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I. MATHEMATICS IN OUR WORLD Core Idea: Mathematics is a useful way to think about nature and the world. Learning Objectives: Identify patterns in nature Articulate the importance of mathematics in one’s life Argue about the nature of mathematics, what is it, how...
I. MATHEMATICS IN OUR WORLD Core Idea: Mathematics is a useful way to think about nature and the world. Learning Objectives: Identify patterns in nature Articulate the importance of mathematics in one’s life Argue about the nature of mathematics, what is it, how it is expressed, represented, and used. Express appreciation for mathematics as a human endeavor What is mathematics? Where is mathematics? What is mathematics for? What is mathematics all about? What role does mathematics play in our world? Human mind and culture – develop a formal system of thoughts for recognizing, classifying, and exploiting patterns Great secret: NATURE’S PATTERNS - rules that govern natural processes Johannes Kepler – The Six-Cornered Snowflake Nature’s Patterns Patterns - regular, repeated, or recurring forms or designs In nature Numerical patterns – Fibonacci sequence, flowers Geometric patterns - snowflakes Wave patterns - sand dunes, stripes, spots Patterns of movement – flight of birds, motion of heavenly bodies Chaos – clouds, rivers networks, mountains, trees Fibonacci Sequence - Golden Rectangle Geometric Patterns What role does mathematics play in our world? Mathematics is a useful and systematic way to think about and understand nature. Examples Storm is a functions of temperature and pressure Plant growth is a function of nutrients and sunlight Success is a function of good character, positive mind set and discipline What is mathematics all about? Patterns Numbers Operations and Functions Processes Data structure Proof – presents a story about mathematics that works. What do we want mathematics to tell us about nature? How do things happen. Why things happen. What is mathematics for? Mathematics helps organize patterns and regularities in the world Helps us to solve puzzles. It is a systematically way of digging out the rules and structures that lie behind some observed pattern or regularity Rules and structures are then used to explain what’s going on. What is mathematics for? Helps organize the underlying patterns and regularities Helps predict behavior of nature and phenomena Helps control nature and occurrences for our own ends Make practical use of what we have learned about our world END Section 2.1 VARIABLES A variable is sometimes thought of as a mathematical “John Doe” because you can use it as a placeholder when you wan to talk about something but either: 1. you imagine that it has one or more values but you don’t know what they are, or 2. you want whatever you say about it to be equally true for all elements is a given set, and so you don’t want to be restricted to considering only a particular, concrete value for it. Example: 1. Is there a number with the following property: doubling it and adding 3 gives the same result as squaring it? 2. Are there numbers with the property that the sum of their squares equals the square of their sum? 3. Given any real number, its square is nonnegative. Kinds of Mathematical Statements UNIVERSAL STATEMENT – says that a certain property is true for all elements in a set. (Ex. All positive numbers are greater than 0) A CONDITIONAL STATEMENT – says that if one thing is true then some other things also has to be true. (Ex. If 378 is divisible by 18, then 378 is also divisible by 6.) Given a property that may or may not be true, an EXISTENTIAL STATEMENT says that there is at least one thing for which the property is true. (Ex. There is a prime number that is even.) UNIVERSAL CONDITIONAL STATEMENT Universal statements contain some variations of the words “for all” and conditional statements contain versions of the words “if – then”. A Universal conditional statement is a statement that is both Universal and Conditional. Ex: For all animals x, if x is a dog, then x is a mammal. If x is a dog, then x is a mammal. Or: If an animal is a dog, then the animal is a mammal. For all dogs x, x is mammal Or: All dogs are mammals. Examples: Rewriting a Universal Conditional Statement Fill in the blanks to rewrite the following statement: For all real numbers 𝑥, if 𝑥 is a nonzero then 𝑥 2 is positive. a. If a real number is nonzero, then its square is ____. b. For all nonzero real numbers 𝑥,____________. c. If 𝑥 _______, then ________. d. The square of any nonzero real number is ______. e. All nonzero real numbers have _____. TRY: For all real numbers 𝑥, if 𝑥 is greater than 2, then 𝑥 2 is greater than 