Mathematics Project Coordinator: Jean-Paul Groleau

Transcription

1 L ogic MTH

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3 LOGIC

4 Mathematics Project Coordinator: Jean-Paul Groleau Author: Serge Dugas Content Revision: Jean-Paul Groleau Pedagocical Revision: Jean-Paul Groleau English Translation: Ministère de l'éducation Services à la communauté anglophone Direction de la production en langue anglaise Linguistic Revision: Johanne St-Martin Electronic Publishing: P.P.I. inc. Société de formation à distance des commissions scolaires du Québec All rights for translation and adaptation, in whole or in part, reserved for all countries. Any reproduction, by mechanical or electronic means, including microreproduction, is forbidden without the written permission of a duly authorized representative of the Société de formation distance des commissions scolaires du Québec (SOFAD). Legal Deposit 2009 Bibliothèque et Archives nationales du Québec Bibliothèque et Archives Canada ISBN

5 TABLE OF CONTENTS Introduction to the Program Flowchart Program Flowchart How to Use This Guide General Introduction Intermediate and Terminal Objectives of the Module Diagnostic Test on the Prerequisites Answer Key for the Diagnostic Test on the Prerequisites Analysis of the Diagnostic Test Results Information for Distance Education Students UNITS 1. Propositions and al Connectives Truth Value of a Compound Proposition Tautologies, Contradictions and Implications Equivalences Negation of a Compound Proposition Propositions, Propositional Forms and Solution Sets of Propositional Forms Truth Value of a Quantified Compound Propositional Form Final Summary Answer Key for the Final Summary Terminal Objectives Self-Evaluation Test Answer Key for the Self-Evaluation Test Analysis of the Self-Evaluation Test Results Final Evaluation Answer Key for the Exercises Glossary List of Symbols Bibliography Review Activities

6 INTRODUCTION TO THE PROGRAM FLOWCHART WELCOME TO THE WORLD OF MATHEMATICS This mathematics program has been developed for adult students enrolled either with Adult Education Services of school boards or in distance education. The learning activities have been designed for individualized learning. If you encounter difficulties, do not hesitate to consult your teacher or to telephone the resource person assigned to you. The following flowchart shows where this module fits into the overall program. It allows you to see how far you have come and how much further you still have to go to achieve your vocational objective. There are three possible paths you can take, depending on your goal. The first path, which consists of Modules MTH (MTH-314) and MTH (MTH-416), leads to a Secondary School Vocational Diploma (SSVD) and certain college-level programs for students who take MTH The second path, consisting of Modules MTH (MTH-426), MTH (MTH-436) and MTH (MTH-514), leads to a Secondary School Diploma (SSD), which gives you access to certain CEGEP programs that do not call for a knowledge of advanced mathematics. Lastly, the path consisting of Modules MTH (MTH-526) and MTH (MTH-536) will lead to CEGEP programs that require a thorough knowledge of mathematics in addition to other abilities. Good luck! If this is your first contact with the mathematics program, consult the flowchart on the next page and then read the section "How to Use this Guide." Otherwise, go directly to the section entitled "General Introduction." Enjoy your work! 0.4

7 PROGRAM FLOWCHART CEGEP MTH You ar e here MTH-536 MTH MTH Complement and Synthesis II Introduction to Vectors MTH Geometry IV MTH-514 MTH MTH MTH-526 Optimization II Probability II MTH MTH MTH MTH Trigonometric Functions and Equations Exponential and Logarithmic Functions and Equations Real Functions and Equations Conics MTH Statistics III MTH Optimization I Trades DVS MTH-436 MTH MTH Complement and Synthesis I The Four Operations on Algebraic Fractions MTH-426 MTH MTH MTH MTH MTH Sets, Relations and Functions Quadratic Functions Straight Lines II Factoring and Algebraic Functions Exponents and Radicals MTH-416 MTH MTH MTH MTH Statistics II Trigonometry I Geometry III Equations and Inequalities II MTH Straight Lines I MTH-314 MTH MTH Geometry II The Four Operations on Polynomials MTH-216 MTH MTH Statistics and Probabilities I Geometry I MTH Equations and Inequalities I MTH Decimals and Percent MTH-116 MTH MTH The Four Operations on Fractions The Four Operations on Integers 25 hours = 1 credit 50 hours = 2 credits 0.5

8 HOW TO USE THIS GUIDE Hi! My name is Monica and I have been asked to tell you about this math module. What s your name? I m Andy. You ll see that with this method, math is a real breeze! Whether you are registered at an adult education center or pursuing distance education, you have probably taken a placement test which tells you exactly which module you should start with. My results on the test indicate that I should begin with this module. Now, the module you have in your hands is divided into three sections. The first section is the entry activity, which contains the test on the prerequisites. By carefully correcting this test using the corresponding answer key, and recording your results on the analysis sheet

