MATH 1

1MATH 1

QUANTITATIVE REASONING

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2Why take this course?Quantitative concepts and skills are crucial to meeting the difficult challenges in this rapidly changing technological world.The ability to reason with quantitative information is an essential component of decision making.

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Course Description:Logical, quantitative and mathematical thinking

Course Goals:At the end of this course, you will gain:a comprehensive, integrated yet synoptic understanding of mathematics as a system of knowledge and as a way of knowing; and the ability to think critically and logically, and reason quantitatively on issues that are relevant to yourself and to society.

Course Objectives:At the end of this course, you should be able to:

describe the nature and functionality of quantitative literacy;examine real issues and problems from a variety of perspectives;describe the different levels of mathematical understanding;challenge wrong attitudes and misconceptions about mathematics;recognize the basic principles of reasoning;

differentiate inductive and deductive arguments;analyze real-life arguments;recognize the relationships of logic, science, and mathematics in the search for knowledge;describe the value and limitation of logic in the search for knowledge;integrate quantitative reasoning with logical reasoning by investigating the concept of numbers;

examine uncertainty and use numbers with precision;apply thinking skills to the solution of quantitative problems andenumerate and illustrate the processes involved in problem solving.

Course Outline:

Part I. THINKING LOGICALLY

A. Literacy in the Modern World 1. Interdisciplinary thinking2. What is quantitative literacy?3. Challenging misconceptions about mathematics4. What is mathematics?5. The road to quantitative literacy

Arguments and reasoning1. Logic: the study of reasoning2. The value of logical arguments3. Propositions the building block of arguments4. Deductive /Inductive arguments5. Arguments in the real world6. Fallacies of relevance7. Fallacies of numbers and statistics

C. The search for truth and knowledge 1. The search for truth2. The limitations of logic3. Logic and science4. Paradoxes

10Part 1. Thinking Logically

What it actually depicts is the so-called Helix Nebula, described by astronomers as "a trillion-mile-long tunnel of glowing gases." At its center is a dying star which has ejected masses of dust and gas to form tentacle-like filaments stretching toward an outer rim composed of the same material. Our own sun may look like this in several billion years.

Image Credit: NASA, WIYN, NOAO, ESA, Hubble Helix Nebula Team, M. Meixner (STScI), & T. A. Rector (NRAO).

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111.1 Literacy for the Modern WorldLiteracy in todays world demands not only reading and writing but also an understanding of the fundamental concepts of logic, mathematics and science.

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12The focus of this course is

Quantitative Literacy -

literacy in terms of information involving mathematical ideas or numbers.

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13 Quantitative literacy is fundamental to nearly every discipline of study and to different issues in society which an individual faces.

Virtually every major issue in the modern world involves mathematical ideas.

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14Interdisciplinary ThinkingIdeas from disciplines other than mathematics are also essential to understanding current issues.

Important issues, whether personal or societal, are interdisciplinary in nature. They can best be understood when examined from various perspectives.

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15Issues are better studied in an approach that recognizes how the various branches of human knowledge are interconnected.

The danger of compartmentalized education is the lack of perspective and the inability to see the big picture.

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16What is quantitative literacy?

Quantitative literacy is literacy in terms of information involving mathematical ideas or numbers. It is the ability to interpret and reason with quantitative information. Quantitative reasoning is the process of interpreting and reasoning with quantitative information.

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17What is innumeracy? It is the lack of quantitative literacy.

Since quantitative literacy is a survival skill, the lack of it can lead to financial trouble and personal problems.

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18At the societal level, decisions involving government spending, resource utilization, environmental protection, wars and social justice require quantitative reasoning.

Innumeracy leads to a misunderstanding of logic, probability and statistics, thus to an inability to distinguish between legitimate science and fraudulent science.

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19Advantages of Quantitative LiteracyIt enriches the appreciation of both ancient and modern culture.

