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Mathematics in the Modern World A Worktext for GEC104 Academic Year 2020 – 2021 Mindanao State University – Main Campus, Marawi City Mathematics in the Modern World Mahid M. Mangontarum Mark P. Laurente Salma L. Naga-Ma...

Mathematics in the Modern World A Worktext for GEC104 Academic Year 2020 – 2021 Mindanao State University – Main Campus, Marawi City Mathematics in the Modern World Mahid M. Mangontarum Mark P. Laurente Salma L. Naga-Marohombsar Norlailah M. Madid Aslayn H. Datu-dacula Rodelito M. Aldema Ednelyn Ranas-Cantallopez Janet B. Macoy Charles B. Montero Redeemtor R. Sacayan Gloria A. Rosalejos Erneta A. Intod Merry Jane S. Benitez Amerkhan G. Cabaro Alyssa Fatmah S. Mastura Raicah R. Cayongcat-Rakim Mathematics Department College of Natural Sciences and Mathematics 1  About the Cover The main image in the cover page shows the inner chambers of a nautilus shell. Through the years, mathematicians have utilized the well -known Fibonacci sequence to describe certain shapes and patterns that appear in nature including the one found inside a nautilus shell. This has become a symbol which represents the connection between mathematics and the world we live in. The image can be viewed online using the following link: https://www.pinterest.ph/pin/527273068847969697/ 2  Preface Mathematics in the Modern World is the worktext for the course GEC104 – Mathematics in the Modern World for the First Semester of the Academic Year 2020 – 2021 in Mindanao State University (MSU) – Main Campus, Marawi City. It was prepared by a committee composed of select faculty members from the Mathematics Department of the MSU Main Campus by virtue of Office Order No. 2, Series of 2020, issued by the Chairperson of the Department. GEC104 is one of the nine (9) General Education courses listed in CHED Memorandum Order No. 20, Series of 2013, which is mandatory for all undergraduate students to take regardless of their major. This emanated from the implementation of the K to 12 Curriculum in the Philippines per Republic Act No. 10533, otherwise known as the “Enhanced Basic Education Act or 2013”. The worktext Mathematics in the Modern World is also in line with the Implementing Rules and Regulations for the Flexible Learning Options (FLO) of the Mindanao State University – Main Campus as approved by the MSUS President Habib W. Macaayong, DPA, on September 4, 2020. The purpose is to ensure continued delivery of quality education to the university constituents amidst the COVID-19 pandemic in accordance with the guidelines of the Inter-Agency Task Force for the Management of Emerging Infectious Diseases (IATF) and advisories from the Commission on Higher Education (CHED). The main intent of the text is to promote interest in and simplify the study of Mathematics, especially to undergraduate students. Hence, it provides discussions on the relevance and importance of mathematics, and presents methods in solving practical problems in the real world. In order to fully appreciate this workbook, as a prerequisite, a thorough understanding of basic algebra and statistics–including operations, sets, data collection and sampling techniques is of great use and importance. MAHID M. MANGONTARUM Co-Chairperson, GEC104 Committee 3  Message from the Chairperson Greetings of Peace MSU students. Today, the world of education is undergoing massive changes brought about by the global pandemic. Teachers, students and parents are faced with more challenges during these unprecedented times. Because of the so-called “new normal”, the faculty members of the Mindanao State University – Main Campus, the Mathematics Department in particular, will be engaging the students using the virtual platform and other flexible learning options. The road may be difficult but is still an exciting one since this is an opportunity to for everyone to learn new things. It has been two years since the planning for a worktext for GEC104 (Mathematics in the Modern World). We have been dreaming of this moment since the day we had our training for the teaching of the new General Education Curriculum back in 2018. I would like to express my heartfelt congratulations to the working committee who made this dream into reality. This is a job well-done. The introduction of this GEC104 worktext is very timely given the current situation. I storngly believe that every instructor, students and even parents who will be given a chance to read this text will have the opportunity to dive into the complex but rich world of Mathematics, and might develop new ideas that will be benificial to them in the future. Mathematics plays a vital role in our lives that is why we should learn and continue to appreciate mathematics. I encourage all to work together in making this First Semester, AY 2020 – 2021, fruitful and meaningful. Thank you very much. MARK P. LAURENTE, PhD, CSPE, LPT Department Chairperson 4 Table of Contents Chapter 0. Classroom Orientation............................................................ 6 Brief Description of the Course.............................................................................. 6 Learning Outcomes.................................................................................................. 6 Grading System......................................................................................................... 7 Grading Scale............................................................................................................ 7 The Mindanao State University............................................................................... 8 Chapter 1. Mathematics in our World....................................................... 9 Getting to know more about Mathematics............................................................. 9 Fibonacci Sequence................................................................................................ 14 Chapter 2. Mathematical Language and Symbols.................................... 18 Importance of Language........................................................................................ 18 Characteristics of Mathematical Language......................................................... 19 Nouns versus Sentences......................................................................................... 19 Synonyms and Antonyms....................................................................................... 20 Conventions............................................................................................................. 21 Chapter 3. Logic...................................................................................... 23 A Brief History........................................................................................................ 23 Logic Statement...................................................................................................... 24 Simple and Compound Statements....................................................................... 26 Universal and Existential Statements................................................................... 32 Quantifiers and Negation...................................................................................... 32 Conditional Statements.......................................................................................... 36 Derived Conditionals and Biconditional Statements.......................................... 40 Chapter 4. Problem Solving and Reasoning............................................ 44 Inductive and Deductive Reasoning..................................................................... 44 Problem-Solving Strategies.................................................................................... 47 Games that Promote Problem-Solving Skills....................................................... 60 Chapter 5. Elementary Statistics............................................................. 65 Statistical Measures of Data.................................................................................. 65 Normal Probability Distribution........................................................................... 97 About the Authors................................................................................ 107 5 Chapter 0. Classroom Orientation Brief Description of the Course The Course Code is “GEC104” and the Course Description is “Mathematics in the Modern World”. This is a three-unit course that deals with the nature of mathematics, appreciation of its practical, intellectual, and aesthetic dimensions, and application of mathematics in daily life. Learning Outcomes At the end of the course, the students should be able to: 1. Discuss and argue about the nature of mathematics, what it is, how it is expressed, represented, and used; 2. Use different types of reasoning to justify statements and arguments made about mathematics and mathematical concepts; 3. Discuss the language and symbols of mathematics; 4. Use a variety of statistical tools to process and manage numerical data; 5. Analyze codes and coding schemes used for identification, privacy and security purposes; 6. Use mathematics in other areas such as finance, voting, health and medicine, business, environment, arts and design, and recreation; 7. Appreciate the nature uses of mathematics in everyday life; and affirm honesty and integrity in the application of mathematics to various human endeavors. 