9 ... you can tell if you re well enough prepared to do all the activities in the module. And if I m not, if I need a little review before moving on, what happens then? In that case, before you start the activities in the module, the results analysis chart refers you to a review activity near the end of the module. Good! In this way, I can be sure I have all the prerequisites for starting. Exactly! The second section contains the learning activities. It s the main part of the module. START The starting line shows where the learning activities begin.? The little white question mark indicates the questions for which answers are given in the text. The target precedes the objective to be met. The memo pad signals a brief reminder of concepts which you have already studied. Look closely at the box to the right. It explains the symbols used to identify the various activities.? The boldface question mark indicates practice exercises which allow you to try out what you have just learned. The calculator symbol reminds you that you will need to use your calculator.? The sheaf of wheat indicates a review designed to reinforce what you have just learned. A row of sheaves near the end of the module indicates the final review, which helps you to interrelate all the learning activities in the module. FINISH Lastly, the finish line indicates that it is time to go on to the self-evaluation test to verify how well you have understood the learning activities. 0.7

10 There are also many fun things in this module. For example, when you see the drawing of a sage, it introduces a Did you know that... A Did you know that...? Yes, for example, short tidbits on the history of mathematics and fun puzzles. They are interesting and relieve tension at the same time. Must I memorize what the sage says? No, it s not part of the learning activity. It s just there to give you a breather. It s the same for the math whiz pages, which are designed especially for those who love math. They are so stimulating that even if you don t have to do them, you ll still want to. And the whole module has been arranged to make learning easier. For example. words in boldface italics appear in the glossary at the end of the module... Great!... statements in boxes are important points to remember, like definitions, formulas and rules. I m telling you, the format makes everything much easier. The third section contains the final review, which interrelates the different parts of the module. There is also a self-evaluation test and answer key. They tell you if you re ready for the final evaluation. Thanks, Monica, you ve been a big help. I m glad! Now, I ve got to run. See you! Later... This is great! I never thought that I would like mathematics as much as this! 0.8

11 GENERAL INTRODUCTION LOGIC To begin with, logic (from the Greek word logos meaning reason) is the study of the formal rules that all accurate deductions must respect. It is, since Antiquity, one of the great disciplines of philosophy along with ethics and metaphysics. Furthermore, during the 20 th century, we saw the sudden development of a mathematical and computer approach to logic. Since the 20 th century, it has found many applications in the fields of engineering, linguistics, cognitive psychology, analytical psychology and communications. Mathematical logic has therefore reprised the objective of logic, that is to say studying reasoning, but restricting itself to statements expressed mathematically which have the advantage of being extremely standardized. More specifically, you will learn in this module how to use logical quantifiers, construct the truth table of a compound proposition and determine its value. You will need to differentiate between a tautology (always true) and a contradiction (always false). You will be asked to examine the notions of implications and equivalences, determine the negation of a compound proposition and list the solution set of a compound propositional form. After this, you will study the notions of existential and universal quantifiers. Finally, you will have to determine the negation of a compound propositional form and calculate the value of a compound propositional form. There are many topics to look into, but rest assured, you will see that they are very enriching. 0.9

12 INTERMEDIATE AND TERMINAL OBJECTIVES OF THE MODULE Module MTH contains 12 objectives and requires 25 hours of study distributed as follows. The terminal objectives appear in boldface. Objectives Number of Hours* % (Evaluation) 1 to % 4 to % % 8 and % 10 to % * Two hours are allotted for the final evaluation. 1. Identifying Propositions Identify the propositions in a list of five to ten simple verbal and mathematical statements. 2. al Connectives (1) Given a proposition expressed as a verbal or mathematical statement, determine whether it is a negation, a conjunction, a disjunction (inclusive or exclusive), a conditional proposition or a biconditional proposition on the basis of its logical connective. (2) Transcribe this proposition, using one of the following symbols to represent the logical connective: 0.10

13 for a negation (not) for a conjunction (and) for a disjunction (or) for a conditional proposition (if... then) for a biconditional proposition (if and only if) Simple statements must be selected. 3. Truth Value of a Compound Proposition Given the truth table for each type of proposition (negation, conjunction, disjunction, conditional proposition and biconditional proposition), determine the truth value of a proposition consisting of up to three simple propositions, by making sure to follow the proper sequence for operations involving logical connectives. The truth value of each simple proposition is known. The given compound proposition must be written symbolically and should consist of no more than three logical connectives. The steps in the solution must be shown. 4. Tautologies and Contradictions Set up a truth table for a proposition consisting of up to three simple propositions and three logical connectives in order to determine if that proposition is a tautology or a contradiction. The proposition is a tautology if it is always true whatever the truth value of each of its components. The proposition is a contradiction if it is always false whatever the truth value of each of its components. These given compound propositions must be written symbolically and all possibilities must be included in the truth table. The steps in the solution must be shown. 0.11