4. a. If a real number is greater than 2, then its square is ____. b. For all real numbers greater than 2, _____. c. If 𝑥 _____, then ______. d. The square of any real number greater than 2 is ____. e. All real numbers greater than 2 have ____. UNIVERSAL EXISTENTIAL STATEMENTS A universal existential statement is a statement that is universal because its first part says that a certain property is true for all objects of a given type, and is existential because its second part asserts the existence of something. Ex. Every real number has an additive inverse. All real numbers have additive inverses. Or: For all real numbers r, there is an additive inverse for r. Or: For all real number r, there is a real number s such that s is an additive inverse for r. Example: Rewriting a Universal Existential Statement Every pot has a lid a. All pots ____. b. For all pots P, there is _____. c. For all pots P, there is a lid L such that ______. TRY: All bottles have cap a. Every bottle _____. b. For all bottles B, there ______ c. For all bottles B, there is a cap C such that _____ An existential universal statement is a statement that is existential because its first part asserts that a certain object exists and is universal because its second part says that he object satisfies a certain property for all things of a certain kind. For example: There is a positive integer that is less than or equal to every positive integer. Some positive integer is less than or equal to every positive integer. Or: There is a positive integer 𝑚 that is less than or equal to every positive integer. Or: There is a positive integer 𝑚 such that every positive integer is greater or equal to m. Or: There is a positive integer 𝑚 with the property that for all positive integers 𝑛, 𝑚 ≤ 𝑚𝑛. Example: Rewriting an Existential Universal Statement There is a person in my class who is at least as old as every person in my class. a. Some _____ is at least as old as ____ b. There is a person p in my class such that p is ____. c. There is a person p in my class with the property that for every person q in my class, p is ____. TRY: There is a bird in this flock that is at least as heavy as every bird in the flock. a. Some _____ is at least as heavy as ____. b. There is a bird b in this flock such that b is _____. c. There is a bird b in this flock with the property that for every bird d in the flock, b is ____. Exercise 2.1 Page 29 Fill in the blanks using the variable or variables to rewrite the given statement. 1. Is there a real number whose square is -1? a. Is there a real number x, such that ___ ? b. Does there exist ___ such that 𝑥 2 = −1? 2. Is there an integer that has a remainder 2 when it is divided by 5 and a remainder of 3 when it is divided by 6? a. Is an integer n such that n has ______? b. Does there exist ___ such that if n is divided by 5 the remainder is 2 and if ____? Exercise 2.1 Page 29 Fill in the blanks to rewrite the give statement. 8. For all objects J, if J is a square than J has four sides. a. All squares _____. b. Every square ____. c. If an object is a square, then it _____. d. If J ___, then J ____ e. For all squares J, ____. \ \ 2.1 THE LANGUAGE OF SETS Use of the word set as a formal mathematical term was introduced in 1879 by George Cantor (1845- 1918) The Language of Sets Notation: If S is a set, the notation 𝑥 ∈ 𝑆, means than x is an element of S. The notation 𝑥 ∉ 𝑆 means x is not an element of S. A set may be specified using the set-roster notation by writing all of its elements between braces. For example, {1,2,3} denote as the set whose elements are 1,2 and 3. A variation of the notation is sometimes used to describe a very large set as when we write,{1,2,3,…,100}, refers to a set of integers from 1 to 100. A similar notation can also describe an infinite set as when we write, {1,2,3,…}, refers to the set of all positive integers. EXAMPLES Using the Set-Roster Notation 1. Let 𝐴 = 1,2,3 , 𝐵 = 2,3,1 , 1. A, B and C have exactly the same three elements: 1, 𝐶 = 1 , 1, 2, 3, 3, 3,. What are the elements of 2, and 3. Therefore A,B and C are simply different ways to represent the same set. A, B and C? How are A, B, and C related? 2. They are not equal. Because {0} is a set with one 2. Is {0} = 0? element, namely 0, whereas 0 is just zero. 3. How many elements are there in set {1, {1} }? 3. It has 2 elements: 1 and the set whose only element is 1. 4. For each nonnegative integer n, 4. 𝑈1 = 1, −1 , 𝑈2 = 2. −2 , 𝑙𝑒𝑡 𝑈𝑛 = 𝑛, −𝑛. Find 𝑈1 , 𝑈2 , 𝑎𝑛𝑑 𝑈0. 𝑈0 = 0, 0 = {0} CHECK YOUR PROGRESS Using the Set-Roster Notation 1. Let 𝑋 = 𝑎, 𝑏, 𝑐 , 𝑌 = 𝑐, 𝑎, 𝑏 , 𝑍 = 𝑎, 𝑎, 𝑏, 𝑏, 𝑐, 𝑐, 𝑐,. What are the elements of X, Y and Z? How are X, Y, and Z related? 