Sets - Chapter 2 PDF

CHAPTER 2 SETS  Section 2.1 Basic Properties of Sets Objectives of the day: At the end of the lesson, the students will be able to: 1. use three methods to represent sets; 2. define the empty set and use the symbols  and ; 3. apply set notation to sets of real numbers and...

CHAPTER 2 SETS  Section 2.1 Basic Properties of Sets Objectives of the day: At the end of the lesson, the students will be able to: 1. use three methods to represent sets; 2. define the empty set and use the symbols  and ; 3. apply set notation to sets of real numbers and its subsets; 4. determine a set’s cardinal number; 5. recognize and apply equivalent sets, equal sets, subsets and proper subsets; and 6. distinguish between finite and infinite sets. 3 4  What is a SET? A set is a well-defined collection of distinct objects. The terms “set,” “collection,” and “family” are synonymous. Individual objects are called elements or members of a set. Usually, we denote sets with capital letters while elements with small letters. 5 A set maybe finite or infinite. For example, the set consisting of months that begin with the letter M has March and May as its elements. This is a finite set. On the other hand, the set consisting of positive odd integers is an example of an infinite set. 6 Methods for Representing Sets 1. Statement form method - well-defined description of the elements of the set is given. 2. Roster method - listing or enumerating the members. Commas are used to separate the elements of the set while braces are used to designate that the enclosed elements form a set. 3. Set-Builder Notation - especially useful when describing infinite sets. 7  The table below gives two examples of sets which are being described using the statement form method and the roster method. Statement form Method Roster Method The set of first seven { 2 , 3 , 5 , 7 , 11 , 13 , 17 } prime numbers The set of days of the week { Mon, Tues, Wed, Thurs, Fri, Sat, Sun} Describing the Sets Using Statement Form Method and Roster Method Table 2.1.1 8 Example 2.1.1 Use roster method to write each of the given sets. a. The set of all letters in the word MATHEMATICS. b. The set of the three major island groups in the Philippines. 9 Example 2.1.1 Use roster method to write each of the given sets. a. The set of all letters in the word MATHEMATICS. b. The set of the three major island groups in the Philippines. Solution a. The set of all letters in the word MATHEMATICS is {M, A, T, H, E, I, C, S}. b. The set of the three major island groups in the Philippines is {Luzon, Visayas, Mindanao} Note: When writing sets, the order of the elements does not matter. 10 Example 2.1.2 Use statement form method to describe each of the given sets. a. { a, e , i ,o ,u } b. { 2, 4, 6, 8, 10} Solution a. The set of vowels in the English Alphabet. b. The set of first five positive even integers. 11 Common Number Sets SYMBOL DESCRIPTION 𝑵 NATURAL NUMBERS/ COUNTING NUMBERS 𝑵 = {1, 2, 3, 4, 5, 6, 7,…} 𝑾 WHOLE NUMBERS The number zero together with the natural numbers. 𝑾 = {0, 1, 2, 3, 4, 5, 6, 7,…} 𝒁 INTEGERS The positive integers, negative integers and zero. 𝒁 = {…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,…} 12 Common Number Sets SYMBOL DESCRIPTION 𝑸 RATIONAL NUMBERS Rational numbers are numbers of the form where p and q are integers and q ≠ 0. These numbers are either terminating or repeating decimals. Examples: , 𝑰 IRRATIONAL NUMBERS Irrational numbers are numbers which cannot be written as a simple fraction. These are numbers which are nonterminating and nonrepeating decimals. Examples: 𝝅 = 𝟑. 𝟏𝟒𝟏𝟓𝟗 … , 𝟐 = 1.41421… 𝑹 REAL NUMBERS Real numbers are either rational numbers or irrational numbers. 13 The symbol is read “is an element of ” or “ it belongs to ”. On the other hand, the symbol  is read “is not an element of ” or “ it does not belong to ”. Example 2.1.3. Determine whether each statement is true or false. a. -5 b. Solution a. There are no negative natural numbers, so the statement is false. b. Since is not an integer, so the statement is true. 14 Another method of representing a set is set- builder notation. Set-builder notation is especially useful when describing infinite sets. For instance, in set-builder notation, the set of natural numbers greater than 7 is written as follows: 15 Example 2.1.4 Use set-builder notation to write the set of integers less than –3. solution and x < -3} 16 Definition 2.1.1 The empty set is the set that has no elements in it. It is also called null set or void set. The symbol  or { } is used to represent the empty set. Example 2.1.5 The set of negative natural numbers is an example of an empty set. 17 Definition 2.1.2 A set is finite if the number of elements in a set is finite. In other words, a finite set is a set which you could in principle count and finish counting. Definition 2.1.3 An infinite set is a set whose elements cannot be counted. In other words, an infinite set is a set that has no last element. 