14 5. Implications ( ) Given two compound propositions linked together to form a conditional proposition, set up a truth table and determine if that conditional proposition is always true whatever the truth value of each of its components. If such is the case, connect the two compound propositions using the symbol for an implication ( ). These given compound propositions must be written symbolically and each should consist of no more than three simple propositions and three logical connectives. All possibilities must be included in the truth table and the steps in the solution must be shown. 6. Equivalences ( ) Given two compound propositions linked together to form a biconditional proposition, set up a truth table and determine if that biconditional proposition is always true whatever the truth value of each of its components. If such is the case, connect the two compound propositions using the symbol for an equivalence( ). These given compound propositions must be written symbolically and each should consist of no more than three simple propositions and three logical connectives. All possibilities must be included in the truth table and the steps in the solution must be shown. 7. Negation of a Compound Proposition Determine the negation of a compound proposition written symbolically. The negation is determined by rewriting the given proposition so that only the simple propositions bear the negation symbol. Each compound proposition should consist of no more than three simple propositions and five logical connectives. The steps in the solution must be shown. 0.12

15 8. Propositions and Propositional Forms Identify the propositions and the propositional forms in a list of five to ten verbal and mathematical sentences, some of which contain variables. 9. Solution Sets of Propositional Forms Given a universe containing five to ten elements, list the elements of the solution set of a simple propositional form or the elements of the solution set of a propositional form consisting of two simple propositional forms linked together by a logical connective. In the latter case, the answer must also include the solution set of each simple propositional form. The propositional forms must be expressed mathematically. 10. Existential and Universal Quantifiers (1) Given a list of five to ten quantified verbal sentences, indicate those that contain an existential quantifier and those that contain a universal quantifier. (2) Transcribe these statements, using one of the following symbols to represent the quantifier: for the existential quantifier (There is at least one...)! for the unique existential quantifier (There is only one...) for the universal quantifier (For all...) 11. Negation of a Quantified Compound Propositional Form. Determine the negation of a quantified compound propositional form expressed either verbally, mathematically or symbolically. The negation is determined by rewriting the given proposition so that only the simple propositions bear the negation symbol. This compound propositional form should consist of no more than three simple propositional forms and three logical connectives. The steps in the solution must be shown. 0.13

16 12. Truth Value of a Quantified Propositional Form Given a universe containing five to ten elements, determine the truth value of a quantified compound propositional form, making sure to follow the proper sequence for operations involving logical connectives. This compound propositional form should consist of no more than three simple propositional forms expressed mathematically and three logical connectives. The steps in the solution and the solution set of each propositional form must be indicated. 0.14

17 The 12 objectives in this module are covered in 7 units as outlined below. Unit Objective 1 Identifying Propositions 1 al connectives 2 2 Truth Value of a Compound Proposition 3 3 Tautologies and Contradictions 4 Implications 5 4 Equivalences 6 5 Negation of a Compound Proposition 7 6 Propositions and Propositional Forms 8 Solution Sets of Propositional Forms 9 7 Existential and Universal Quantifiers 10 Negation of a Quantified Compound Propositional Form 11 Truth Value of a Quantified Compound Propositional 12 Form 0.15

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19 DIAGNOSTIC TEST ON THE PREREQUISITES Instructions 1. Answer as many questions as you can. 2. You may use a calculator. 3. Write your answers on the test paper. 4. Don't waste any time. If you cannot answer a question, go on to the next one immediately. 5. When you have answered as many questions as you can, correct your answers using the answer key which follows the diagnostic test. 6. To be considered correct, your answers must be identical to those in the key. In addition, the various steps in your solution should be equivalent to those shown in the answer key. 7. Transcribe your results onto the chart which follows the answer key. This chart gives an analysis of the diagnostic test results. 8. Do only the review activities that apply to each of your incorrect answers. 9. If all your answers are correct, you may begin working on this module. 0.17

20 1. Indicate: a) All the prime numbers below b) All the factors or divisors of c) All the multiples of d) All the divisors of e) All the even prime numbers Solve the following equations, then verify your results. a) 4x + 8 = 40 Verification b) 6 3x = 12 Verification

21 c) 3x 7 = 2x + 8 Verification

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23 ANSWER KEY FOR THE DIAGNOSTIC TEST ON THE PREREQUISITES 1. a) 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 b) 1, 2, 3, 5, 6, 10, 15, 30 c) 0, 6, 12, 18, 24, 30, d) 1, 2, 5, 10, 25, 50 e) 2 2. a) 4x + 8 = 40 Verification 4x = x + 8 = 40 4x = 32 4(8) + 8 = 40 x = = = 40 b) 6 3x = 12 Verification 3x = x = 12 3x = 6 6 3( 2) = 12 x = = = 12 c) 3x 7 = 2x + 8 Verification 3x 2x = x 7 = 2x + 8 x = 15 3(15) 7 = 2(15) = =

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25 ANALYSIS OF THE DIAGNOSTIC TEST RESULTS Question Answer Review Correct Incorrect Section Page 1. a) Unit 1 b) Unit 1 c) Unit 1 d) Unit 1 e) Unit 1 2. a) Unit 1 b) Unit 1 c) Unit 1 Before going on to If all your answers are correct, you may begin working on this module. For each incorrect answer, find the related section listed in the Review column. Do the review activities for that section before beginning the units listed in the right-hand column under the heading Before going on to. 0.23