6 Grading System Attendance 5% Classroom Performance 35% (quizzes, chapter exams, group/individual activities) Homework/Project 30% (assignments, integrating project) Midterm Exam 15% Final Exam 15% TOTAL 100.0% Grading Scale Percentage Grade Remark 96 – 100 1.00 Excellent 92 – 95 1.25 88 – 91 1.50 Very Good 84 – 87 1.75 80 – 83 2.00 Good 74 – 79 2.25 69 – 73 2.50 Satisfactory 64 – 68 2.75 60 – 63 3.00 Pass Below 60 5.00 Failed 7 The Mindanao State University Mandate As enshrined in its Charter (RA 1387), MSU was established in Marawi City on September 1, 1961 to achieve the following mandate: 1. Educate the youth of Mindanao, Sulu and Palawan (MINSUPALA) by offering degree programs in various fields of learning; 2. Support businesses and industries in the region by providing their manpower requirements; and 3. Integrate Muslims and other cultural minorities into the mainstream of national life. Philosophy The MSUS is committed to the total development of man and to the search for truth, virtue and academic excellence. Vision The MSUS aspires to be the Premier Supra Regional University in the MINSUPALA region. Mission Committed to the attainment of peace and sustainable development in the MINSUPALA region, the MSUS will set the standards of excellence in Science, Arts, Technology and other fields; accelerate the economic, cultural, socio-political and agro- industrial development of the Muslim and other cultural groups, thereby facilitating their integration into the national community; preserve and promote the cultural heritage of the region and conserve its rich natural resources; and infuse moral and spiritual values. For collaborative efforts, for diplomatic relations and for international recognition as a leading institution of higher learning, the MSUS will pursue vigorously linkages with foreign agencies. 8 Chapter 1. Mathematics in our World The field of Mathematics is a diverse discipline that deals not only with arithmetic and geometry, but also with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems. Mathematics reveals hidden patterns that help us understand nature, the world around us, and the universe in general. Through the years, many scientists and mathematicians have discovered mathematical concepts and patterns in nature which are also observed in various phenomena. In this chapter, we will learn more of what is mathematics. Getting to know more about Mathematics What is Mathematics? To help us understand the world of mathematics, we have to first understand the meaning of mathematics. The following are some common definitions of mathematics: 1. Mathematics is all about the unbelievable patterns of numbers formed by nature and of the universe; 2. Mathematics is all about language in different forms like patterns, shapes, music, among others; 3. Mathematics is all about what our eyes can see, what our ears can hear and even what we can perceive in our physical environment; 4. Mathematics is a language we understand. Aside from these definitions, British mathematician and writer Ian Nicholas Stewart, also defines mathematics in his book “Nature’s numbers: The Unreal Reality of Mathematical Imagination” as a formal system of thought for recognizing and exploiting patterns. The book takes readers on an exciting, lucid voyage of discovery as Stewart investigates patterns of form, number, shape and movement in the world around us. Summing up these ideas lead to the discovery of the great secret uncovered by mathematics: Nature’s patterns are not just there to be admired, they are vital clues to the rules that govern natural processes. 9  Mathematics is usually associated with the following notions: 1. Numbers and counting (operations). People get to know mathematics through counting numbers and performing operations associated with numbers. 2. Numeric and Geometric patterns. Patterns done either numerically or geometrically are also one of those common perceptions to mathematics. 3. Patterns of movement. There are moving bodies which follow a pattern that can be mathematically represented. Indeed, any person can recognize the beauty of nature’s mathematical patterns without any mathematical training. Recognize Deduce the the clues of underlying patterns in rules and nature regularities The figure above shows how mathematics is related to nature. In this diagram, we see that in order to understand nature, we may use mathematics by recognizing the clues in the patterns present in nature. Where is Mathematics? Mathematics is indeed everywhere. 1. We see hints or clues of it in nature. Philosopher Galileo once said “The laws of nature are written in the language of mathematics”. Mathematics can be seen in nature if we look close enough. The things in nature that we see, hear, touch, smell and taste can be explained through the help of mathematics. 2. In our daily routine. In our daily routine, mathematics has more involvement than we know. Students in particular use mathematics in simple activities like budgeting their weekly allowance, making schedules for studying during exams, or even when deciding what time to wake up after an overnight movie marathon. 3. In our work. The use of mathematics is not limited only to mathematicians or mathematics educators. For instance, teachers from other fields make use of mathematics in computing grades. Airplane pilots and ship captains require knowledge in computation and algebra in order to properly navigate. Most 10  factory workers nowadays need to understand basic algebra and trigonometry to operate complex manufacturing electronic equipment. 4. In people and communities. Mathematics plays an important role in people’s life. For instance, mathematics is very helpful in understanding various features of the human body starting from body structures to cellular formations. Populations and livelihood of different communities also make use of mathematics. 5. In events. Mathematics helps us understand past and present events so that we may be able to predict a possible recurrence. This is particularly important in the case of preventing calamities. What is Mathematics for? Mathematics is on the list of underappreciated disciplines by most people. Many are still unaware of the need for every individual to know and appreciate the role that mathematics plays, some of which are listed as follows. 1. To help us unravel the puzzles of nature, and provides a useful way to think about nature. Mathematics help provide solutions to anything concerning the nature. This helps human beings build a better connection to what surrounds them. 2. Organize patterns and regularities as well as irregularities. With mathematics, we are able to make arrangements and understand patterns of various behaviors. 3. To be able to predict. By examining previous occurrences and patterns, we are able to predict future outcomes using various mathematical tools. 4. To help us control weather and epidemics. Meteorologists make use of mathematical concepts in studying and understanding the behavior of the atmosphere in order to prevent a disaster from happening. Also, mathematics help provide effective solutions in preventing the widespread of epidemics. 5. Provides tool for calculations. Mathematics is best-known for calculation. 6. Provides new questions to think about. Mathematics gives precise information about anything. Thus, a single idea may lead to a series of questions which can develop into further research. What is Mathematics all about? Mathematics one broad discipline. The following list enumerate some of the things which are usually associated with mathematics. 1. Numbers, symbols, notations. When people are asked what mathematics is all about, they usually associate mathematics with numbers, symbols and notations. 11  2. Operations, equations, functions. People recognize mathematics when looking at expressions involving operations, or mathematical equations and functions. 3. Processes and “thingification” of processes (that are abstractions). There are a few people with adequate knowledge in mathematics who may say that mathematics is all about the study of certain processes, and giving concrete representations to things that are usually abstract in form. 4. Proof – a story rather than a sequence of statements. Mathematics also serves to clarify and render precise information which can be used as a proof in testing the credibility of an idea being studied. How is Mathematics done? For us to understand the world of mathematics, there is a need for us to understand how it is done. The following is a list of guidelines that may help one to study mathematics: 1. With curiosity. All mathematical study are motivated by the author’s curiosity. The need to find answers to questions pushes mathematicians to dig deeper and conduct more studies. 2. With a penchant for seeking patterns and generalities (inquisitiveness). Mathematicians do not only depend on a piece of idea. They in fact seek patterns and combine pieces of information to make generalizations of particular cases. 3. With a desire to know the truth. One major factor for the continuous development of mathematics is the desire of individuals in seeking explanations to various phenomena. 4. With trial and error. Like in any scientific discovery or invention, there are times when mathematics requires us to undergo numerous trials and experience failures before arriving at a successful end. 5. Without fear of facing more questions and problems to solve. When studying mathematics, one must be fearless. Fearless in the sense that when one person attempts to acquire more knowledge about something, he usually faces more questions and more problems in the process. Who uses Mathematics? 