2. How many elements are there in set {a, {a,b}, {a} }? 3. For each positive integer x, 𝑙𝑒𝑡 𝐴𝑥 = 𝑥, 𝑥 2. Find 𝐴1 , 𝐴2 , 𝑎𝑛𝑑 𝐴3. CERTAIN SET OF NUMBERS THAT ARE FREQUENTLY USED Symbol Set ℝ Set of all real numbers ℤ Set of all integers ℚ Set of all rational numbers, or quotients of integers SET- BUILDER NOTATION Let S denote a set and let 𝑃 𝑥 be a property that elements of S may or may not satisfy. We may define a new set to be 𝒕𝒉𝒆 𝒔𝒆𝒕 𝒐𝒇 𝒂𝒍𝒍 𝒆𝒍𝒎𝒆𝒏𝒕𝒔 𝒙 𝒊𝒏 𝑺 𝒔𝒖𝒄𝒉 𝒕𝒉𝒂𝒕 𝑷 𝒙 𝒊𝒔 𝒕𝒓𝒖𝒆. We denote this set as follows: {𝑥 ∈ 𝑆Τ𝑃 𝑥 } EXAMPLES Describe the following sets Answers a. An open interval of real numbers (strictly) a. {𝑥 ∈ 𝑅| − 2 < 𝑥 < 5} between -2 and 5.It is pictured using the number line. b. {𝑥 ∈ 𝑍| − 2 < 𝑥 < 5} b. {−1, 0, 1, 2, 3, 4} c. {𝑥 ∈ 𝑍 + | − 2 < 𝑥 < 5} c. {1, 2, 3, 4} CHECK YOUR PROGRESS Describe the following sets a. {𝑥 ∈ 𝑅| − 5 < 𝑥 < 1} b. {𝑥 ∈ 𝑍| − 1 ≤ 𝑥 < 6} c. {𝑥 ∈ 𝑍 − | − 4 ≤ 𝑥 ≤ 0} SUBSETS If A and B are sets, then A is called a subset of B, written 𝐴 ⊆ 𝐵, if and only if, every element of A is also an element of B. 𝐴 ⊆ 𝐵 means that For all elements x, if 𝑥 ∈ 𝐴, 𝑡ℎ𝑒𝑛 𝑥 ∈ 𝐵 𝐴 ⊆ 𝐵 means that For all elements x, if 𝑥 ∈ 𝐴, 𝑡ℎ𝑒𝑛 𝑥 ∉ 𝐵 𝐴 𝑖𝑠 𝑎 𝒏𝒐𝒕 𝒂 𝒔𝒖𝒃𝒔𝒆𝒕 𝑜𝑓 𝐵 means that there is at least one element of A that is not an element of B. 𝐴 𝑖𝑠 𝒑𝒓𝒐𝒑𝒆𝒓 𝒔𝒖𝒃𝒔𝒆𝒕 𝑜𝑓 𝐵 if, and only if, every element of B is in B but there is at least one element of B that is not in A. EXAMPLES Let 𝑨 = 𝒁+ , 𝑩 = 𝒏 ∈ 𝒁 0 ≤ 𝑛 ≤ 100 , 𝑎𝑛𝑑 Answers C = 100,200,300,400,500. a. False. Zero is not a positive integer. Thus 0 is in B, but it is not in A. 𝐵 ⊆ 𝐴 Evaluate the truth and falsity of the following statements: b. True. Each element in C is a positive integer and, hence, is in A, but there are a. 𝐵 ⊆ 𝐴 elements in A that are not in C. b. C is a proper subset of A c. C and B have at least one element in c. True. For example, 100 is in both C common and B d. 𝐶 ⊆ 𝐵 d. False. For example, 200 is in C but not e. 𝐶 ⊆ 𝐶 in B e. True. Every element in C is in C. CHECK YOUR PROGRESS Let 𝑨 = {𝟐, 𝟐 , 𝟐)𝟐 , 𝑩 = {2, 2 , 2 }, 𝑎𝑛𝑑 𝐶 = {2} Evaluate the truth and falsity of the following statements: a. A ⊆ 𝐵 b. 𝐵 ⊆ 𝐴 c. A is a proper subset of B d. 𝐶 ⊆ 𝐵 e. C is a proper subset of A EXAMPLES Distinction between ∈ , ⊆ a. 2 ∈ {1,2,3} b. 2 ∈ 1, 2,3 c. 2 ⊆ 1, 2, 3 d. 2 ⊆ 1, 2, 3 e. 2 ⊆ 1, 2 f. 2 ∈{ 1 , 2 } CARTESIAN PRODUCT Given sets A and B, the Cartesian product of A and B, denoted AxB read “A cross B”, is the set of all ordered pairs (a, b), where a is in A and b is in B. Symbolically: AxB ={ (a,b)/ a A and b B } EXAMPLES Cartesian Products Answers Let 𝐴 = 1, 2, 3 , 𝑎𝑛𝑑 𝐵 = 𝑢, 𝑣 , 𝑓𝑖𝑛𝑑 a. AxB b. BxA c. BxB d. How many elements are in AxB, BxA, and BxB? CHECK YOUR PROGRESS Cartesian Products: Let Y = 𝑎, 𝑏, 𝑐 , 𝑎𝑛𝑑 𝑍 = 1,2 , 𝑓𝑖𝑛𝑑 a. Y x Z b. Z x Y c. Y x Y d. How many elements are in Y x Z, Z x Y, and Y x Y? Operations on Sets Objectives: Union - Define and describe the union and Intersection intersection of sets and the complement of a set. Complement - Use Venn diagrams to represent sets, subsets, and set operations. Difference - Solve problems involving sets using Venn diagram. 9/23/2024 O per at i ons on Set s 2 The universal set is a general set that contains all elements under discussion. U John Venn (1843 – 1923) created Venn diagrams to show the visual relationship among sets. Universal set is represented by a rectangle Subsets within the universal set are depicted by circles, or sometimes ovals A’ or other shapes. 9/23/2024 O per at i ons on Set s 3 Example 1: Determining Sets From a Venn Diagram Use the Venn diagram to determine each of the following sets: a. U U = { O , ∆ , $, M, 5 } b. A Subtitle A= { O,∆ } c. The set of elements in U that are not in A. {$, M, 5 } Representing Two Sets in a Venn Diagram Disjoint Sets: Two sets that have Equal Sets: If A = B then AB no elements in common. and B A. Proper Subsets: All elements of Sets with Some Common Elements set A are elements of set B. Some means “at least one”. The representing the sets must overlap. 9/23/2024 O per at i ons on Set s 5 Example 2: Determining sets from a Venn Diagram Use the Venn Diagram to determine: a. U Solutions: b. B a. U = { a, b, c, d, e, f, g } c. The set of elements in A but not B b. B = {d, e } d. The set of elements in U that are not in B c. {a, b, c } e. The set of elements in both A and B. d. {a, b, c, f, g } e. {d} 9/23/2024 O per at i ons on Set s 6 The Intersection and Union of Sets The intersection of sets A and B, written A∩B, is the set of elements common to both set A and set B. This definition can be expressed in set-builder notation as follows: A∩B = { x | x A and xB} The union of sets A and B, written AUB is the set of elements are in A or B or in both sets. This definition can be expressed