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27 INFORMATION FOR DISTANCE EDUCATION STUDENTS You now have the learning material for MTH and the relevant homework assignments. Enclosed with this material is a letter of introduction from your tutor indicating the various ways in which you can communicate with him or her (e.g. by letter, telephone) as well as the times when he or she is available. Your tutor will correct your work and help you with your studies. Do not hesitate to make use of his or her services if you have any questions. DEVELOPING EFFECTIVE STUDY HABITS Distance education is a process which offers considerable flexibility, but which also requires active involvement on your part. It demands regular study and sustained effort. Efficient study habits will simplify your task. To ensure effective and continuous progress in your studies, it is strongly recommended that you: draw up a study timetable that takes your working habits into account and is compatible with your leisure time and other activities; develop a habit of regular and concentrated study. The following guidelines concerning the theory, examples, exercises and assignments are designed to help you succeed in this mathematics course. 0.25

28 Theory To make sure you thoroughly grasp the theoretical concepts: 1. Read the lesson carefully and underline the important points. 2. Memorize the definitions, formulas and procedures used to solve a given problem, since this will make the lesson much easier to understand. 3. At the end of an assignment, make a note of any points that you do not understand. Your tutor will then be able to give you pertinent explanations. 4. Try to continue studying even if you run into a particular problem. However, if a major difficulty hinders your learning, ask for explanations before sending in your assignment. Contact your tutor, using the procedure outlined in his or her letter of introduction. Examples The examples given throughout the course are an application of the theory you are studying. They illustrate the steps involved in doing the exercises. Carefully study the solutions given in the examples and redo them yourself before starting the exercises. 0.26

29 Exercises The exercises in each unit are generally modelled on the examples provided. Here are a few suggestions to help you complete these exercises. 1. Write up your solutions, using the examples in the unit as models. It is important not to refer to the answer key found on the coloured pages at the end of the module until you have completed the exercises. 2. Compare your solutions with those in the answer key only after having done all the exercises. Careful! Examine the steps in your solution carefully even if your answers are correct. 3. If you find a mistake in your answer or your solution, review the concepts that you did not understand, as well as the pertinent examples. Then, redo the exercise. 4. Make sure you have successfully completed all the exercises in a unit before moving on to the next one. Homework Assignments Module MTH contains three assignments. The first page of each assignment indicates the units to which the questions refer. The assignments are designed to evaluate how well you have understood the material studied. They also provide a means of communicating with your tutor. When you have understood the material and have successfully done the pertinent exercises, do the corresponding assignment immediately. Here are a few suggestions. 0.27

30 1. Do a rough draft first and then, if necessary, revise your solutions before submitting a clean copy of your answer. 2. Copy out your final answers or solutions in the blank spaces of the document to be sent to your tutor. It is preferable to use a pencil. 3. Include a clear and detailed solution with the answer if the problem involves several steps. 4. Mail only one homework assignment at a time. After correcting the assignment, your tutor will return it to you. In the section "Student s Questions", write any questions which you may wish to have answered by your tutor. He or she will give you advice and guide you in your studies, if necessary. In this course Assignment 1 is based on units 1 to 4. Assignment 2 is based on units 5 to 7. Assignment 3 is based on units 1 to 7. CERTIFICATION When you have completed all the work, and provided you have maintained an average of at least 60%, you will be eligible to write the examination for this course. 0.28

31 START UNIT 1 PROPOSITIONS AND LOGICAL CONNECTIVES 1.1 SETTING THE CONTEXT If It's Tuesday, This Must Be Belgium! For several weeks now, a new television quiz show that tests contestants' knowledge of geography has proven to be very popular. The prizes are out of this world and the rules are simple: after reading six statements, the quiz master asks the two contestants to say which statements are true and which are false. To make things interesting, the quiz master will include a few statements that cannot be classified as either true or false.? What would your answers have been during last Tuesday's quiz? Statement True False??? 1. Winnipeg is a Canadian city. 2. Québec is west of Toronto. 3. It is the highest peak in the world. 4. Havana is the capital of Cuba. 5. There are 60 American states. 6. That river is located in France. 1.1

32 It is easy to determine if Statements 1 and 4 are true. We can also say that Statements 2 and 5 are false. Statement 3 is true if we are talking about Mount Everest, otherwise it is false. Thus, we cannot say whether this statement is true or false. The same applies to Statement 6. If we are talking about the Seine, for instance, the statement is true, whereas it is false if we are talking about the Richelieu River. Your answers should have been those indicated in the table. Statement True False??? 1. Winnipeg is a Canadian city. 2. Quebec City is west of Toronto. 3. It is the highest peak in the world. 4. Havana is the capital of Cuba. 5. There are 60 American states. 6. That river is located in France. A statement that is either true or false, but not both, is called a proposition. Therefore, Statements 1, 2, 4 and 5 are propositions. We say that a proposition has only one truth value. Finding the truth value of a proposition means determining if the proposition is true or false. To achieve the objectives of this unit, you must be able to identify the propositions in a list of simple verbal and mathematical statements. In addition, you should be able to determine if the proposition is a negation, a conjunction, a disjunction, a conditional proposition or a biconditional proposition on the basis of its logical connective and then transcribe the proposition into the appropriate symbolic form. 1.2