1. Mathematicians who specializes in either Pure or Applied fields. 2. Scientists studying either natural or social sciences. 3. Practically everyone uses mathematics in dealing with everyday life activities. Every individual uses different mathematics at different times for different purposes using different tools with different attitudes (diversity and universality). 12 Why is Mathematics important to know? Mathematics is intertwined with every element of our lives either directly or indirectly. In order to help those who seek to understand the importance of mathematics and its connections with the real world, mathematics educators/teachers must convince them first to see mathematics as something necessary in life. This will help inspire them to hone their mathematics skills. The following are some of the reasons why it is important to know mathematics: 1. It puts order in disorder. With the help of mathematics, we can organize well. There are available methods in mathematics that can be utilized in assigning courses to teachers, arranging schedule of classes for students, deciding where to build hospitals, finding strategic areas to place CCTV’s and more. When we are faced with disarray, one may look at possible solutions in mathematics. 2. It helps us become better persons. Adequate knowledge in mathematics may help us become better persons as it can be used as guide in making decisions. For instance, a student with knowledgeable in operations can easily make use of mathematical computations in budgeting his allowance to ensure that it will be sufficient. 3. It helps make the world a better place to live in. Mathematics help explain everything in nature which in return help people build a strong relationship with their surroundings. ACTIVITY Watch the following videos: 1. “Nature by Numbers” by Cristóbal Vila Link: https://www.youtube.com/watch?v=kkGeOWYOFoA 2. “The Fingerprint of God: Fibonacci numbers & Golden ratio” by Happy2bmuslim93 Link: https://www.youtube.com/watch?v=7Uo4Oond1e8 Write a 1-page essay about what you have learned after watching the said videos. 13 Fibonacci Sequence Fibonacci (1170–1250), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano (which means Leonardo the Traveller from Pisa) is one of the best-known mathematicians of medieval Europe. In 1202, after a trip that took him to several Arab and Eastern countries, Fibonacci wrote the book Liber Abaci. In this book, he introduced the so- called modus Indorum (method of the Indians), today known as the Hindu–Arabic numeral system, and explained why the Hindu-Arabic numeral system that he had learned about during his travels was a more sophisticated and efficient system than the Roman numeral system. Moreover, in the said book, Fibonacci http://totallyhistory.com/fibonacci/ solved a problem that he himself created. The problem is concerned with the birth rate of rabbits based on idealized assumptions. Here is a statement of Fibonacci’s rabbit problem. “At the beginning of a month, you are given a pair of newborn rabbits. After a month the rabbits have produced no offspring; however, every month thereafter, the pair of rabbits produces another pair of rabbits. The offspring reproduce in exactly the same manner. If none of the rabbits dies, how many pairs of rabbits will there be at the start of each succeeding month?” The solution of this problem is a sequence of numbers that we now call the Fibonacci sequence. This solution can be represented by the following figure that shows the numbers of pairs of rabbits on the first day of each of the first six months. The larger rabbits represent mature rabbits that produce another pair of rabbits each month. The numbers in the blue region—1, 1, 2, 3, 5, 8—are the first six terms of the Fibonacci sequence. 14  In this diagram, it can be seen that the number of pairs of rabbits for any month after the first two months can be determined by adding the numbers of pairs of rabbits in each of the two previous months. For instance, the number of pairs of rabbits at the start of the sixth month is 3+5=8. This can be used as a basis in giving a formal definition to the sequence. A recursive definition for any sequence is one in which each successive term of the sequence is defined by using some of the preceding terms. In this case, if we use the notation 𝐹𝑛 to represent 𝑛𝑡ℎ Fibonacci number, then the sequence of numbers 𝐹𝑛 , where 𝑛 = 1,2, … , is defined by the linear recurrence equation 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 , for 𝑛 ≥ 3 and with 𝐹1 = 𝐹2 = 1. As a convention, we set 𝐹0 = 0. Example. Use the definition of the Fibonacci numbers to find the 7th and 8th Fibonacci numbers. Solution. Recall that the first six terms in the Fibonacci sequence are 1, 1, 2, 3, 5, and 8. Using the recursive definition of 𝐹𝑛 , we have 𝐹7 = 𝐹6 + 𝐹5 = 8 + 5 = 13 15 and 𝐹8 = 𝐹7 + 𝐹6 = 13 + 8 = 21. Thus, the 7th and 8th Fibonacci numbers are 13 and 21, respectively. Another way of finding the terms in the Fibonacci sequence is through the use of Binet’s formula given by 𝑛 𝑛 11 + √5 1 − √5 𝐹𝑛 = [( ) −( ) ]. √5 2 2 This formula can be used to find the 𝑛𝑡ℎ term of the Fibonacci sequence in an explicit and non-recursive manner. The advantage of this formula over the recursive definition is that you can determine the 𝑛𝑡ℎ Fibonacci number without finding the two preceding Fibonacci numbers. It was named after mathematician Jacques Philippe Marie Binet. Binet derived this formula by using the roots of the characteristic equation 𝑥 2 − 𝑥 − 1 = 0. Binet’s formula can also be written as 𝛼 𝑛 − 𝛽𝑛 𝐹𝑛 = 𝛼−𝛽 1+√5 1−√5 where 𝛼 = and 𝛽 =. Here, 𝛼 is also known as the golden ration. 2 2 Example. Use Binet’s formula to find the 20th, 30th, and 40th Fibonacci numbers. Solution. To solve for the 20th Fibonacci number, we only need to substitute 𝑛 = 20 in Binet’s formula to get 20 20 1 1 + √5 1 − √5 𝐹20 = [( ) −( ) ]. √5 2 2 We may use a calculator to get 𝐹20 = 6765. Now, for the 30th and 40th Fibonacci numbers, we have 30 30 1 1 + √5 1 − √5 𝐹30 = [( ) −( ) ] = 832040 √5 2 2 16 and 40 40 1 1 + √5 1 − √5 𝐹40 = [( ) −( ) ] = 102334155. √5 2 2 EXERCISE 1. Use the Binet’s formula to find the 8th Fibonacci number without using a calculator. REFERENCES R. N. Aufmann, J. S. Lockdown, R. D. Nation, & D. K. Clegg, Mathematical Excursions, 3rd Edition, Brooks/Cole Cengage Learning, 2013. E. Kilic, The Binet formula, sums and representations of generalized Fibonacci p- numbers, European J. Combin. 29 (3), (2008), 701–711. I. Stewart, Nature’s Number: The Unreal Reality of Mathematics Imagination, BasicBook: A Division of Harper Collins Publishers, 1995. https://artofproblemsolving.com/wiki/index.php/Binet%27s_Formula 17 Chapter 2. Mathematical Language and Symbols Importance of Language According to Schiro (1997), one very important element in a student’s mathematical success is his competence to communicate mathematically. Hermann Weyl (1922) contributes that “Just as everybody must strive to learn language and writing before he can use them freely for expression of his thoughts, here too there is only one way to escape the weight of formulas. It is to acquire such power over the tool that, unhampered by formal technique, one can turn to the true problems”. Language is a necessary tool in the communication process. Good communication requires understanding. Everyone needs to firstly understand the language used in communicating to successfully transmit ideas from one to another. Hypothetical Situation: Imagine the following scenario: you’re in a math class and the instructor passes a piece of paper to each student. It is announced that the paper contains Study Strategies for Students of Mathematics and that you are to read it and make comments. Upon glancing at the paper, however, you observe that it is written in a foreign language that you do not understand. Is the instructor being fair? Will the student be able to receive the ideas being transmitted by the instructor? Definitely, the answer to these questions is a big NO! The instructor is probably trying to make a point. Although the ideas in the paragraph may be simple, there is no access to the ideas without a knowledge of the language in which the ideas are expressed. This situation has a very strong analogy in mathematics. People frequently have trouble understanding mathematical ideas: not necessarily because the ideas are difficult, but because they are being presented in a foreign language ― the language of Mathematics. 18 Characteristics of Mathematical Language Like any other language, mathematical language also has unique characteristics that help mathematicians to easily convey the thoughts or ideas they want to express. These characteristics also help every individual to communicate mathematically without the fear of being misinterpreted. Some important characteristics of mathematical language are as follows 1. Precise. It enables shared ideas/thoughts to be interpreted accurately with very fine distinctions. It prevents ambiguous interpretation of the transmitted ideas/thought. 2. Concise. It enables ideas/thoughts to be expressed completely and briefly. 3. Powerful. It enables to one express complex thoughts with relative case. 4. Devoid of emotional content. Unlike other languages, it doesn’t give importance to associating emotions/feelings in expressing thoughts or ideas. In communicating mathematically, sender doesn’t need emotions to successfully transmit ideas to its receiver. 