in set-builder notation as follows: AUB = { x | x A or xB} For any set A: 1. A∩Ø = Ø 2. AUØ = A 9/23/2024 O per at i ons on Set s 7 Example 3: Finding the Intersection of Two Sets Find each of the following intersections: a. {7, 8, 9, 10, 11} ∩ {6, 8, 10, 12} {8, 10} b. {1, 3, 5, 7, 9} ∩ {2, 4, 6, 8} Ø c. {1, 3, 5, 7, 9} ∩ Ø Ø 9/23/2024 O per at i ons on Set s 8 Example 4: Finding the Union of Two Sets Find each of the following Unions: a. {7, 8, 9, 10, 11} ∪ {6, 8, 10, 12} {6, 7, 8, 9, 10,11, 12} b. {1, 3, 5, 7, 9} ∪ {2, 4, 6, 8} {1, 2, 3, 4, 5, 6, 7, 8} c. {1, 3, 5, 7, 9} ∪ Ø {1, 3, 5, 7, 9} 9/23/2024 O per at i ons on Set s 9 The Complement of a Set The complement of set A, symbolized by A’ is the set of all elements in the universal set that are not in A. This idea can be expressed in set-builder notation as follows: A’ = {x | x U and x A} The shaded region represents the complement of set A. This region lies outside the circle. 9/23/2024 O per at i ons on Set s 10 Example 5 Finding a Set’s Complement 1.Let U = { 1, 2, 3, 4, 5, 6,7, 8, 9} and A = {1, 3, 4, 7 }. Find A’. 1. Solution: Set A’ contains all the elements of set U that are not in set A. Because set A contains the elements 1,3,4,and 7, these elements cannot be members of set A’: A’ = {2, 5, 6, 8, 9} 9/23/2024 O per at i ons on Set s 11 Example 6: Performing Set Operations Always perform any operations inside parenthesis first! Given: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 1, 3, 7, 9 } B = { 3, 7, 8, 10 } 1. Find a.(A U B)’ Solution: A U B = {1, 3, 7, 8, 9, 10} (A U B)’ = {2, 4, 5, 6} 9/23/2024 O per at i ons on Set s 12 Given: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 1, 3, 7, 9 } B = { 3, 7, 8, 10 } b.A’ ∩ B’ Solution A’ = {2, 4, 5, 6, 8, 10} B’ = {1, 2, 4, 5, 6, 9} A’ ∩ B’ = {2, 4, 5, 6 } 9/23/2024 O per at i ons on Set s 13 Example 7 Determining Sets from a Venn Diagram Set to Description of Set Regions in Venn Determine Diagram a. A B set of elements in A or B or Both I,II,III b. (A B)’ set of elements in U that are not in A B IV c. A B set of elements in both A and B II d. (A B)’ set of elements in U that are not in A B I, III, IV e. A’ B set of elements that are not in A and are in B III f. A B’ set of elements that are in A or not in B or both I,II, IV 9/23/2024 O per at i ons on Set s 14 The Difference of Two Sets The difference of set B from set A, symbolized by A-B. is the set A B of all elements in A, but not in B. A-B 9/23/2024 O per at i ons on Set s 15 Example 8: Find the difference between two sets Given: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 1, 3, 7, 9 } B = { 3, 7, 8, 10 } 1. Find a. A -B b. B -A Solution: Solution: A - B = {1,9} B - A = {8,10} 9/23/2024 O per at i ons on Set s 16 Example 9: Find the difference between two sets Given: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 1, 3, 7, 9 } C = { 1,2,3,4,5} B = {2,4,6,8,10} D={} Find 1. 𝐴 ∪ 𝐵 5. 𝐵 − 𝐶 2. 𝐴 ∩ 𝐵 6. (𝐴 ∩ 𝐶)′ 3. 𝐶′ 7. (𝐴 ∩ 𝐶) ∪ 𝐷 4. 𝐴′ ∩𝐵 8. 𝐶 ∩ 𝐷′ 9/23/2024 O per at i ons on Set s 17 Example 8: The Cardinal Number of the Union of Two Finite Sets 1. Some of the results of the campus blood drive survey indicated that 490 students were willing to donate blood, 340 students were willing to help serve a free breakfast to blood donors, and 120 students were willing to do both. How many students were willing to donate blood or serve breakfast? 9/23/2024 O per at i ons on Set s 18 Example 9: Fifty people were asked about the pets they keep at home. The Venn diagram shows the result. D = {people who have dogs}, F = {people who have fish}, and C = {people who have cats} How many people have a. dogs? b. dogs and fish? c. dogs or cats? d. Fish and cats but not dogs? e. Dogs or fish but not cats? f. All three? g. Neither one of the three? 9/23/2024 O per at i ons on Set s 19 Example 10: A survey asked 100 students whether they play Mobile Legends(ML) or Call Of Duty (COD) mobile. Eighteen students play ML and COD, 41 play ML and 51 play COD. A. Draw a two-set Venn Diagram that displays the result. B. How many students play COD only? C. How may students do not play either of the online games? 