18  Definition 2.1.4 The cardinality of a finite set is the number of distinct elements in the set. It is also called the cardinal number of the set. The cardinal number of a finite set A is denoted by Free templates for all your presentation needs the notation or. Example 2.1.6 For PowerPoint and 100% free for personal Ready to use, Blow your audience Google Slides or commercial use professional and away with attractive Elements that are listed more than once are customizable visuals counted only once. Thus, = 3. 19 Definition 2.1.5 Set A is equal to set B, denoted by A = B if and only if A and B have exactly the same elements. Example 2.1.7 A = { 5, 7, 0, 5} and B = {0, 5, 7} Since the two sets have exactly the same elements, so they are equal. 20 Definition 2.1.6 Set A is equivalent to set B denoted by A B if and only if A and B have the same number of elements. Note: All equal sets are equivalents sets. However, the converse of this statement is false. Example 2.1.8 = {d, f, h, k, x, v} and Each set has exactly six elements, so the sets are equivalent. 21  Subset and Proper Subset 22 Definition 2.1.7 Set A is a subset of set B, denoted by A B, if and only if every element of set A is an element of set B. Example 2.1.9 This statement is true because every element of the first set is also an element of the second set. 23 23 Subset Relationships a. b. The notation is used to denote that set A is not a subset of set B. Example 2.1.10 0 is a whole number, but 0 is not a natural number, so this statement is true. 24 24  Definition 2.1.8 Set A is a proper subset of set B, denoted by A B, if every element of set A is an element of set B, and A B. Example 2.1.11 Free templates for all your presentation needs A = { 1, 2, 3, 4} , B = { 5, 2, 4, 1, 3} The elements of set A are also elements of set B and A B, so A B. For PowerPoint and 100% free for personal Ready to use, Blow your audience Google Slides or commercial use professional and away with attractive customizable visuals 25 Theorem 2.1.1 A set with n elements has 2n subsets. Theorem 2.1.2 A set with n elements has 2n - 1 proper subsets. 26  Example 2.1.12 Set Y shows the four popular soft drinks that are sold in a school canteen. Y = {coke, pepsi, 7-up, sprite} Free templates for all your presentation needs List all the subsets of set Y. For PowerPoint and 100% free for personal Ready to use, Blow your audience Google Slides or commercial use professional and away with attractive customizable visuals 27 Solution An organized list shows the following subsets. {} Subset with 0 element {coke}, {pepsi}, {7-up}, {sprite} Subsets with 1 element {coke, pepsi}, {coke, 7-up}, {coke, sprite}, {pepsi, 7-up}, Subsets with 2 elements {pepsi, sprite}, {7-up, sprite} 28 {coke, pepsi, 7-up}, {coke, pepsi, sprite}, Subsets with 3 elements {coke, 7-up, sprite} {pepsi, 7-up, sprite}, {coke, pepsi, 7-up, sprite} Subsets with 4 elements Therefore, a set with 4 elements has 16 subsets and 15 proper subsets. 29 Example 2.1.13 A newly opened bakery sells breads for which you can choose from eight toppings. a. How many different variations of breads can the bakery serve? b. What is the minimum number of toppings the bakery must provide if it wishes to advertise that it offers over 2000 variations of its breads? 30  Solution The bakery can serve a bread with no topping, one topping, two toppings, three toppings, four toppings and so forth up to all eight toppings. Let T be Free the set consisting templates for allofyour the presentation eight toppings. The needs elements in each subset of T describe exactly one of the variations of toppings that the bakery can serve. For PowerPoint and 100% free for personal Ready to use, Blow your audience Google Slides or commercial use professional and away with attractive customizable visuals 31 Consequently, the number of different variations of breads that the restaurant can serve is the same as the number of subsets of T. Thus, the bakery can serve 28 = 256 different variations of its breads. b. Use the Freemethod templates of for guessing all your and checking presentation to find needs the smallest natural number n for which 2n > 2000. 29 = 512 210 = 1024 211 = 2048 The bakery must For PowerPoint and provide a minimum 100% free for personal of 11 toppings Ready to use, Blow your audience if itGoogle wishes Slides to offer over 2000 or commercial use variations of professional and customizable its away with attractive visuals breads. 32 REFERENCES 1. Aufmann, R.N.(2018). Mathematics in the Modern World. Rex Book Store, Inc. 2. Daligdig, R.M. (2019). Mathematics in the Modern World.Free Lorimar Publishing, templates Inc. presentation needs for all your 3. Carpio, J.N. and Peralta, B.D. (2018). Mathematics in the Modern World. Books Atbp. Publishing Corp. 4. Olejan, R.O., Veloria, E.V., Bonghanoy, G.B., Ondaro,andJ.E.,and For PowerPoint Sumalinog, 100% free for personal J.D. (2018). Ready to use, Mathematics Blow your audience Google Slides in the Modern orWorld. commercial use professional and MUTYA Publishing customizable away with attractive House, Inc. visuals 5. Manlulu, E.A. and Hipolito, L.M. (2019). A Course Module for Mathematics in the Modern World. Rex Book Store, Inc. 33