5. Nontemporal. It emphasizes that there is no past, present, or future time in the process of mathematical communication. Everything is in “is-form”. It is important to keep in mind that the language of Mathematics can easily be learned, but it requires the efforts needed to learn any foreign language. Nouns versus Sentences In the English language, a part of speech necessary in constructing an idea expressing a complete thought is the noun, and any idea that expresses a complete thought is called sentence. In Mathematics, we also have nouns and sentences. Expressions are the mathematical version of “noun”. In Mathematics, it is also defined as a name given to a mathematical object or interest. This however does not state a complete thought. On the other hand, the mathematical version of “sentences” is called Sentences and these express facts or opinions that gives complete thought. For illustration, we will try to compare the nouns and sentences in mathematical language to that in English language. See the following table: 19  Examples of Nouns and Sentences English Mathematics 1 Nouns Lisa, Blackpink, Marawi City, birthday 5, 2+3, , 𝑥 − 1 2 1. Lisa is a member of the K-Pop girl 2 + 3 = 5. group, Blackpink. 2. The capital of the Philippines is 5 − 2 = 4. Sentences Marawi City. 𝑥 − 1 = 0. 3. Jenny invites Rose to her birthday tomorrow. It can be observed from this table that all sentences numbered 1 are true sentences, whereas those sentences numbered 2 are false. The third examples are sentences that states an opinion which is either true of false. It is true that Jenny invites Rose to her birthday party tomorrow if Jenny really does, but if she doesn’t, then that sentence is false. Similarly, 𝑥 − 1 = 0 is true when 𝑥 = 1. However, this is false when 𝑥 ≠ 1. This simply means that sentences are ideas that can either be true or false but not both. Synonyms and Antonyms In the English language, we have synonyms and antonyms. Synonyms are defined as words having the same meaning while words having opposite meanings are called antonyms. Similarly, in Mathematics, we also have synonyms and antonyms. Expressions that can be expressed differently are called synonyms. Examples of these are the following: 16 1. The expression 8 can also be expressed as 5 + 3, , 23 , (10 − 3) + 1, 2 + 2 2 + 2 + 2, 2 × 4, and so on. 2. The expression 5 × 2 can also be expressed as 2 × 5, 1 × 2 × 5, and so on. 20  Antonyms in mathematics exist and are called “inverses”. There are two different types: the additive inverse and the multiplicative inverse. Note every real number, always has an additive and multiplicative inverse. Examples. 1 1. The real number 8 has additive inverse −8 and the multiplicative inverse. 8 3 3 2. The additive inverse of the expression − is while its multiplicative 7 7 7 inverse is −. 3 We denote the additive inverse of any real number 𝑎 as −𝑎 and its multiplicative inverse by 𝑎−1. Multiplicative inverses are commonly known as reciprocals. Conventions Languages have conventions. In English, for example, it is conventional to capitalize proper names like “Ana” and “Japan”. This convention makes it easy for a reader to distinguish between a common noun (like “carol”, a Christmas song) and a proper noun (like “Carol”, a person). Mathematics also has its conventions, which help readers to distinguish between different types of mathematical expressions. Mathematical conventions are defined as sets of facts, names, notations which are widely-used in the area of Mathematics. Mathematicians abide by conventions in order to allow readers to understand what they write without constantly redefining basic terms. For instance, the fact that one evaluates multiplication before addition in the expression 2 + 3 × 4 is merely conventional since PEMDAS is mathematical convention. Another one is letter conventions. In Mathematics, letters often have special uses. For example, letters at the beginning of English alphabet (such as 𝑎, 𝑏, and 𝑐) usually mean constants or fixed values. In the general equation 𝑦 = 𝑎𝑥 + 𝑏. In this given equation, people may assume that 𝑎 and 𝑏 are fixed values and 𝑥 is the one that changes which in turn makes 𝑦 change. Aside from that, letters from 𝑖 to 𝑛, that is, 𝑖, 𝑗, 𝑘, 𝑙, 𝑚, 𝑛, usually mean positive integers and letters at the end of English Alphabet, that is … , 𝑥, 𝑦, 𝑧, usually mean variables or unknowns. Note that letter conventions are not rules but they are often used as such in the world of Mathematics. 21 ACTIVITY Watch the following video: Math isn’t hard, it’s a language|Randy Palisoc|TEDxManhattanBeace Link: https://www.youtube.com/watch?v=V6yixyiJcos Write a 1-page essay about what you have learned after watching the said video. REFERENCES The Language of Mathematics by Dr. Carol JVF Burns, http://onemathematicalcat.org https://artofproblemsolving.com/wiki/index.php/Mathematical_convention https://www.mathsisfun.com/mathematics-language.html 22 Chapter 3. Logic A Brief History “Logic will get you from A to B. Imagination will take you everywhere." – Albert Einstein One area of mathematics that has its roots deep in philosophy is the study of logic. Logic is defined as the study of formal reasoning based upon statements or propositions, but it is also commonly defined as a science of correct, critical reasoning. Additionally, logic is also said to be a belief that is supported by factual evidences. Aristotle identified some simple patterns in human reasoning while Leibniz dreamt of reducing reasoning to calculation. As a viable mathematical subject, however, logic is relatively recent: the 19th century pioneers were Bolzano, Boole, Cantor, Dedekind, Frege, Peano, C.S. Pierce, and E. Schroder. From our perspective we see their work as leading to boolean algebra, set theory, propositional logic, predicate logic, as clarifying the foundations of the natural and real number systems, and as introducing suggestive symbolic notation for logical operations. Also, their activity led to the view that logic together with set theory can serve as a basis for all of mathematics. Gottfried Wilhelm Leibniz (1646–1716) was one of the first mathematicians to make a serious study of symbolic logic. Leibniz tried to advance the study of logic from a merely philosophical subject to a formal mathematical subject. Leibniz never completely achieved this goal; however, several mathematicians, such as Augustus De Morgan (1806–1871) and George Boole (1815–1864), contributed to the advancement of symbolic logic as a mathematical discipline. Boole published The Mathematical Analysis of Logic in 1848. In this pamphlet, he argued persuasively that logic should be allied with mathematics, not philosophy. Following this, he published the more extensive work, An Investigation of the Laws of Thought in 1854. Concerning this document, the mathematician Bertrand Russell stated that, “Pure mathematics was discovered by Boole in a work which is called ‘The Laws of Thought’.” 23  The early part of the 20th century was also marked by the so-called foundational crisis in mathematics. A strong impulse for developing mathematical logic came from the attempts during these times to provide solid foundations for mathematics. Mathematical logic has now taken on a life of its own, and also thrives on many interactions with other areas of mathematics and computer science. In the second half of the last century, logic as pursued by mathematicians gradually branched into four main areas: model theory, computability theory (or recursion theory), set theory, and proof theory. The topics in this course are part of the common background of mathematicians active in any of these areas. What distinguishes mathematical logic within mathematics is that statements about mathematical objects and structures are taken seriously as mathematical objects in their own right. More generally, in mathematical logic we formalize (formulate in a precise mathematical way) notions used informally by mathematicians such as: property, statement (in a given language), structure, truth (what it means for a given statement to be true in a given structure), proof (from a given set of axioms), and algorithm. Logic Statement Every language contains different types of sentences, such as statements, questions, and commands. Consider the following statements. “Is the test today?” is a question. “Go get the newspaper” is a command. “This is a nice car” is an opinion. “Denver is the capital of Colorado” is a statement of fact. The symbolic logic that Boole was instrumental in creating applies only to sentences that are statements, the definition of which is given as follows. A statement is a declarative sentence that is either true or false, but not both. Example 1. Determine whether each sentence is a statement. 1. The main campus of the Mindanao State University is in Marawi City. 2. What will be my grade in the course, Mathematics in the Modern World? 3. Open the door. 4. 2 is a negative number. 5. 𝑥 + 1 = 5. 24 Answers: 1. The given sentence is a true statement because it satisfies the definition. It is a declarative sentence that states a fact since the Mindanao State University – Main Campus is in Marawi City. 2. The given sentence is not a statement since it is not a declarative sentence but an interrogative sentence. 3. Since the given sentence is a command, it is not a declarative sentence. It is an imperative sentence; thus, it is not a statement. 