9/23/2024 O per at i ons on Set s 20 Sets and Precise Use of Everyday English Set operations and Venn diagrams provide precise ways of organizing, classifying, and describing the vast array of sets and subsets we encounter every day. Or refers to the union of sets And refers to the intersection of sets 9/23/2024 O per at i ons on Set s 21 The way to get started is to quit talking and begin doing. Walt Disney 9/23/2024 O per at i ons on Set s 22 Thank you 9/23/2024 O per at i ons on Set s 23 sajava Language of Relations and Functions Section2.3 https://www.youtube.com/watch?v=GEg1q Page 39 (Mathematics in the Modern World) Y91bc4 https://www.youtube.com/watch?v=Uz0Mt FlLD-k sajava Learning Objectives ✓ Discuss the language and symbols and conventions of mathematics ✓ Explain the nature of mathematics as a language ✓ Perform operations on mathematical expression correctly ✓ Acknowledge that mathematics is a useful language sajava sajava Challenge Jack said to Kris, “Your are my father.” Kris said to Dionesia, “You are my mother.” Dionesia said to Manny’, “ You are my son”. What is the relationship between Jack and Manny? sajava Historical background C.S Peirce (19th century) – American mathematician and logician introduced this formal definition of relation. sajava Relation Definition Let A and B be sets. A relation R from A to B is a subset of AxB. Given an ordered pair (x,y) in AxB, x is related to y by R, written xRy, if and only if, (x,y) is in R. The set A is called the domain of R and the set B is called its co- domain. sajava Example ▪ Let A = 0,1,2 and B =1,2,3 Recall AxB = (0,1),(0,2),(0,3),(1,1),(1,2),(1,3),(2,1),(2,2),(2,3) ▪ Relation R from A to B Let xA and yB. A is related to B only if “x is less than y”. Thus R = (0,1),(0,2),(0,3),(1,2),(1,3),(2,3) ▪ What is the domain of R? 0,1,2 ▪ What is the co-domain of R? 1,2,3 Remark: A relation R is as subset of AxB. sajava Example: Let A = {1,2} and B={1,2,3} and define a relation R from A to B as follows. Given any (x,y) AxB. 𝒙−𝒚 (x,y) R means that is an integer. 𝟐 a. Find the elements of R. b. What are the domain and co-domain of R? c. Construct the arrow diagram of R. sajava Solutions a. List the elements of A x B = { (1,1) , (1,2), (1,3), (2,1), (2,2), (2,3) } Examine each ordered pair in AxB to see whether its elements satisfy the defining condition for R. 𝑥−𝑦 1−1 (1,1) ; = = 0, which is an integer, (1,1) R or 1R1. 2 2 𝑅 = { 1,1 , 1,3 , 2,2 } sajava Answers: 𝑎. 𝑅 = { 1,1 , 1,3 , 2,2 } 𝑏. 𝐷𝑜𝑚𝑎𝑖𝑛 = {1, 2} 𝐶𝑜 − 𝐷𝑜𝑚𝑎𝑖𝑛 = {1, 2,3} 𝑐. 𝐴𝑟𝑟𝑜𝑤 𝐷𝑖𝑎𝑔𝑟𝑎𝑚 sajava Check your Progress Let 𝑌 = 0,1,2 𝑎𝑛𝑑 𝑍 = {0,1} and define a relation R from Y to Z as follows: Given any 𝑥, 𝑦 ∈ 𝑌 × 𝑍, 𝑥+𝑦 𝑥, 𝑦 ∈ 𝑅 𝑚𝑒𝑎𝑛𝑠 𝑡ℎ𝑎𝑡 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟. 2 a. Find the elements of R b.What are the domain and co-domain of R? c. Construct an arrow diagram of R. sajava Functions A function F from set A to set B is a relation with domain A and co-domain B that satisfies the following two properties. 1.For every element x in A, there is an element y in B such that (x, y) F. 2.For all elements x in A and y and z in B, if (x , y ) F and (x, z) F. then y = z. sajava Functions **A relation F from A to B is a function if, and only if: 1.Every element of A is the first element of an ordered pair of F. 2.No two distinct ordered pairs in F have the same first element. Notation: If A and B are sets and F is a function from A to B, then given any element x in A, the unique element in B is related to x by F is denoted F(x), which is read “F of x.” sajava Example: 1. Given relation R = { (1,2) , (2,2) , (3, 3 ) } R is a function (many to one) 2. Given relation T = { ( 1, 2) , ( 1,3) , (2, 4) } T is not a function(one to many) sajava Example: 3. Let 𝐴 = 2,4,6 𝑎𝑛𝑑 𝐵 = 1,3,5. Which of the relations R, S, and T defined below are functions from A to B? a. 𝑅 = 2,5 , 4,1 , 4,3 , 6,5 b. For all 𝑥, 𝑦 ∈ 𝐴 × 𝐵, 𝑥, 𝑦 ∈ 𝑆 𝑚𝑒𝑎𝑛𝑠 𝑡ℎ𝑎𝑡 𝑦 = 𝑥 + 1 c. T defined by the arrow diagram 2 1 4 3 6 5 sajava Answers: 1. R is not a function because it does not satisfy property (2). The ordered pairs (4,1) and (4,3) have the same first element. 2. S is not a function because it does not satisfy property (1). 3. T is a function. sajava Check your Progress Let 𝑋 = 𝑎, 𝑏, 𝑐 𝑎𝑛𝑑 𝑦 = {1,2,3,4}. Which of the relations A, B, C defined below are functions from X to Y a. 𝐴 = 𝑎, 1 , 𝑏, 2 , 𝑐, 3 b.For all 𝑥, 𝑦 ∈ 𝑋 × 𝑌, 𝑥, 𝑦 ∈ 𝐵 𝑚𝑒𝑎𝑛𝑠 𝑥 𝑖𝑠 𝑎 𝑣𝑜𝑤𝑒𝑙 𝑎𝑛𝑑 𝑦 𝑖𝑠 𝑒𝑣𝑒𝑛 c. C is defined by the arrow diagram a 1 2 b 3 4 c sajava sajava Evaluating functions sajava Various Ways of Presenting a Function ✓as set of ordered pairs ✓as arrow diagram (mapping of two sets) ✓as function machine ✓as an equation (defined by formulas) ✓as a graph sajava End The Lord is good, a stronghold in the day of trouble and He knows them that trust in Him. Nahum 1:7 sajava 9/24/2024 PROBLEM SOLVING CHAPTER 3 Problem Solving Section 3.1 Inductive and Deductive Reasoning Problem Solving 9/24/2024 Inductive Reasoning Problem Solving 9/24/2024 Inductive Reasoning ◦The type of reasoning that forms a conclusion based on the examination of specific examples is called inductive reasoning. The conclusion formed by using inductive reasoning is a conjecture, since it may or may not be correct. ◦When you examine a list of numbers and predict the next