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It helps you appreciate the substantial contribution to mathematics, science and technology of achievers in various fields.

E = mc2Andrew Wiles (Fermats Last Theorem)

It provides greater chances for employment.

Wanted!!!SecretaryMust be computer literate , logical, organized and with pleasing personality.

LOOK WHO STUDIED MATHEMATICS!Virginia Wade, Wimbledon Champion, B.S. Math, Sussex Univ.

Edwin Moses, 3-time Olympic champion in the 400-m hurdles, math major in Morehouse College

David Robinson, basketball star, B.S. Math, U.S. Naval AcademyWilliam Perry, U.S. Secretary of Defense, Ph.D. in Math. Penn. State Univ.

Alexander Solzhenitsyn, Nobel Prize winning Russian author, degrees in math from the Univ. of Rostov

Bram Stoker, author of Dracula, studied Math in Trinity Univ, Dublin

Florence Nightingale, pioneer in nursing, studied math and applied it to her workSally Ride, the first American woman in space, studied math in Stanford Univ.

26Laurence Tribe, Harvard Law professor, Summa cum laude in Math, Harvard Univ.

Corazon AquinoFormer Phil. President, math degree, Mount Saint Vincent College,NY

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Art Garfunkel, musician, MA in Math Columbia Univ.

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BRIDGE OVER TROUBLED WATER

When you're wearyFeeling smallWhen tears are in your eyesI will dry them all

I'm on your sideWhen times get roughAnd friends just can't be foundLike a bridge over troubled waterI will lay me downLike a bridge over troubled waterI will lay me down

When you're down and outWhen you're on the streetWhen evening falls so hardI will comfort you

I'll take your partWhen darkness comesAnd pain is all aroundLike a bridge over troubled waterI will lay me downLike a bridge over troubled waterI will lay me down

Sail on Silver Girl,Sail on byYour time has come to shineAll your dreams are on their way

See how they shineIf you need a friendI'm sailing right behindLike a bridge over troubled waterI will ease your mindLike a bridge over troubled waterI will ease your mind

29Four levels of quantitative literacy1. Theoretical mathematics- discovery of entirely new mathematical principles

[mathematicians, theoretical scientists]

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302. Applied mathematics-known mathematical tools are applied to problems of immediate interest such as analyzing risk in insurance policies, developing mathematical models to assess human impact on the environment, or teaching mathematics[engineers, scientists, teachers, statisticians, business analysts]

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31Applied mathematics can be viewed as a central resource for addressing and solving problems in a wide and growing variety of disciplines such as business management, economics, engineering, biology/ecology, computer science/artificial intelligence, physics, chemistry, medicine and physiology, and even in psychology and sociology.

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323. Vocational mathematics the use of mathematical tools routinely such as computer programming, and accounting and banking wherein mathematical methods are used to analyze financial records and investment strategiesdoes not involve discovery of new principles or application of principles in new ways [computer programmers, accountants]

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334. Basic Quantitative literacy

necessary for everyone

survival skill in todays technological society

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34MisconceptionsMath requires special or inherent intellectual abilities.

Math is gender dependent, ethnic based, hereditary.

Math in modern issues is too complex for the average person to understand. It is difficult and dull.

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35Math makes you less sensitive to the romantic and aesthetic aspects of life.

Math makes no allowance for creativity.

Math provides exact answers.

Math is irrelevant to my life.

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The word mathematics is derived from the Greek word mathematikos which means inclined to learn (thus literally, to be mathematical is to be curious, open-minded, and interested in always learning more)

We look at mathematics in three different ways:

What is mathematics?

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371. Mathematics as the sum of its branches

logic- study of the principles of reasoning arithmetic- study of methods for operating on numbersalgebra- study of methods for working with unknown quantitiesgeometry-study of size and shapetrigonometry-study of triangles and their usesprobability- study of chancestatistics study of methods for analyzing datacalculus - study of quantities that change

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382. Mathematics as a way to model the worldMathematics can be viewed as a tool for creating models, representations that allow us to study real phenomena.