4. We all know that 2 is a positive number. This means the sentence “2 is a negative number” is false. However, this is a declarative sentence for it states an opinion. Hence, this is a statement, specifically a false statement. 5. The given sentence is a declarative sentence for it states and opinion. Additionally, we all know that it is true for 𝑥 = 4 and it is false for any other values of 𝑥. This means the sentence is either true or false depending on the value given to 𝑥, and can never be both true and false. Therefore, this is a statement, which is known as open sentence/statement. An open sentence is a statement that involves one or more variables and which becomes true or false when the variables are assigned to specific values. A truth set of an open sentence is the set of all values that will make the open sentence true. Example 2. Determine the truth set of the following open sentence. 1. 𝑥 + 1 = 5 2. 2𝑥 − 3 = 9 3. 𝑥 is a negative number such that 𝑥 2 = 4 4. 𝑥 is an integer such that 𝑥 2 = −16 5. 𝑥 2 − 25 = (𝑥 − 5)(𝑥 + 5), where 𝑥 is real Answers: 1. The only value of 𝑥 that makes the open sentence true is 4. Hence, its truth set is {4}. 2. Similarly, 6 is the only value of 𝑥 that can make the open sentence true. Hence, the truth set is {6}. 25  3. In this given open sentence, we found out that there are two values of 𝑥 that can make “𝑥 2 = 4” correct, that is, 2 and −2. However, it is stated in the sentence that 𝑥 must be negative. This means, 2 is rejected. Hence, the truth set of the given open sentence is {−2}. 4. Noting “𝑥 2 = −16”, 4𝑖 and −4𝑖 are the only values of 𝑥 that can make the equation true. However, the obtained values are not integers. Hence, the truth set for this open sentence is {} or ∅. 5. Any number that is to be substituted to 𝑥 can make the open sentence true. Hence, the truth set is set of real numbers. Simple and Compound Statements A simple statement is a statement that conveys a single idea. If a statement conveys two or more ideas, then it is a compound statement. Connecting simple statements with words and phrases such as “and”, “or”, “if... then”, and “if and only if” creates a compound statement. For instance, “I will attend the meeting or I will go to school” is a compound statement. It is composed of the two simple statements, “I will attend the meeting” and “I will go to school”. The word “or” is a connective for the two simple statements. The truth value of a simple statement is either true (T) or false (F). On the other hand, the truth value of a compound statement depends on the truth values of its simple statements and its connectives. A truth table is a table that shows the truth value of a compound statement for all possible truth values of its simple statements George Boole used symbols such as 𝑝, 𝑞, 𝑟, and 𝑠 to represent simple statements and the symbols ˄, ˅, ~, →, and ↔ to represent connectives. See the following table for a summary: 26  Logic Connectives and Symbols Statement Connective Symbolic form Type of statement not 𝒑 not ~𝑝 Negation 𝒑 and 𝒒 and 𝑝∧𝑞 Conjunction 𝒑 or 𝒒 or 𝑝∨𝑞 Disjunction If 𝒑, then 𝒒 If…then 𝑝→𝑞 Conditional 𝒑 if and only if 𝒒 if and only if 𝑝↔𝑞 Biconditional The negation of an statement 𝑝, denoted by ~𝑝 (to be read as “not 𝑝”), is a statement that says the opposite of 𝑝. This asserts that 𝑝 is false. Thus, if 𝑝 is true, then ~𝑝 is false, and if 𝑝 is false, then ~𝑝 is true. Truth table for ~𝒑 𝒑 ~𝒑 T F F T Example 3. Write the negation of the following statements. 1. 𝑝: Today is Friday. 2. 𝑞: Sheila has two extra pens. 3. 𝑟: The ABS-CBN franchise renewal was not approved by the Philippine Congress. 4. 𝑠: 3𝑥 + 1 = 5 5. 𝑡: Five is greater than 6. Answers. 1. ~𝑝: Today is not Friday. 2. ~𝑞: Sheila does not have two extra pens. 3. ~𝑟: The ABS-CBN franchise renewal was approved by the Philippine Congress. 4. ~𝑠: 3𝑥 + 1 ≠ 5 5. ~𝑡: Five is less than or equal to 6. 27  The conjunction of two statements 𝑝 and 𝑞, denoted by 𝑝 ∧ 𝑞 (and read as “𝑝 and 𝑞”), is true if and only if both 𝑝 and 𝑞 are true. The disjunction of 𝑝 and 𝑞, denoted by 𝑝 ∨ 𝑞 (and read as “𝑝 or 𝑞”), is true if and only if at least one of 𝑝 or 𝑞 is true. Truth Table for 𝒑 ∧ 𝒒 𝒑 𝒒 𝒑∧𝒒 T T T T F F F T F F F F Truth table for 𝒑 ∨ 𝒒 𝑝 𝑞 𝑝∨𝑞 T T T T F T F T T F F F Note. The word “but” is also a conjunction. It is sometimes used to precede a negative phrase. For instance, the statement “I've fallen and I can't get up" means the same as "I've fallen but I can't get up." In either case, if 𝑝 is "I've fallen" and 𝑞 is "I can get up" the conjunction above is symbolized as p ∧ ~q. 28 Example 4. Consider the following simple statements. 𝑝: Today is Friday. 𝑞: It is raining. 𝑟: I am going to a movie. 𝑠: I am not going to the basketball game. Write the following compound statements in symbolic form: 1. Today is Friday and it is raining. 2. It is not raining and I am going to a movie. 3. I am going to the basketball game or I am going to a movie. 4. Today is not Friday and it is not raining. Answers. 1. 𝑝 ∧ 𝑞 3. ∼ 𝑠 ∨ 𝑟 2. ∼ 𝑞 ∧ 𝑟 4. ~𝑝 ∧ ~𝑞 In the next example, we translate symbolic statements into English sentences. Example 5. Consider the following statements: 𝑝: The game will be played in MSU. 𝑞: The game will be shown on ABS-CBN. 𝑟: The game will not be shown on GMA. 𝑠: The Sultans are favored to win. Write each of the following symbolic statements in words: 1. 𝑞˄𝑝 3. ~𝑞 ∧ ~𝑟 2. ~𝑟˄𝑠 4. ~𝑝 ∨ 𝑠 Answers. 1. The game will be shown on ABS-CBN and the game will be played in MSU. 2. The game will be shown on GMA and the Sultans are favored to win. 3. The game will not be shown on ABS-CBN, but the game will be shown on GMA. (This is also correct, “The game will not be shown on ABS-CBN and the game will be shown on GMA.”) 4. The game will not be played in MSU or the Sultans are favored to win. 29 Example 6. Determine whether each statement is true or false. 1. 7 ≥ 5 2. 5 is a whole number and 5 is an even number. 3. 2 is a prime number and 2 is an even number. 4. 0.5 is an integer or a whole number. Solution. 1. 7 ≥ 5 means 7 > 5 or 7 = 5. Now, 𝑝: 7 > 5 is true while 𝑞: 7 = 5 is false. Thus, noting the connective “or”, the statement 7 ≥ 5 is a true statement. 2. The statement “𝑝: 5 is a whole number” is true, and the statement “𝑞: 5 is an even number” is false. Hence, keeping in mind the connective “and”, the truth value of the given statement is false. 3. This is a true statement because each simple statement is true, that is, “𝑝: 2 is a prime number” and “𝑞: 2 is an even number” are both true. 4. The given statement means 0.5 is an integer or 0.5 is a whole number. Hence, this is a false statement because “𝑝: 0.5 is an integer” and “𝑞: 0.5 is a whole number” are both false. Remark. When statements 𝑝, 𝑞, 𝑟, … involves a variable 𝑥, we may also use the notations 𝑝𝑥 , 𝑞𝑥 , 𝑟𝑥 , …, respectively. Negation of Conjunctions and Disjunctions STATEMENT TRUTH SET NEGATION TRUTH SET 𝑝𝑥 ∧ 𝑞𝑥 𝑃∩𝑄 ∼ (𝑝𝑥 ∧ 𝑞𝑥 ) ≡∼ 𝑝𝑥 ∨∼ 𝑞𝑥 𝑃𝐶 ∪ 𝑄𝐶 𝑝𝑥 ∨ 𝑞𝑥 𝑃∪𝑄 ∼ (𝑝𝑥 ∨ 𝑞𝑥 ) ≡∼ 𝑝𝑥 ∧∼ 𝑞𝑥 𝑃𝐶 ∩ 𝑄𝐶 Example 7. Given 𝑈 = {−4, −3, −2, −1,0,1,2,3,4}. Find the negation of the following and determine the truth sets of each negation: 1. 𝑝𝑥 ∧ 𝑞𝑥 : 𝑥 2 = 4 and (𝑥 + 5)(𝑥 − 2) = 0 2. 𝑝𝑥 ∨ 𝑞𝑥 : 𝑥 + 4 = 4 or 𝑥 2 < 5. 3. 𝑝𝑥 ∨ 𝑞𝑥 : 𝑥 − 1 < 2 or 3𝑥 − 2 = 0. 30 Solution. 1. The negation of the given statement is ∼ 𝑝𝑥 ∨∼ 𝑞𝑥 : 𝑥 2 ≠ 4 or (𝑥 + 5)(𝑥 − 2) ≠ 0. Now, to determine its truth set, we need to obtain the truth sets 𝑃 and 𝑄. To start with, we solve for the values of 𝑥 that will satisfy the given open sentences. For 𝑝𝑥 : 𝑥 2 = 4, we know that the only values of 𝑥 that will make it true are 2 and −2. Thus, 𝑃 = {2, −2}, and 𝑃𝐶 = {−4, −3, −1, 0, 1, 3, 4}. On the other hand, for 𝑞𝑥 : (𝑥 + 5)(𝑥 − 2) = 0, the only values of 𝑥 that satisfy the given open sentence are −5 and 2. This means that 𝑄 = {2} and 𝑄𝐶 = {−4, −3, −2, −1, 0, 1, 3, 4}. Hence, the truth set of the negation is 𝑃𝐶 ∪ 𝑄𝐶 = {−4, −3, −2, −1, 0, 1, 3, 4}. 2. The negation of the given statement is ∼ 𝑝𝑥 ∧∼ 𝑞𝑥 : 𝑥 + 4 ≠ 4 and 𝑥 2 > 5. To determine its truth set, we need to obtain the truth sets 𝑃 and 𝑄. To start with, we solve for the values of 𝑥 that will satisfy the given open sentences. For 𝑝𝑥 : 𝑥 + 4 = 4, we know that the only value of 𝑥 that will make it true is 0. This means that 𝑃 = {0} and 𝑃𝐶 = {−4, −3, −2, −1,1,2,3,4}. On the other hand, for 𝑞𝑥 : 𝑥 2 < 5, the only values of 𝑥 that satisfy the given open sentence are −2, −1, 0, 1, and 2. This means 𝑄 = {−2, −1,0, 1,2} and that 𝑄𝐶 = {−4, −3, 3, 4}. Hence, the truth set of the negation is 𝑃𝐶 ∩ 𝑄𝐶 = {−4, −3,3,4}. 3. The negation of the given statement is ∼ 𝑝𝑥 ∧∼ 𝑞𝑥 : 𝑥 − 1 > 2 or 3𝑥 − 2 ≠ 0. Applying same method we did in the first two examples, we obtain 𝑃 = {−4, −3, −2, −1,0,1,2} since the only values of 𝑥 that will make the statement 𝑝𝑥 : 𝑥 − 1 < 2 are within the interval (−∞,2], and 𝑄 = ∅ since the only value that can make the 2 statement 𝑞𝑥 : 3𝑥 − 2 = 0 true is , which is not an element of 𝑈. This follows 3 𝐶 𝐶 that 𝑃 = {3,4} and 𝑄 = 𝑈. Hence, the truth set of the negation is 𝑃𝐶 ∩ 𝑄𝐶 = {3,4}. 