number in the list according to some pattern you have observed, you are using inductive reasoning. Problem Solving 9/24/2024 Example 1 – Use Inductive Reasoning to Predict a Number ◦Use inductive reasoning to predict the next number in each of the following lists. ◦a. 3, 6, 9, 12, 15, ? b. 1, 3, 6, 10, 15, ? ◦Solution: a. Each successive number is 3 larger than the preceding number. Thus we predict that the next number in the list is 3 larger than 15, which is 18. Problem Solving 9/24/2024 Example 1 – Solution ◦ b. The first two numbers differ by 2. The second and the third numbers differ by 3. ◦ It appears that the difference between any two numbers is always 1 more than the preceding difference. ◦ Since 10 and 15 differ by 5, we predict that the next number in the list will be 6 larger than 15, which is 21. Problem Solving 9/24/2024 Galileo Galilei (1564 -1642) Period of Pendulum in Length of Pendulum heartbeats 1 1 4 2 9 3 16 4 25 5 36 6 49 ? Problem Solving 9/24/2024 Example 3 – A tsunami is a sea wave produced by underwater earthquakes. The height of a tsunami as it approaches land depends on the velocity of the tsunami. Velocity of tsunami, Height of tsunami, a. What happened to the height in feet per second in feet of a tsunami when its velocity is 6 4 doubled? 9 9 12 16 b. What should be the height of 15 25 a tsunami if its velocity is 30 18 36 feet per second? 21 49 24 64 Problem Solving 9/24/2024 Example: The diagram below shows a series of square tiles. Figure 1 2 3 4 5 6 10 15 No. Of tiles Problem Solving 9/24/2024 Counterexamples: ◦ A statement is a true statement provided that it is true in all cases. If you can provide one case for which a statement is not true, called the counterexample, then the statement is a false statement. ◦ Ex. For all numbers x 𝑎. 𝑥 > 0 b. 𝑥 2 > 𝑥 c. 𝑥 2 = 𝑥 𝑥 𝑥+3 d. =1 e. =𝑥+1 f. 𝑥 2 + 16 = 𝑥 + 4 𝑥 3 Problem Solving 9/24/2024 9/24/2024 DEDUCTIVE REASONING Problem Solving Deductive Reasoning ◦Another type of reasoning is called deductive reasoning. ◦Deductive reasoning is distinguished from inductive reasoning in that it is the process of reaching a conclusion by applying general principles and procedures. Problem Solving 9/24/2024 Example 5 – Use Deductive Reasoning to Establish a Conjecture ◦Use deductive reasoning to show that the following procedure produces a number that is four times the original number. ◦Procedure: Pick a number. Multiply the number by 8, add 6 to the product, divide the sum by 2, and subtract 3. Problem Solving 9/24/2024 Example 5 – Solution ◦Let n represent the original number. ◦ Multiply the number by 8: 8𝑛 ◦ Add 6 to the product: 8𝑛 + 6 ◦ Divide the sum by 2: 8𝑛 + 6 = 4𝑛 + 3 2 ◦ Subtract 3: 4𝑛 + 3 − 3 = 4n ◦We started with n and ended with 4n. The procedure given in this example produces a number that is four times the original Problem number. Solving 9/24/2024 Example 6 – Use Deductive Reasoning to Establish a Conjecture ◦Procedure: Pick a number. Multiply the number by 6, add 10 to the product, divide the sum by 2, and subtract 5. Problem Solving 9/24/2024 Problem Solving 9/24/2024 Inductive vs Deductive Reasoning ◦ Determine whether each of the following arguments is an example of inductive or deductive reasoning. 1. During the past ten years, a tree has produced plums every other year. Last year the tree did not produce plums, so this year the tree will produce plums. 2. All home improvements cost more than the estimate. The contractor estimated that my home improvement will cost ₱800,000. Thus my home improvement will cost more than ₱800,000. Problem Solving 9/24/2024 Inductive vs Deductive Reasoning ◦ Determine whether each of the following arguments is an example of inductive or deductive reasoning. 3. All Gillian Flynn novels are worth reading. The novel Gone Girl is a Gillian Flynn novel. Thus, Gone Girl is worth reading. 3. All home improvements cost more than the estimate. The contractor estimated that my home improvement will cost ₱800,000. Thus my home improvement will cost more than ₱800,000. Problem Solving 9/24/2024 Intuition, Proof and Certainty ◦ Intuition: There are a lot of definition of an intuition and one of these is that it is an immediate understanding or knowing something without reasoning. It does not require a big picture or full understanding of the problem, as it uses a lot of small pieces of abstract information that you have in your memory to create a reasoning leading to your decision just from the limited information you have about the problem in hand. Intuition comes from noticing, thinking and questioning. As a student, you can build and improve your intuition by doing the following: a. Be observant and see things visually towards