Models allow us to gain insight into otherwise intractable problems; and point to areas where further research is needed.

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Illustrations of Models

40Law of Natural growth and decay

LetP be the population at any time t andA be the initial population.The rate of increase/decrease of the population is proportional to the present population.

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42Advertising and salesExample: A company finds that the number y of dollars per week that it must spend on advertising in order to sell x units of its product is

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43Animal PhysiologyIf W is the weight of an average animal of a certain species at age t, it is often found that

where A and B are certain constants.

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44Predator-Prey ModelLet x and y be the sizes of 2 populations, one of which of which preys on the other. At any time t, x and y satisfy

where a, h and k are certain constants.

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45Telephone Demand ExampleDuring business hours (8 A.M. -5 P.M.), the number of telephone calls per minute passing through a switchboard vary according to the formula

where t is time in hours measured from 8 A.M..

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http://en.wikipedia.org/wiki/Normal_distribution Normal Distribution Curve

The Learning Curve

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http://www.theweatherforecast.info/afbeeldingen/filipijnen_neerslag.gif&imgrefurlAverage Monthly Rainfall in the Phils.

49Mathematical models take the form of tables, graphs, equations, etc. We will learn numerous techniques for building mathematical models and use them to study meaningful problems.

However, models are not the real thing, they are only as good as the equations and observations from which they are made.

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503. Mathematics as a languageIt is the language of nature because it is very useful for modeling the natural world.

It has its own grammar and vocabulary.

You must be fluent in this language. Quantitative literacy is the level of mathematical fluency required for success in todays world.

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y = f(x)As x becomes large, f(x) also becomes large without bound.

5y = g(x)As x becomes large, g(x) gets closer and closer to 5.

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52How to attain quantitative literacyIdentify your personal goals and strategies. Goal - an end toward which effort is directed Strategy - a plan or method for achieving a goalBreak down your psychological barriers.Set your course.

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There is a need to understand the general principles of reasoning before one can learn to reason with quantitative information.Possessing skills in reasoning can be advantageous.1.2 Arguments and Reasoning53

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It gives confidence in ones ability to learn, ask questions and figure out things.It helps us understand the many forces of persuasion and fallacy shown in media. It is an important ingredient in sound decision making.54

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Outline1.2.1 What is Logic?1.2.2 Logical Arguments1.2.3 Propositions 1.2.4 Deductive Arguments1.2.5 Inductive Arguments1.2.6 Analyzing Real-life Arguments 1.2.7 The Forces of Persuasion1.2.8 Fallacies of Relevance1.2.9 Fallacies of Numbers and Statistics1.2.10 Fallacies Involving Percentages55

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1.2.1 Logic: The Study of ReasoningMany people make decisions for many different reasons.Decisions may be made from emotion or from sound reasoning.In reasoning, you carefully analyze the potential impacts of your decision on your personal goals.56

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Reasoning ability, together with personal background and emotional state, shapes beliefs and opinions.Reasoning ability is crucial to the individuals evaluation of beliefs, opinions and decisions of others.Quantitative reasoning is the ability to reason with quantitative or mathematical information.57

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What is Logic?Logic is the study of the methods and principles used to distinguish correct reasoning from wrong reasoning.58

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Time-out to think:

Do you believe that all decisions should be based on logic? When should logic be used and when should other factors be considered?

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1.2.2 Logical Arguments

In logic, the word argument has a more precise meaning than it does in ordinary English language.

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Consider the following argument.Jose: Soldiers are immoral.Maria: No, they are not.Jose: Yes they are! They should go to hell.Maria: You dont even know what youre talking about!Jose: I know a lot more than you know. Ive watched lots of film about it. Maria: I cant talk to you. Youre not a UP student. Youre an idiot!61

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A heated conversation indeed but not a logical argument

It has no structure, and neither Jose nor Marias conclusion is established from their statements.

