 has 4 elements, it follows that it has 2 = 16 subsets. Answer: 𝐴 has 16 subsets. The subsets are as follows. 7.1. Introduction to Sets 20  Mathematics Unit 7: Sets and the Set of Real Numbers {} {0} {1} {2} {3} {0, 1} {0, 2} {0, 3} {1, 2} {1, 3} {2, 3} {0, 1, 2} {0, 1, 3} {0, 2, 3} {1, 2, 3} {0, 1, 2, 3} D. Instructions: Answer the problem below. Set 𝐴 consists of all positive odd integers less than 10, and set 𝐵 consists of counting numbers less than 10. Is set 𝐴 a subset of set 𝐵? Explain your answer. Solution: Set 𝐴 in roster form is 𝐴 = {1, 3, 5, 7, 9} and set 𝐵 in roster form is 𝐵 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Note that all elements in 𝐴 are in 𝐵. Thus, 𝐴 ⊆ 𝐵. Answer: 𝐴 is a subset of 𝐵 Worksheet 3 A. Instructions: Write the following sets in roster form. 1. Set 𝐴 is the set of all consonants in the word SUPERCALIFRAGILISTICEXPIALIDOCIOUS. Solution: The consonants in the word SUPERCALIFRAGILISTICEXPIALIDOCIOUS are C, D, F, G, L, P, R, S, T, and X. Answer: 𝐴 = {𝐶, 𝐷, 𝐹, 𝐺, 𝐿, 𝑃, 𝑅, 𝑆, 𝑇, 𝑋} 7.1. Introduction to Sets 21  Mathematics Unit 7: Sets and the Set of Real Numbers 2. Set 𝐵 is the set of all positive even counting numbers less than or equal to 30. Solution: The positive even counting numbers less than or equal to 30 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, and 30. Answer: 𝐵 = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30} 3. Set 𝐶 is the set of composite numbers less than 40. Solution: The composite numbers less than 40 are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, and 39. Answer: 𝐶 = {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39} B. Instructions: Write the following sets in set-builder notation. 1. 𝐴 = {Emilio Aguinaldo, Manuel Quezon, Jose Laurel, Sergio Osmena, Manuel Roxas} Solution: The people in the set are the first five presidents of the Philippines. Answer: 𝐴 = {𝑥 | 𝑥 is one of the first five presidents of the Philippines} 2. 𝐵 = {21, 28, 35, 42, 49} Solution: The numbers 21, 28, 35, 42, and 49 are multiples of 7 between 20 and 50. Answer: 𝐵 = {𝑥 | 𝑥 is a multiple of 7 from 20 to 50} 7.1. Introduction to Sets 22  Mathematics Unit 7: Sets and the Set of Real Numbers 3. 𝐶 = {5, 15, 25, 35, 45, 55, 65} Solution: The numbers 5, 15, 25, 35, 45, 55, and 65 are odd multiples of 5 that are less than 70. Answer: 𝐶 = {𝑥 | 𝑥 is an odd multiple of 5 that is less than 70} C. Instructions: Determine the number of subsets of each of the following sets and list down all of its subsets. 1. 𝐴 = {0, 2, 4, 6} Solution: 4 Since 𝐴 has 4 elements, it follows that it has 2 = 16 subsets. Answer: 𝐴 has 16 subsets. The subsets are as follows: {} {0} {2} {4} {6} {0, 2} {0, 4} {0, 6} {2, 4} {2, 6} {4, 6} {0, 2, 4} {0, 2, 6} {0, 4, 6} {2, 4, 6} {0, 2, 4, 6} 2. 𝐵 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢} Solution: 5 Since 𝐵 has 5 elements, it follows that it has 2 = 32 subsets Answer: 𝐵 has 32 subsets. The subsets are as follows: {} {𝑎} {𝑒} {𝑖} {𝑜} {𝑢} {𝑎, 𝑒} {𝑎, 𝑖} {𝑎, 𝑜} {𝑎, 𝑢} {𝑒, 𝑖} {𝑒, 𝑜} {𝑒, 𝑢} {𝑖, 𝑜} {𝑖, 𝑢} {𝑜, 𝑢} {𝑎, 𝑒, 𝑖} {𝑎, 𝑒, 𝑜} 7.1. Introduction to Sets 23  Mathematics Unit 7: Sets and the Set of Real Numbers {𝑎, 𝑒, 𝑢} {𝑎, 𝑖, 𝑜} {𝑎, 𝑖, 𝑢} {𝑎, 𝑜, 𝑢} {𝑒, 𝑖, 𝑜} {𝑒, 𝑖, 𝑢} {𝑒, 𝑜, 𝑢} {𝑖, 𝑜 𝑢} {𝑎, 𝑒, 𝑖, 𝑜} {𝑎, 𝑒, 𝑖, 𝑢} {𝑎, 𝑒, 𝑜, 𝑢} {𝑎, 𝑖, 𝑜, 𝑢} {𝑒, 𝑖, 𝑜, 𝑢} {𝑎, 𝑒, 𝑖, 𝑜, 𝑢} D. Instructions: Answer the problem below. Set 𝐴 consists of all even integers less than 10, and set 𝐵 consists of all the positive factors of 120. Is set 𝐴 a subset of set 𝐵? Explain your answer. Solution: Set 𝐴 in roster form is 𝐴 = {2, 4, 6, 8} and 𝐵 = {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120} Note that all elements in 𝐴 are also in 𝐵. Thus, 𝐴 ⊆ 𝐵. Answer: 𝐴 is a subset of 𝐵 Synthesis Wrap-up 1. What are the basic concepts of sets learned in this lesson? Possible answer: “Basic concepts include the definition of sets, elements, subsets, proper subsets, finite and infinite sets, equal sets, null sets, universal sets, and cardinality.” 2. How do subsets differ from proper subsets? Possible answer: “A subset can be equivalent to the set it comes from, while a proper subset is always smaller than the set it comes from.” 7.1. Introduction to Sets 24  Mathematics Unit 7: Sets and the Set of Real Numbers Application and Values Integration 1. How can the concept of sets be applied in organizing data in real life? Possible answer: “Sets help in categorizing and structuring data, like sorting books in a library or organizing groceries.” 2. What values can be learned from studying sets? Possible answer: “Studying sets teaches the value of organization, analytical thinking, and clarity in categorization.” Bridge to the Next Topic 1. How might understanding sets help in learning about set operations? Possible answer: “Understanding sets forms the basis for grasping more complex concepts like union, intersection, and set differences.” 2. What are you curious about regarding set operations? Possible answer: “I am interested in how sets combine or interact.” Possible Answers to the Essential Questions 1. Why is it important to understand the concept of subsets? Possible answer: “Understanding sets is crucial to analyze relationships between different sets and understand how sets can be part of larger sets. 2. How does the concept of cardinality help in comparing sets? Possible answer: “It helps determine the size of sets and compare them quantitatively.” 7.1. Introduction to Sets 25  Mathematics Unit 7: Sets and the Set of Real Numbers References “Definition and Representation of a Set.” n.d. eMathZone. Accessed December 5, 2023. https://www.emathzone.com/tutorials/algebra/definition-and-representation-of-set.html Pierce, Rod. 2023. “Set-Builder Notation.” Math is Fun. August 10, 2023. https://www.mathsisfun.com/sets/set-builder-notation.html “Set Builder Notation.” n.d. Cuemath. Accessed December 5, 2023. https://www.cuemath.com/algebra/set-builder-notation/ “What is a Set?” n.d. BYJU’s. Accessed December 5, 2023. https://byjus.com/maths/what-is-a-set/ 7.1. Introduction to Sets 26