33 1.1.1 Identifying Propositions Is there an easy way to determine whether a statement is a proposition? We should be able to say without a doubt that a statement is true or false after observing reality or checking the facts. When we say, "Dakar is the capital of Senegal," we can conclude that this statement is a proposition, even if we do not know its truth value offhand, because we have only two choices: Dakar is or is not the capital of Senegal. For the moment, it doesn't really matter whether the proposition is true or false, since this information can always be checked later on.? Find the truth value of the above proposition.... It is true that Dakar is the capital of Senegal. We can check this statement in a dictionary or an encyclopedia. Let's see if the following types of sentences are propositions. Interrogative sentences: "What time is it?" "Where are you?" The answer to these questions is neither " true" nor "false." Thus, interrogative sentences are not propositions. Exclamations: "Show me your driver's license!" "Not so fast!" These sentences are not propositions, because we cannot give them a truth value. Sentences involving personal opinion or matters of taste: "It's the most beautiful room in the house." "Paul is always nice to people." Stating that Paul is always nice to people is expressing a personal opinion. Perhaps some people think he is not nice at all. We cannot determine whether this statement is true or false, so it is not a proposition. 1.3

34 Sentences we cannot categorize as true or false because the subject or object of the sentence is not specified: "It's the largest planet." We have to know which planet we are talking about before concluding that it is the largest. As a result, this type of sentence is not a proposition. Incomplete sentences: "Saturday, February 24th." This is not even a sentence, let alone a proposition. Declarative sentences that have one and only one truth value: "Montréal is an island." "Mozart is not dead." The first of these statements is true, whereas the second is false. These statements can easily be checked and are propositions. A proposition is a statement that is either true or false, but not both. State whether or not the following sentences are propositions.? 1. Paul's uncle is a musician....? 2. Somewhere, over the rainbow....? 3. How sad Venice is!...? 4. A litre of milk costs nearly a dollar....? 5. How is your sister?...? 6. The sun sets at 7:30 p.m. in the spring....? 7. Pass me the butter....? 8. Québec is the capital of Canada.... Sentences 1, 4, 6 and 8 are propositions. 1.4

35 Mathematical statements can also be propositions. For example, = 8 is a true proposition. The inequality 18 > 25 is also a proposition, but this one is false.? Is the statement 8x = 72 a proposition?... This statement is not a proposition. If x = 9, the proposition is true, but if x is not equal to 9, the proposition is false. The definition of proposition is very precise: it is a statement that is either true or false, but not both. The statement 8x = 72 is called a propositional form. In order to understand the rest of this unit, it is important to grasp the difference between a proposition and a propositional form. There is no room for ambiguity in logic. Just as a door is either open or closed, a proposition must be true or false. Remember that when a proposition is true, we say its truth value is true and when the opposite is the case, its truth value is false. Example 1 a) The statement "8 is not a square" is a proposition. The truth value of this proposition is true. b) The statement "14 6" is not a proposition, so we cannot give it a truth value. c) The statement "19 < 15" is a proposition. Its truth value is false. d) The statement "3x 2 = 10" is not a proposition. We cannot give it a truth value because the value of x is unknown. This equation is a propositional form. 1.5

36 Exercise 1.1 By answering yes or no, indicate whether the following statements are propositions. When you encounter a proposition, state its truth value = is a very large number A square has 4 right angles The cube of There is a country named Italy x + 4x = 7x , 2 and 5 are divisors of A triangle always has two congruent angles Why not me? A leap year has 365 days Different Types of Propositions and al Connectives is a science that studies the principles of reasoning using simple propositions like those we have just seen, and many others, as we shall see later. At this point, we are concerned only with the form of a sentence and not its content. Don't be surprised if early on you encounter such sentences as: "Paula is my sister if and only if = 6." We will examine the content of propositions a little later in the course when defining the mathematical concepts you will need for subsequent mathematics courses. 1.6

37 Now let's examine another type of proposition known as a compound proposition. Here are some everyday examples of compound propositions. If I fail my examination, then I'll have to take it over. Hockey and baseball are my favorite sports. I am going to have a piece of pie or some yogurt. The summer will not end soon. I will go to the movies if and only if you come with me. Compound propositions are also used in mathematics. 8 is greater than 6 and less than 12. A triangle is right-angled if and only if it has an angle of 90. The square root of 25 is 5 or 5. If a quadrilateral is a square, then it is a rectangle. The number 9 is not a prime number. We can form compound propositions by using the words and, or, if... then and if and only if to connect two simple propositions. The negation of a proposition, not, also creates a compound proposition. The expressions not, and, or, if... then, and if and only if are called logical connectives. We all know that the process of adding and subtracting numbers is represented by plus and minus signs, formally known as "addition" and "subtraction." Similarly, the logical connectives used to form compound propositions are given symbols and formal names. 1.7