with your critical thinking. b. Make your own manipulation on the things that you have noticed and observed. c. Do the right thinking and make a connections with it before doing the solution. Problem Solving 9/24/2024 Example: ◦ Based on the given picture below, which among of the two yellow lines is longer? Is it the upper one or the lower one? Problem Solving 9/24/2024 Now, let us test your intuition. We have here a set of problems. Make your own conclusion based on the given problem without solving it mathematically. ◦1. Which of the two have the largest value? Explain it accurately towards to correct conclusion. ◦ 103 ; 310 ◦2. Which among of the following has a largest product? ◦ 34 × 12 ; 21 × 43 ; 54 × 31 Problem Solving 9/24/2024 ◦3. Look at the figure below. Are two lines a straight line?. What is your intuition? Problem Solving 9/24/2024 ◦ B. Proof and Certainty ◦ Another equally important lesson that the student should be learned is on how to deal with mathematical proof and certainty. By definition, a proof is an inferential argument for a mathematical statement while proofs are an example of mathematical logical certainty. A mathematical proof is a list of statements in which every statement is one of the following: ◦ (1) an axiom ◦ (2) derived from previous statements by a rule of inference ◦ (3) a previously derived theorem Problem Solving 9/24/2024 ◦ A mathematical proof is a list of statements in which every statement is one of the following: ◦ (1) an axiom ◦ (2) derived from previous statements by a rule of inference ◦ (3) a previously derived theorem ◦ There is a hierarchy of terminology that gives opinions about the importance of derived truths: ◦ (1)A proposition is a theorem of lesser generality or of lesser importance. ◦ (2) A lemma is a theorem whose importance is mainly as a key step in something deemed to be of greater significance. ◦ (3) A corollary is a consequence of a theorem, usually one whose proof is much easier than that of the theorem itself. Problem Solving 9/24/2024 ◦ TWO WAYS ON HOW TO PRESENT THE PROOF ◦ a. Outline Form Proposition: If P then Q. ◦ 1. Suppose/Assume P ◦ 2. Statement ◦ 3. Statement... Statement Therefore Q. ◦ b. Paragraph Form Proposition: ◦ If P then Q. Assume/Suppose P. ____________. ___________. _____________________. ____________... _____________. _______________. _________________. Therefore Q. Problem Solving 9/24/2024 Example: Prove (in outline form) that “If x is a number with 5x + 3 = 33, then x = 6” ◦ Proof: ◦ 1. Assume that x is a number with 5x + 3 = 33. ◦ 2. Adding -3 both sides of an equation will not affect the equality of the two members on an equation, thus 5x + 3 – 3 = 33 – 3 ◦ 3. Simplifying both sides, we got 5x = 30. ◦ 4. Now, dividing both member of the equation by 5 will not be affected the equality so 5𝑥 5 = 30 5. ◦ 5. Working the equation algebraically, it shows that x = 6. ◦ Therefore, if 5x + 3 = 33, then x = 6. Problem Solving 9/24/2024 Prove (in paragraph form) that “If x is a number with 5x + 3 = 33, then x = 6” ◦ Proof: ◦ If 5x + 3 = 33, then 5x + 3 − 3 = 33 − 3 since subtracting the same number from two equal quantities gives equal results. 5x + 3 − 3 = 5x because adding 3 to 5x and then subtracting 3 just leaves 5x, and also, 33 − 3 = 30. Hence 5x = 30. That is, x is a number which when multiplied by 5 equals 30. The only number with this property is 6. Therefore, if 5x + 3 = 33 then x = 6. Problem Solving 9/24/2024 Logic Puzzles ◦Logic puzzles, can be solved by using deductive reasoning and a chart that enables us to display the given information in a visual manner. Problem Solving 9/24/2024 Example: Solve a Logic Puzzle ◦Each of four neighbors, Sean, Maria, Sarah, and Brian, has a different occupation (editor, banker, chef, or dentist). From the following clues, determine the occupation of each neighbor. ◦1. Maria gets home from work after the banker but before the dentist. ◦2. Sarah, who is the last to get home from work, is not the editor. ◦3. The dentist and Sarah leave for work at the same time. ◦4. The banker lives next door to Brian. Problem Solving 9/24/2024 Example – Solution ◦From clue 1, Maria is not the banker or the dentist. In the following chart, write X1 (which stands for “ruled out by clue 1”) in the Banker and the Dentist columns of Maria’s row. ◦From clue 2, Sarah is not the editor. Write X2 (ruled out by clue 2) in the Editor column of Sarah’s row. Problem Solving 9/24/2024 Example – Solution ◦We know from clue 1 that the banker is not the last to get home, and we know from clue 2 that Sarah is the last to get home; therefore, Sarah is not the banker. Write X2 in the Banker column of Sarah’s row. ◦From clue 3, Sarah is not