31 Universal and Existential Statements Suppose you are talking with your friend Jenny, and she is describing two clubs that she has joined. While describing the people in the first club, she says the following: “There exists a member of Club 1 such that the member has red hair”. In describing the second club, she says the following: “For all members in Club 2, the member has red hair.” Based on these two statements, what can you tell about the members' hair color in Club 1 and Club 2? Well, let's take a look at her statements, and pick them apart. In Mathematics, the statements “There exists a member of Club 1 such that the member has red hair” and “For all members in Club 2, the member has red hair” are two different logic statements which can be categorized as an existential statement and a universal statement, respectively. A universal statement is defined as a statement that states a property that is true to all. Examples of this are the following: 1. All positive numbers are greater than zero. 2. For every even integer 𝑥, 𝑥 is divisible by 2. 3. Ang bawat isa ay may pag-asa. 4. Ang lahat ng halaman ay nakakain. On the other hand, an existential Statement is a statement which states that there is at least one thing for which the property is true. For instance, we have the following: 1. There is a prime number that is even. 2. There exists a number which is divisible by any number except itself. 3. Mayroon isang tao sa mundo na magmamahal saiyo. 4. Mayroon isang mag aaral sa MMW na hindi makakapasa. Quantifiers and Negation The phrases “there exists” and “for all” play a huge role in logic and logic statements. In fact, they are so important that they have a special name: quantifiers. Quantifiers are words, expressions, or phrases that indicate the number of elements that a statement pertains to. In mathematical logic, there are two quantifiers: “there exists” and “for all”. In a statement, the word “some” and the phrases “there exists” and “at least one” are called existential quantifiers. Existential quantifiers are used as prefixes to 32 assert the existence of something. In symbol, we use the notation “∃𝑥” to be read as “for some 𝑥”, “there exists 𝑥 such that…”, or “there is some 𝑥 such that…” to represent existential quantifiers. Note: The symbol ∃ should be used in extension to several variables, and as part of the verbalization of a symbolic existential statement, the appropriate pronunciation should include the phrase “… such that…”. On the other hand, in a statement, the words “none”, “no”, “all”, and “every” are called universal quantifiers. The universal quantifiers “none” and “no” deny the existence of something, whereas the universal quantifiers “all” and “every” are used to assert that every element of a given set satisfies some condition. The symbolic representation of universal quantifiers is “∀𝑥” read as “for every 𝑥”, “for all 𝑥”, “for each 𝑥”, or “given any 𝑥”. Note: The symbol ∀ is also be used with an extension to several variables. Example 8. Determine the truth value in each of the following: 1. ∀𝑥 ∈ {1,2,3}: 𝑥 2 is less than 10. 2. ∃𝑥 ∈ {10,15,20}: 3𝑥 + 1 is odd. 3. ∀𝑥 ∈ {1,2,3}: 𝑥 + 𝑦 is prime. 4. ∃𝑥 ∈ {2,4,6}: 2𝑥 − 5 = 5. Answers. 1. Notice that the quantifier used in this statement is ∀𝑥, a universal quantifier. This means that all given values of 𝑥 must be true for all for 𝑝𝑥 : 𝑥 2 𝑖𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 10. To find the truth value of this universal statement, we will assign each given values of 𝑥 to 𝑝𝑥 and determine their corresponding truth value. That is, by assigning the values 1, 2, and 3 to 𝑥 in the statement “𝑥 2 𝑖𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 10”, we found out that these all led to true statements. Hence, the truth value of the statement “∀𝑥 ∈ {1,2,3}: 𝑥 2 is less than 10” is true. 2. In this example, we notice that the quantifier used in this statement is ∃𝑥, an existential quantifier. This means that there must be at least one value of 𝑥 (but not all) on the given values that will make 𝑝𝑥 : 3𝑥 + 1 𝑖𝑠 𝑜𝑑𝑑 true. Similarly, to find the truth value of this existential statement, we will assign each given values of 𝑥 to 𝑝𝑥 and determine their corresponding truth value. That is, by assigning the values 10, 15, and 20 to 𝑥 in the statement “3𝑥 + 1 𝑖𝑠 𝑜𝑑𝑑”, we found out that the statement is true only for 𝑥 = 10 and 20. 33  Hence, the truth value of the statement “∃𝑥 ∈ {10,15,20}: 3𝑥 + 1 is odd” is true. 3. For this example, since we used two variables in the extension of ∀, we need to use the concept of ordered pairs (𝑥, 𝑦). This means we have the following ordered pairs: (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2) and (3, 3). Substituting the values in the ordered pair to 𝑥 and 𝑦 in 𝑝𝑥 : 𝑥 + 𝑦 𝑖𝑠 𝑝𝑟𝑖𝑚𝑒 turns out that there are some for which the statement is true. Thus, the truth value of “∀𝑥 ∈ {1,2,3}: 𝑥 + 𝑦 is prime” is false. 4. This is done through doing similar method to what we did for example (b). Thus, its truth value is false since none of the given values of 𝑥 made 𝑝𝑥 : 2𝑥 − 5 = 5 true. Like any other logic statements, we can also negate universal and existential statements. Recall that the negation of a false statement is a true statement and that the negation of a true statement is a false statement. It is important to remember this fact when forming the negation of a quantified statement. For instance, what is the negation of the false statement, “All dogs are mean”? You may think that the negation is “No dogs are mean,” but this is also a false statement. Thus the statement “No dogs are mean” is not the negation of “All dogs are mean.” The negation of “All dogs are mean,” which is a false statement, is in fact “Some dogs are not mean,” which is a true statement. The statement “Some dogs are not mean” can also be stated as “At least one dog is not mean” or “There exists a dog that is not mean”. Negation of ∀𝒙: 𝒑 and ∃𝒙: 𝒑 STATEMENT NEGATION ∀𝑥: 𝑝 ~(∀𝑥: 𝑝) ≡ ∃𝑥: ~𝑝 ∃𝑥: 𝑝 ~(∃𝑥: 𝑝) ≡ ∀𝑥: ~𝑝 To help us easily negate any quantified statement, the following tables are useful: STATEMENT NEGATION All 𝑋 are 𝑌. Some 𝑋 are not 𝑌. No 𝑋 are 𝑌. Some 𝑋 are 𝑌. Some 𝑋 are not 𝑌. All 𝑋 are 𝑌. Some 𝑋 are 𝑌. No 𝑋 are 𝑌. 34 Example 9. Write the negation of each of the following statements. 1. Some airports are open. 2. All movies are worth the price of admission. 3. No odd numbers are divisible by 2. 4. Some students are not permitted to submit their research assignment. Answers: 1. No airports are open. 2. Some movies are not worth the price of admission. 3. Some odd numbers are divisible by 2. 4. All students are permitted to submit their research assignment. Example 10. State the negation of the following sentences and determine the truth values of each negation: 1. ∀𝑥 ∈ {0,1,2,3} : 𝑥 3 – 1 is an odd integer. 2. ∃𝑥,𝑦 ∈ {−1,0,1} : 𝑥 = 𝑦 + 1. 3. ∃𝑥 ∈ {−1, 0, 1}: 𝑥 2 = 𝑥 Answers: 1. The negation of the given statement is “∃𝑥 ∈ {0,1,2,3}: 𝑥 3 − 1 is an even integer”. To determine the truth value of this negation, we simply evaluate 𝑥 3 − 1 at each given value of 𝑥. Since 𝑥 3 − 1 is even when 𝑥 = 3, the negation is a true statement. 2. The negation of the given statement is “∀𝑥 ∈ {−1,0,1}: 𝑥 ≠ 𝑦 + 1”. To determine the truth value of the negation, we substitute the given values of 𝑥 and 𝑦 in the expression 𝑥 ≠ 𝑦 + 1. The negation is a universal statement, all values for 𝑥 and 𝑦 in the set {−1,0,1} must satisfy the condition that 𝑥 ≠ 𝑦 + 1. Observe that when 𝑥 = 1 and 𝑦 = 0, it is false that 𝑥 ≠ 𝑦 + 1. Thus, the negation is false. 3. The negation of the given statement is “∀𝑥 ∈ {−1, 0, 1}: 𝑥 2 ≠ 𝑥”. To determine the truth value of this negation, we simply evaluate 𝑥 2 ≠ 𝑥 at each given value of 𝑥. Since the negation of the given existential statement is true only for 𝑥 = −1, thus, its truth value is false. 35 Conditional Statements “If you don’t get in that plane, you’ll regret it. Maybe not today, maybe not tomorrow, but soon, and for the rest of your life.” The above quotation is from the movie Casablanca. Rick, played by Humphrey Bogart, is trying to convince Ilsa, played by Ingrid Bergman, to get on the plane with Laszlo. The sentence, “If you don’t get in that plane, you’ll regret it,” is a conditional statement. Conditional statements can be written in “if 𝑝, then 𝑞” form or in “if 𝑝, 𝑞” form. The following are conditional statements. 1. If we order pizza, then we can have it delivered. 2. If you go to the movie, you will not be able to meet us for dinner. 3. If 𝑛 is a prime number greater than 2, then 𝑛 is an odd number. In any conditional statement represented by “If 𝑝, then 𝑞” or by “If 𝑝, 𝑞,” the 𝑝 statement is called the antecedent while the 𝑞 statement is called the consequent. Example 11. Identify the antecedent and consequent in the following statements: 1. If our school was this nice, I would go there more than once a week. —The Basketball Diaries 2. Kung mahal mo ako, nangangahulugang kailangan mo ako 3. If you strike me down, I shall become more powerful than you can possibly imagine. —Obi-Wan Kenobi, Star Wars, Episode IV, A New Hope 4. If 378 is divisible by 18, then 378 is divisible by 6. 5. If 𝑥 2 = 9, then 𝑥 > 0. Answers: 1. Antecedent: our school was this nice Consequent: I would go there more than once a week 2. Antecedent: mahal mo ako Consequent: kailangan mo ako 3. Antecedent: you strike me down Consequent: I shall become more powerful than you can possibly imagine 4. Antecedent: 378 is divisible by 18 Consequent: 378 is divisible by 6 5. Antecedent: 𝑥 2 = 9 Consequent: 𝑥 > 0 Every conditional statement can be stated in many equivalent forms. It is not even necessary to state the antecedent before the consequent. For instance, the 36 conditional “If I live in Boston, then I must live in Massachusetts” can also be stated as “I must live in Massachusetts, if I live in Boston”. The table below lists some of the various forms that may be used to write a conditional statement. Equivalent forms of Conditional statements CONDITIONAL STATEMENTS If 𝑝 then 𝑞. Every 𝑝 is a 𝑞. If 𝑝, 𝑞. 