38 The five logical connectives and their symbols are presented in the following table. Expression al Connective Symbol (Formal Name) not and or if... then if and only if negation conjunction disjunction conditional connective biconditional connective A compound proposition consists of two or more simple propositions linked together by means of any of the following logical connectives: negation, conjunction, disjunction, conditional connective, biconditional connective. In logic, the lower case letters p, q, r and s are generally used to represent propositions, just as the letters x and y are used in algebra to represent unknown or variable quantities and the capital letters A, B and C are used in geometry to indicate the vertices of a triangle. It is merely a matter of convention. Let's see how we can go about translating verbal or mathematical statements into symbolic language using these logical connectives. Example 2 p: the weather is nice, q: I am going out. Using the letters p and q along with logical connectives we can "translate" various propositions into logical language. 1.8

39 Everyday Language Symbol Meaning The weather is not nice. p not p It is nice and I go out. p q p and q It is nice or I go out. p q p or q If it is nice, then I go out. p q if p, then q It is nice if and only if I p q p if and only if q go out. In addition, we can create compound propositions consisting of more than one logical connective. Example 3 We are given the same propositions p and q as in Example 2. a) It is nice or I do not go out: p q. b) If I do not go out, then it is nice: q p. c) I go out and it is not nice: q p. d) I go out if and only if it is not nice: q p. Here are some exercises to help you review these basic concepts. Exercise For each of the following compound propositions, indicate the symbol for the logical connective and give its name. Symbol Name a) If a number is even, then it can be divided by

40 b) You are allowed in the discotheque if and only if you are 18 years old c) One swallow does not make a spring d) I will have an apple or a pear e) I like the metro and the bus p: I am thirsty, q: I drink some juice. What do the following compound propositions mean? a) q p :... b) p q :... c) q p :... d) q p :... e) q : p: 18 is an even number, q: 2 9 = 18. Translate the following propositions into logical symbols. a) If 2 9 = 18, then 18 is an uneven number.... b) 18 is not an even number.... c) 2 9 = 18 and 18 is an even number

41 So, you see, logic is not so difficult after all! We will now take a closer look at each of the logical connectives by putting them in a table, as we do with multiplication tables. These tables are called truth tables because they make it possible to determine whether a compound proposition is true or false. Let p: cats have four paws. Negation We can negate this proposition in several ways: cats do not have four paws; it is not true to say that cats have four paws; it is false to say that cats have four paws. The symbol for all of these compound propositions is written p.? What is the truth value of the proposition 12 = ?...? What is the truth value of the proposition ?...? What is the truth value of the proposition 30 = 18?...? What is the truth value of the proposition 30 18?... In reply to the above questions, you should have said that the first and last propositions are true, and that the other two are false. The negation of a true proposition is a false proposition. The negation of a false proposition is a true proposition. A table will make this idea easier to understand. This table is called the truth table for negations. 1.11

42 In this table the first column represents the truth value of proposition p and the second column, the truth value of the compound proposition p. Note that "T" represents true and "F" represents false. Truth Table for the Negation p p T F F T p The second line shows that proposition The third line shows that proposition p is false if p is true. p is true if p is false. It is always this truth table that applies when we encounter a negation. Conjunction p: an apple is a fruit, q: a carrot is a vegetable. Use the connective and to join these two propositions, then write them using symbolic logic. An apple is a fruit and a carrot is a vegetable. p q Find the truth value of each of the following compound propositions.? 1. An apple is a fruit and a carrot is a vegetable....? 2. An apple is a fruit and a carrot is not a vegetable....? 3. An apple is not a fruit and a carrot is a vegetable....? 4. An apple is not a fruit and a carrot is not a vegetable

43 Only the first compound proposition is true because it is the only one in which the two simple propositions are true. In the second compound proposition, the second simple proposition is not true. In the third compound proposition, the first simple proposition is false. In the last compound proposition, both simple propositions are false. We can use what we know about the four propositions above to construct a truth table for the conjunction of two simple propositions. Truth Table for the Conjunction p q p q p q T T T T F F F T F F F F The conjunction of two simple propositions is true only when both simple propositions are true. We can create compound propositions consisting of both a negation and a conjunction. Example 4 p: 6 2 = 4, q: 4 is an even number. The proposition "6 2 4 and 4 is an even number" is written: p q. The proposition p number." q is equivalent to "6 2 = 4 and 4 is not an even 1.13