the dentist. Write X3 for this condition. There are now Xs for three of the four occupations in Sarah’s row; therefore, Sarah must be the chef. Problem Solving 9/24/2024 Example – Solution ◦Place a in that box. Since Sarah is the chef, none of the other three people can be the chef. Write X3 for these conditions. There are now Xs for three of the four occupations in Maria’s row; therefore, Maria must be the editor. ◦Insert a to indicate that Maria is the editor, and write X3 twice to indicate that neither Sean nor Brian is the editor. Problem Solving 9/24/2024 Example – Solution ◦From clue 4, Brian is not the banker. Write X4 for this condition. See the following table. Since there are three Xs in the Banker column, Sean must be the banker. ◦Place a in that box. Thus Sean cannot be the dentist. Write X4 in that box. Since there are 3 Xs in the Dentist column, Brian must be the dentist. Place a in that box. ◦Sean is the banker, Maria is the editor, Sarah is the chef, and Brian is the dentist. Problem Solving 9/24/2024 Example: Brianna, Ryan,, Tyler and Ashley were recently elected as the new class officers (president, vice president, secretary, treasurer) of the freshmen college organization. From the following clues, determine which position each holds. 1. Ashley is younger than the president but older than the treasurer. 2. Brianna and the secretary are both the same age, and they are the youngest member of the group. 3. Tyler and the secretary are next-door neighbors. Problem Solving 9/24/2024 KenKen® Puzzles KenKen® is an arithmetic-based logic that was invented by a Japanese math teacher Tetsuya Miyamoto in 2004. The noun “ken” has “knowledge” and “awareness” as synonyms. So Kenken translates as “knowledge squared” or “awareness squared”. Problem Solving 9/24/2024 6+ 3× 5+ 2 Problem Solving 9/24/2024 6× 7+ 2 8× 4× 12 × 1− 1 Problem Solving 9/24/2024 Magic Squares 2 13 10 11 6 12 4 15 1 Problem Solving 9/24/2024 THANK YOU! Problem Solving 9/24/2024 Solving Problem Strategies “Polya’s Four Steps in Solving Problem” George Polya is one of the foremost recent mathematicians to make a study of problem solving. He was born in Hungary and moved to the United States in 1940. He is also known as “The Father of Problem Solving”. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory. He is also noted for his work in heuristics and mathematics education. Heuristic, a Greek word means that "find" or "discover" refers to experience- based techniques for problem solving, learning, and discovery that gives a solution which is not guaranteed to be optimal. 1.Understand the Problem 2.Devise a Plan 3.Carry Out a plan 4.Review the Solution Example: The sum of three consecutive positive integers is 165. What are these three numbers? Step 1. Understand the problem When we say consecutive numbers, these are like succeeding numbers. Say, 4, 5, 6 are three consecutive numbers for single-digit numbers. For the two digit number, example of these three consecutive is 32, 33, and 34. Noticing that the second number added by 1 from the first number and the third number is increased by 2 from the first number. Step 2. Devise a plan From the previous discussion of this lesson, devising a plan is very essential to solve a problem. We could use an appropriate plan for this kind of problem and that is formulating a working equation. Since we do not know what are these three consecutive positive integers, we will be using a variable, say x to represent a particular number. This variable x could be the first number. Now, since it is consecutive, the second number will be increased by 1. So, the possible presentation would be x + 1. The third number was increased by 2 from the first number so the possible presentation would be x + 2. Since, based on the problem that the sum of these three consecutive positive integers is 165, the working equation is: Step 4. Look back and review the solution We need to review our solution to check if the answer is correct. How are we going to do that? Just simply add the identified three consecutive positive integers and the result should be 165. So, adding these three numbers, 54 + 55 + 56 will give us a sum of 165. Step 3. Carry out the plan We already know the working formula. To be able to determine the three positive consecutive integers, we will be using the concept of Algebra here in order to solve the problem. Manipulating algebraically the given equation; Ex 2. There are ten students in a room. If they give a handshake for his classmate once and only once, how many handshakes can be made? 1.Understand the Problem 2.Devise a Plan 3.Carry Out the Plan 4.Review the solution Ex 3. How many squares are in the figure? Thank You!