𝑞, if 𝑝. 𝑝 only if 𝑞. 𝑞 provided that 𝑝. 𝑝 implies 𝑞. 𝑞 is a necessary condition for 𝑝. Not 𝑝 or 𝑞. 𝑝 is a sufficient condition for 𝑞. Example 12. Write each of the following in “If 𝑝, then 𝑞” form: 1. The number is an even number provided that it is divisible by 2. 2. Today is Friday, only if yesterday was Thursday. 3. Jenny is not Lisa’s sister or Jisoo is Lisa’s friend. 4. Being born in Korea is a necessary condition for being a citizen of Korea. Answers: 1. The statement, “The number is an even number provided that it is divisible by 2,” is in “𝑞 provided that 𝑝” form. This means that we have the following: Antecedent: “It is divisible by 2” Consequent: “The number is an even number” Thus its “If 𝑝, then 𝑞” form is “If it is divisible by 2, then the number is an even number”. 2. The statement, “Today is Friday, only if yesterday was Thursday,” is in “𝑝 only if 𝑞” form. This means that we have the following: Antecedent: “Today is Friday” Consequent: “Yesterday was Thursday” Thus its “If 𝑝, then 𝑞” form is “If today is Friday, then yesterday was Thursday”. 3. The statement, “Jenny is not Lisa’s sister or Jisoo is Lisa’s friend,” is in “Not 𝑝 or 𝑞” form. This means that we have the following: Antecedent: “Jenny is Lisa’s sister” 37  Consequent: “Jisoo is Lisa’s friend” Thus its “If 𝑝, then 𝑞” form is “If Jenny is Lisa’s sister, then Jisoo is Lisa’s friend”. 4. The statement, “Being born in Korea is a necessary condition for being a citizen of Korea,” is in “𝑞 is a necessary condition for 𝑝” form. This means that we have the following: Antecedent: “Being a citizen of Korea” Consequent: “Being born in Korea” Thus its “If 𝑝, then 𝑞” form is “If you are a citizen of Korea, then you are born in Korea”. (NOTE: Logic statements require proper sentence construction to be understood clearly.)  Arrow Notation The conditional statement, “If 𝑝, then 𝑞,” can be written using the arrow notation 𝑝 → 𝑞. The arrow notation 𝑝 → 𝑞 is read as “if 𝑝, then 𝑞” or as “𝑝 implies 𝑞.” The truth value of any conditional statement 𝑝 → 𝑞 is determined by the truth value of its antecedent and consequent. The conditional 𝑝 → 𝑞 is false only if 𝑝 is true and 𝑞 is false. It is true in all other cases. Truth table for 𝒑 → 𝒒 𝒑 𝒒 𝒑→𝒒 T T T T F F F T T F F T Note: The conditional ∀𝑥: 𝑝𝑥 → 𝑞𝑥 (or simply 𝑝𝑥 → 𝑞𝑥 ) is true if and only if P ⊆ Q where P and Q are truth sets of 𝑝𝑥 and 𝑝𝑥 respectively. Example 13. Determine the truth value of each of the following. 1. If 2 is an integer, then 2 is a rational number. 2. If 3 is a negative number, then 5 > 7. 3. If 5 > 3, then 2 + 7 = 4 4. If 4 ≥ 3, then 2 + 5 = 6. 5. If 23 and 27 are odd numbers, then these are not divisible by 2. 38 Answers: 1. Since both antecedent and consequent are true, the conditional statement is also true. 2. The truth value of the antecedent and the consequent are both false. Thus, the given conditional statement is a true statement. 3. This is a false statement since the antecedent and the consequent are true and false, respectively. 4. Note that the antecedent can also be written as 4 > 3 or 4 = 3 which is a disjunction. To determine its truth value, we need to consider the connective used. In this case, we have a disjunction, hence, the antecedent is true. Since the consequent is false, the truth value of the given conditional statement is false. 5. Similarly, the antecedent of this conditional statement can also be written as “23 is an odd number and 27 is an odd number”. Thus, taking note of the connective used, the truth value of both the antecedent and the consequent is true. Hence, the given conditional statement is a true statement.  The Negation of the conditional statement We all know that conditional statements have equivalent forms, and one of this forms is “not 𝑝 or 𝑞” which is written symbolically as ~𝑝 ∨ 𝑞. This means that “𝑝 → 𝑞” and “~𝑝 ∨ 𝑞” are equivalent, in symbols, we write 𝑝 → 𝑞 ≡ ~𝑝 ∨ 𝑞. It follows that the negation of the conditional statement 𝑝 → 𝑞, denoted by ~(𝑝 → 𝑞), can equivalently be written as ~(~𝑝 ∨ 𝑞). By one of De Morgan’s laws, it can be expressed as the conjunction 𝑝 ∧ ~𝑞. The negation 𝒑 → 𝒒 ∼ (𝒑 → 𝒒) = 𝒑 ∧∼ 𝒒 39 Example 14. Write the negation of each conditional statement. 1. If they pay me the money, I will sign the contract. 2. If the lines are parallel, then they do not intersect. 3. If I finish the report, I will go to the concert. 4. If the square of 𝑛 is 25, then 𝑛 is 5 or −5. Answers: 1. They paid me the money and I did not sign the contract. 2. The lines are parallel and they intersect. 3. I finished the report and I did not go to the concert. 4. The square of 𝑛 is 25 and 𝑛 is not 5 and −5. Derived Conditionals and Biconditional Statements The statement “𝑝 if and only if 𝑞” is called a biconditional. It is written in symbols as “𝑝 ↔ 𝑞” and is equivalent to the conjunction (𝑝 → 𝑞) ˄ (𝑞 → 𝑝). The Biconditional 𝑝 ↔ 𝑞 𝑝 ↔ 𝑞 ≡ [(𝑝 → 𝑞) ∧ (𝑞 → 𝑝)] Biconditional statements are true only when 𝑝 and 𝑞 have the same truth value. Truth table for 𝒑 ↔ 𝒒 𝒑 𝒒 𝒑↔𝒒 T T T T F F F T F F F T Example 15. State whether each biconditional is true or false. 1. 𝑥 + 4 = 7 if and only if 𝑥 = 3. 2. 𝑥 2 = 36 if and only if 𝑥 = 6. 3. 𝑥 > 7 if and only if 𝑥 > 6. 4. 𝑥 + 5 > 7 if and only if 𝑥 > 2. 40 Answers: 1. Both equations are true when 𝑥 = 3 and both are false when 𝑥 ≠ 3. Since the equations have the same truth value for any value of 𝑥, the biconditional is a true statement. 2. Both equations are true for 𝑥 = 6. However, when 𝑥 = −6, the first equation is true but the second equation is false. Thus, the two equations have the same truth value only for some values of 𝑥. Hence, the biconditional is a false statement. 3. Both equations are true for any values of 𝑥 greater than 7. Likewise, they are both false for any values less than 7. However, when 𝑥 = 7, the first equation turns false while the second equation is true. Hence, the biconditional is a false statement. 4. The two equations have the same truth value for any values of 𝑥. That is, both equations are true when 𝑥 is greater than 2, false when 𝑥 is less than 2, and false when 𝑥 = 2. Thus, this is a true statement. Every conditional statement 𝑝 → 𝑞 has three related statements, namely, the converse, the inverse, and the contrapositive. These are called derived conditionals. The converse of a conditional statement, denoted by 𝒒 → 𝒑, is formed by interchanging the antecedent 𝑝 with the consequent 𝑞. The inverse of a conditional statement, denoted by ~𝒑 → ~𝒒, is formed by negating the antecedent 𝑝 and negating the consequent 𝑞. The contrapositive of a conditional statement, denoted by ~𝒒 → ~𝒑, is formed by negating both the antecedent 𝑝 and consequent 𝑞, then interchanging these negated statements. Statements Related to the Conditional Statement 𝒑 → 𝒒 The converse of 𝑝 → 𝑞 is 𝑞 → 𝑝. The inverse of 𝑝 → 𝑞 is ∼ 𝑝 →∼ 𝑞. The contrapositive of 𝑝 → 𝑞 is ∼ 𝑞 →∼ 𝑝. Example 16. Write the converse, inverse, and contrapositive of the following: 1. If you are good in Mathematics, then you are good in logic. 2. If I get the job, then I will rent the apartment. 3. If 𝑥 is an odd integer, then 𝑥 2 + 2 is even. 4. If we have a quiz today, then we will not have a quiz tomorrow. 41 Answers: 1. Converse: If you are good in logic, then you are good in Mathematics. Inverse: If you are not good in Mathematics, then you are not good in logic. Contrapositive: If you are not good in logic, then you are not good in Mathematics. 2. Converse: If I rent the apartment, then I get the job. Inverse: If I do not get the job, then I will not rent the apartment. Contrapositive: If I do not rent the apartment, then I did not get the job. 3. Converse: If 𝑥 2 + 2 is even, then 𝑥 is an odd integer. Inverse: If 𝑥 is an even integer, then 𝑥 2 + 2 is odd. Contrapositive: If 𝑥 2 + 2 is odd, then 𝑥 is an even integer. 4. Converse: If we will not have a quiz tomorrow, then we have a quiz today. Inverse: If we do not have a quiz today, then we will have a quiz tomorrow. Contrapositive: If we will have a quiz tomorrow, then we do not have a quiz today. The truth value of the converse, inverse, and contrapositive follows the truth value of the conditional statements. The table given below consists of the truth values of the converse, inverse, and contrapositive of a conditional statement. Truth table for the converse, inverse, and contrapositive of 𝒑 → 𝒒 𝒑 𝒒 𝒒→𝒑 ~𝒑 → ~𝒒 ~𝒒 → ~𝒑 T T T T T T F T T F F T F F T F F T T T Note: The conditional statement 𝑝 → 𝑞 and its contrapositive ~𝑞 → ~𝑝 are equivalent. Similarly, the converse of the conditional statement 𝑞 → 𝑝 is equivalent to the inverse of the conditional statement ~𝑝 → ~𝑞. 42 Example 17. Determine the truth value of the converse, inverse and contrapositive of the following conditional statements. 1. If January has 31 days, then June has also 31 days. 2. If 5 > 7, then 5 > 4. 3. If Rodrigo Duterte is the current Philippine president, then Mar Roxas lost the 2016 Presidential election. 4. For all 𝑥 ∈ {−5, −4, −3, −2, −1,0,1,2,3,4,5}: If 𝑥 2 = 25, then 𝑥 = 5. Answers: 1. The converse, inverse and contrapositive of the given conditional statement are all false. 