44 The proposition "it is not true to say that and 4 is an even number" is written ( p q ). In everyday language, many sentences express the meaning of the conjunction and without using the word and. Note the following example. p: the house is grey, q: the house is square. The house is neither grey nor square. means the house is not grey and the house is not square. p q The house is not grey, but it is square. means the house is not grey and the house is square. p q Now let's go on to our third truth table. Your knowledge of the first two should help you construct the disjunction table because you are now more familiar with the language of logical symbols. Disjunction Henry's boss asked him to work Saturday morning or one night this week to update some important files. Naturally, he will be paid overtime! We can describe this situation using the following simple propositions. p: Henry is working Saturday morning, q: Henry is working one night this week. The compound proposition p q represents this situation and means "Henry is working Saturday morning or Henry is working one night this week." 1.14

45 In each of the following four situations, state whether or not Henry is doing what his boss requested. (In other words, find the truth value of the compound propositions.)? 1. Henry has a lot of free time and will work Saturday morning. Since he's afraid he won't finish the work then, he will also work one night this week....? 2. Henry can't work Saturday morning because he has other things to do at that time. However, he will be able to work one night this week....? 3. There's nothing on the agenda Saturday morning, so he will do the work then. However, he is busy every night this week....? 4. Henry doesn't want to work overtime. He told his boss he is busy Saturday morning and every night of the week Yes, Henry has amply fulfilled his boss' request. In fact, it was more than his boss had asked for. 2. Yes. Since his boss had asked him to work one night during the week or Saturday morning, Henry has fulfilled his request. 3. Yes. Since his boss had asked him to work one night during the week or Saturday morning, Henry has fulfilled his request. 4. No, his boss will certainly not be happy with his answer. Have these four situations helped you set up a truth table for a disjunction? If you don't think so, reexamine the situation from the standpoint of Henry's boss. Truth Table for the Disjunction p q p q p q T T T T F T F T T F F F 1.15

46 The disjunction of two simple propositions is always true, except when the two simple propositions are false. It's time to see if you have understood the first three truth tables. Exercise p: Edward is a teacher, q: Edward is 27 years old, r: Edward lives in Laval. Translate each of these compound propositions into everyday language. a) p q... b) p q... c) (p r)... d) p r... e) r q p: a square has 4 sides, q: a triangle has 3 angles. Express each of these compound propositions in symbolic language. a) A square does not have 4 sides or a triangle has 3 angles.... b) It is false to say that a square does not have 4 sides or that a triangle does not have 3 angles

47 c) A square has 4 sides and a triangle does not have 3 angles Give the truth value of the following compound propositions. a) p q when p is true and q is false.... b) q when q is false.... c) p r when p and r are false.... If you have fully understood the first three connectives, then we can begin studying the fourth. Conditional Connective Mary is addicted to buying lottery tickets. Every week she buys at least one ticket hoping to win a big prize. She has promised to take her friend Mark out to dinner if she wins. Occasionally, Mark thinks about Mary's offer and wonders under what circumstances Mary might actually take him out to dinner. In which of the following four propositions does Mary keep her promise?? 1. If Mary wins, then she'll take Mark out to dinner....? 2. If Mary wins, then she won't take Mark out to dinner....? 3. If Mary does not win, then she'll take Mark out to dinner....? 4. If Mary does not win, then she won't take Mark out to dinner

48 1. Yes, that was the agreement. 2. No, because Mary has promised to take him out to dinner if she wins. 3. Yes, because even if she does not win she'll be taking him out to dinner. In this case, she is doing more than just keep her promise. 4. Yes, it's perfectly normal not to take him out to dinner because she did not win the lottery. Given the four answers outlined above, we can construct a truth table for the conditional proposition p q. Truth Table for the Conditional Proposition p q p q p q T T T T F F F T T F F T In the conditional proposition p q, the proposition p is called the antecedent and the proposition q, the consequent. A conditional proposition formed by the connective if... then can be rephrased in several different ways in everyday language. For instance, "If I eat too much, then I get sick." can be rephrased as follows: I get sick if I eat too much. All I need to do to get sick is eat too much. I get sick when I eat too much. 1.18

49 A conditional proposition consisting of two simple propositions is always true, except when the antecedent is true and the consequent is false. Biconditional Connective (if and only if) Tom did not get very good marks on his third report card. He must absolutely get good marks on his fourth report card to be able to go to business college. This is the only way he will be able to get into college. Biconditional propositions generally contain the connective if and only if. Tom has several possibilities to consider. Determine the truth value of the four possible situations.? 1. Tom can go to college if and only if he gets good marks....? 2. Tom can go to college if and only if he doesn't get good marks....? 3. Tom cannot go to college if and only if he gets good marks....? 4. Tom cannot go to college if and only if he doesn't get good marks.... If you concluded that the first and the last propositions were true, bravo! Otherwise, redo the exercise to make sure you have understood it. 1.19

50 Truth Table for the Biconditional Proposition p q p q p q T T T T F F F T F F F T A biconditional proposition consisting of two simple propositions is true when its two component propositions are both true or both false. In fact, a biconditional proposition can be expressed as a conjunction of two conditional propositions. In the preceding example, we could just as well have written: Tom is going to college if he gets good marks. and Tom has good marks if he is going to college. p q q p A little later you will see that in fact these propositions are called equivalent propositions. Everyday language provides other ways of saying if and only if. For instance, the axiom a triangle is equilateral if and only if the three sides are congruent can be expressed as follows: For a triangle to be equilateral, the only requirement is that it should have three congruent sides. A necessary and sufficient condition for a triangle to be equilateral is that it should have three congruent sides. The exercises that follow are related to conditionals and biconditionals and we have added a few negations. 1.20

51 Exercise p: it is raining, q: Jack is going to the movies, r: Karen is taking the metro. Convert each of the following compound propositions into everyday language. a) p q... b) p r... c) q p... d) p q... e) (q r) Use propositions p, q and r given below to translate the following compound propositions into symbolic language. p: Diana has a lot of money, q: Diana has a beautiful house, r: Diana's husband is an electrician. a) If Diana has a beautiful house, then Diana has a lot of money.... b) A necessary and sufficient condition for Diana's husband to be an electrician is that Diana should have a beautiful house.... c) It is false to say that if Diana has a lot of money, it is because her husband is an electrician.... d) If Diana has a lot of money, then her husband is not an electrician

52 3. Give the truth value of each of these compound propositions. a) p q if p is true and q is false.... b) q p if p and q are false.... c) p q if p is false and q is true.... d) q p if p and q are true.... Did you know that a syllogism is a form of logical reasoning that dates back to ancient times? It involves drawing a conclusion from two simple propositions. The best known syllogism is: All men are mortal. Socrates is a man. Therefore, Socrates is mortal. Try making up your own syllogism. 1.22

53 ? 1.2 PRACTICE EXERCISES 1. p: Montréal is a major city, q: Laval is a suburb, r: Québec is a capital. Translate each of these compound propositions into everyday language. a) p q... b) p... c) q r... d) p q... e) (p r ) Use propositions p, q and r given below to translate the following compound propositions into symbolic language. p: 9 is a multiple of 3, q: 3 is a prime number, r: 3 is an odd number. a) 9 is a multiple of 3 and 3 is a prime number.... b) If 3 is an odd number, then it is a prime number.... c) If 9 is a multiple of 3 or 3 is an odd number, then 3 is a prime number

54 d) It is false to say that 3 is an odd number or a prime number.... e) 3 is an odd number if and only if it is a prime number.... f) 3 is neither a prime number nor an odd number In all the following compound propositions, we will assume that p, q and r are true. Give the truth value of these propositions. a) p q... b) r p... c) (p r )... d) p q... e) q r True or false? a) A conditional proposition is true if and only if the two compound propositions are true.... b) The symbol is the symbol for a conjunction.... c) The negation of a true proposition is a false proposition.... d) A conditional proposition is false if the antecedent is true and the consequent is false

55 1.3 SUMMARY ACTIVITY In this first unit we have become familiar with the logical connectives and their truth tables. There are five logical connectives. al Connectives not and or if... then if and only if negation conjunction disjunction conditional connective biconditional connective Truth tables are absolutely necessary to the study of logic. You do not have to know them by heart, since you can always consult them when necessary. Nevertheless, they are easy to remember if you remember their characteristics. Characteristics of Truth Tables The truth value is changed. True only if the two propositions are true. False only if the two propositions are false. False only if the antecedent is true and the consequent is false. True if the two propositions are both true or false. Complete the following truth tables. Then check your answers against the tables given previously or the above table of characteristics. Make sure that your tables are accurate because you will be using them frequently. 1.25

56 The 5 Truth Tables for Compound Propositions Truth Table for the Negation p p p T F Truth Table for the Conjunction p q p q p q Truth Table for the Disjunction p q p q p q T T T T T F T F F T F T F F F F Truth Table for the Conditional Proposition p q p q p q Truth Table for the Biconditional Proposition p q p q p q T T T T T F T F F T F T F F F F For instance, if we want to know the truth value of the proposition p q when p is false and q is true, we look at the third line of the truth table for conditional propositions and we find T in the third column. Consequently, we can say that this proposition is true. In the next unit we will see the steps involved in finding the truth value of much more complicated compound propositions. 1.26

57 1.4 THE MATH WHIZ PAGE A al Riddle John and Charles, two old friends who haven't seen one another since high school happen to meet one day in Montréal. They talk over old times and after awhile Charles asks John how many children he has. "I have three children, " says John. Knowing Charles' fascination with riddles, he adds, "I won't tell you how old they are, but I'll give you a clue: the product of their ages is equal to 36." Charles mulls over the clue, then takes a piece of paper and begins jotting down some numbers. A moment later he says, "I haven't got enough information to tell you how old your children are. Give me another clue." "Look at the building across the street and count the number of windows," says John. The number of windows in that building equals the sum of my children's ages." Charles counts the windows, looks at his calculations and thinks the problem over. He then says, "Listen John, I still can't figure out how old your three children are, but give me one last clue." "The oldest has blue eyes," says John. "Ah, now I've got it figured out!" exclaims Charles. He then gives John the answer. How old are John's children? 1.27

58 1.28