2. The converse and inverse of the given conditional statement are false, while its contrapositive is true. 3. The converse, inverse and contrapositive of the given conditional statement are all true. 4. In this example, the given conditional statement involves open sentences, that is, we labelled as 𝑝𝑥 : 𝑥 2 = 25 and 𝑞𝑥 ∶ 𝑥 = 5. Note that the truth sets of these open sentences are P = {−5,5} and Q = {5}, respectively. Thus, it clearly shows that the given conditional statement is false since 𝑃 ⊈ 𝑄. This also follows that the contrapositive is also false. Moreover, since 𝑄 ⊆ 𝑃, the converse is true which also follows that the inverse is also true since the two are equivalent. REFERENCES R. N. Aufmann, J. S. Lockdown, R. D. Nation, & D. K. Clegg, Mathematical Excursions, 3rd Edition, Brooks/Cole Cengage Learning, 2013. J. Price, J. N. Rath, & W. Leschensky, Pre-algebra, a Transition to Algebra, Lake Forest: Macmillan / McGraw - Hill Publishing Company, 1992. https://www.britannica.com/biography/George-Boole https://www.math.fsu.edu/~wooland/hm2ed/Part2Module1/Part2Module1 https://math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Disc rete_Math/2%3A_Logic/2.7%3A_Quanti%EF%AC%81ers https://study.com/academy/lesson/quantifiers-in-mathematical-logic-types- notation- examples.html#:~:text=Quantifiers%20are%20words%2C%20expressions%2C%20 or,' 43 Chapter 4. Problem Solving and Reasoning Inductive and Deductive Reasoning 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. Inductive reasoning is defined as the process of reaching a general conclusion by examining specific conclusion by examining specific examples. 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. Example 1. Use inductive reasoning to predict the next number in the following lists. 1) 3, 6, 9, 12, 15, ___ Solution. 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. 2) 1, 3, 6, 10, 15, ___ Solution. 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. Note. Inductive reasoning may also be used to make conjectures and generalizations. Example 2. Use inductive reasoning to make a conjecture about an arithmetic procedure. Procedure. Pick a number. Multiply it by 8, add 6 to the product, divide the sum by 2, and subtract 3. 44  Complete the procedure above for several different numbers. Use inductive reasoning to make a conjecture about the relationship between the size of the resulting number and the size of the original number. Solution. Suppose we pick 5 as the original number. Then the procedure would produce the following results: Original number: 5 Multiply by 8: 8  5 = 40 Add 6: 40 + 6 = 46 Divide by 2: 46  2 = 23 Subtract 3: 23 – 3 = 20 We started with 5 and followed the procedure to produce 20. Starting with 6 as our original number produces a final result of 24. Starting with 10 produces a final result of 40. Starting with 100 produces a final result of 400. In each of these cases the resulting number is four times the original number. We conjecture that following the given procedure produces a number that is four times the original number. 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 assumptions, procedures, or principles. Deductive reasoning is the process of reaching a conclusion by applying general principles and procedures. Example 3. 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. Solution: Let n represent the original number. Multiply the number by 8: 8n Add 6 to the product: 8n + 6 45  8𝑛+6 Divide the sum by 2: = 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 number. Inductive Reasoning versus Deductive Reasoning The main difference between inductive and deductive reasoning is that inductive reasoning aims at developing a theory while deductive reasoning aims at testing an existing theory. Inductive reasoning moves from specific observations to broad generalization, and deductive reasoning the other way around. Example 4. Determine whether each of the following arguments is an example of inductive reasoning or deductive reasoning. 1) During the past 10 years, a tree has produced mangoes every other year. Last year the tree did not produce mangoes, so this year the tree will produce mangoes. Answer: This argument reaches a conclusion based on specific examples, so it is an example of inductive reasoning. 2) All home improvements cost more than the estimate. The contractor estimated that my home improvement will cost Php135,000. Thus, my home improvement will cost more than Php135,000. Answer: Because the conclusion is a specific case of a general assumption, this argument is an example of deductive reasoning. 46 Problem-Solving Strategies Ancient Mathematicians such as Euclid and Pappus were interested in Solving Mathematical problems, but they were also interested in heuristics, the study of the methods and rules of discovery and invention. In the 17th century, the mathematician and philosopher Rene Descartes (1596-1650) contributed to the field of heuristics. He tried to develop a universal problem-solving method. Although he did not achieve his goal, he did publish some of his ideas in “Rules for the Direction of the Mind” and his better-known work “Discourse de la methode”. Another mathematician and philosopher, Gottfried Wilhelm Leibnitz (1640-1716) planned to write a book on heuristics titled “Art of Invention”. Of the problem solving process, Leibnitz wrote “Nothing is more important to see the sources of the invention which are in my opinion, more interesting than the inventions themselves.” One of the foremost recent mathematicians to make a study of problem solving was George Polya (1887–1985). He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University. He devised a systematic strategy for solving problems that is now referred to by his name: the Polya Four-Step Problem Solving Strategy. George Polya defines ‘problem- solving’ as an act to: o Find a way out of difficulty, o Find a way around an obstacle, o Find a way where none is known, o Attain a desired end that is not immediately http://authenticinquirymaths.blogspot. com/2013/05/4-steps-with-polya.html attainable by direct means. Polya’s Four-Step Problem Solving Strategy  Understand the Problem This part of Polya’s four-step strategy is often overlooked. You must have a clear understanding of the problem. To help you focus on understanding the problem, consider the following questions. o Can you restate the problem in your own words? o Can you determine what is known about these types of problems? 47  o Is there missing information that, if known, would allow you to solve the problem? o Is there extraneous information that is not needed to solve the problem? o What is the goal?  Devise a Plan Successful problem solvers use a variety of techniques when they attempt to solve a problem. Here are some frequently used procedures. o Make a list of the known information. o Make a list of information that is needed. o Draw a diagram. o Make an organized list that shows all possibilities. o Make a table or a chart. o Work backwards. o Try to solve a similar but simpler problem. o Look for a pattern. o Write an equation. If necessary, define what each variable represents. o Perform an experiment. o Guess at a solution and then check your result.  Carry Out the Plan Once you have devised a plan, you must carry it out. o Work carefully. o Keep an accurate and neat record of all your attempts. o Realize that some of your initial plans will not work and that you may have to devise another plan or modify your existing plan.  Review the Solution Once you have found a solution, check the solution. o Ensure that the solution is consistent with the facts of the problem o Interpret the solution in the context of the problem. o Ask yourself whether there are generalizations of the solution that could apply to other problems. Example 1. During a family gathering last Christmas, Angel was able to collect 12 monetary bills consisting of ₱20 bills and ₱50 bills from her Titos and Titas. She received a total of ₱390 from them. How many of each bill Angel receive? Solution. 48 o Understand the Problem Distribute 12 bills into two portions. o Devise a Plan Perform Guess and Check. There are only 11 ways of splitting 12 into two portions: 11+1, 10+2, and so on. If there are 11 ₱20 bills, then there must be one 50 bills, in which case the total amount is 11(20) + 1(50)=270 pesos. Reversing the amounts, you have 11(50) + 1(20)=570 pesos. Determine which distribution of bills yields the amount ₱390. o Carry Out the Plan Number of ₱20 Number of ₱50 Total amount bills bills 11 1 11(20) + 1(50) = 270 10 2 10(20) + 2(50) = 300 9 3 9(20) + 3(50) = 330 8 4 8(20) + 4(50) = 360 7 5 7(20) + 5(50) = 390 The answer is seven ₱20 bills and five ₱50 bills. o Review the Solution The answer is reasonable as it generates the desired amount of ₱390. To check if this is the only solution, complete the table. Number of ₱20 Number of ₱50 Total amount bills bills 11 1 11(20) + 1(50) = 270 10 2 10(20) + 2(50) = 300 9 3 9(20) + 3(50) = 330 8 4 8(20) + 4(50) = 360 49  7 5 7(20) + 5(50) = 390 6 6 6(20) + 6(50) = 420 5 7 450 4 8 480 3 9 510 2 10 540 1 11 570 A classic alternative is the algebraic process where variables are introduced in the solution. o Devise a Plan. Let 𝑥 be the number of ₱20 bills and 𝑦, the number of ₱50 bills. Since there are 12 paper bills in all, then 𝑥 + 𝑦 = 12 (1) The amount of ₱20 bills is 20 𝑥 while the amount of the ₱50 bills is 50 𝑦, giving the equation 20 𝑥 + 50 𝑦 = 390 (2) Solve the system of equations in two unknowns to find the answer. o Carry out the plan. Equation (1) yields the explicit form 𝑦 = 12 − 𝑥. Using this in equation (2): 20𝑥 + 50(12 − 𝑥) = 390 20𝑥 + 600 − 50𝑥 = 390 −30𝑥 = −210 𝑥=7 And so, 𝑦 = 12 − 7 = 5. As in the first solution, the answer is seven₱20 bills and five₱50 bills. o Review the Solutions Check the total amount:7(20) + 5(50) = 390.