Jose: Soldiers are immoral.Maria: No, they are not.Jose: Yes they are! They should go to hell.Maria: You dont even know what youre talking about!Jose: I know a lot more than you know. Ive watched lots of film about it. Maria: I cant talk to you. Youre not a UP student. Youre an idiot!62

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Let us try to restructure Joses argument.

P:The killing of a human being is immoral.P: Soldiers kill human beings.C: Soldiers are immoral.63

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Marias possible rebuttal:

P: The killing of a human being in self-defense or when his life is in danger is not immoral.C: The killing of a human being is not necessarily immoral.

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P: The killing of a human being is immoral.P: Soldiers kill human beings.C: Soldiers are immoral.P: The killing of a human being in self-defense or when his life is in danger is not immoral.C: The killing of a human being is not necessarily immoral.The arguments are clear and well conceived, the premises lead logically to the conclusion.Establishing logical arguments has a great value: through this process, each side may better understand the other.65

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So what is a logical argument?

A logical argument is made up of building blocks called propositions.It has a premise and a conclusion.66

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A proposition (or statement) is a declarative sentence that makes a distinct claim, such as an assertion or denial; it proposes something to be true or false.

Some of the propositions in the example:Soldiers are immoral.Soldiers are not immoral.1.2.3 Propositions67

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Both premise and conclusion in a logical argument are propositions. Each makes a distinct claim that, depending on your viewpoint, is either true or false.A proposition must have a subject and a predicate.A proposition must be capable of being true or false, but not both at the same time, though we may not know which it is.68

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Types of propositions:Categorical proposition - a proposition that expresses a relationship between two categories or sets, the subject set S and predicate set P.Examples: All politicians are corrupt.No politician is corrupt. Some politicians are corrupt. Some politicians are not corrupt. 69

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The four standard categorical propositions:

1. All S are P. 2. No S are P.( All members of S are members of P.) ( No member of S is a member of P. )

PS

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3. Some S are P.S

P

4. Some S are not P.

P

( Some members of S are members of P.)( Some members of S are not members of P.)S71

Exercise: Rephrase the following categorical propositions into standard form:All UPLB students are smart.Let U be the set of all UPLB students andS be the set of all smart people.Ans. All U are S.72

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Some UPLB students do not like Kris.

Let U be the set of all UPLB students andK be the set of all people who do not like Kris.

Ans. Some U are K.73

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3. UPLB students never cheat during exams.

Let U be the set of all UPLB students andC be the set of all people who cheat during exams.

Ans. No U are C.

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4. Some movie actors do not pay taxes.

Let M be the set of all movie actors andT be the set of all people who pay taxes.

Ans. Some M are not T.75

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A proposition always makes a claim of truth. This claim however is not necessarily true. Although a proposition is capable of being either true or false, determining which it is may not be possible.76

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Claims of truth: A proposition maybeunambiguous - no one can reasonably disagree with its truth or falsity2. unverifiable - would require impossible or impractical procedures to determine its truth or falsity3. matter of opinion - truth can be argued endlessly77

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Illustrations:

The following are unambiguous.

1. The sum of 2 even integers is even.

2. The sum of roots of x2 - 1 = 0 is 0.

3. If today is Tuesday, then tomorrow is Wednesday.78

Illustrations:

The following are unverifiable:

1. It was raining in the Philippines in 300 B.C.

2. I was a butterfly in my first life.

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Illustrations:

The following are a matter of opinion.

1. There is God.

2. Filipino women are better housewives than Australian women.80

Compound proposition a proposition that consists of two or more simple (or prime) propositions joined together by logical connectors.not(negation) and (conjunction) or (disjunction) if then(conditional)if and only if(biconditional)

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pTFpFTLet p be a proposition.Truth Table for a Negation82

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Give the truth value of each statement.1. 2 is not odd.Ans. T (2 is odd.)2. It is not the case that 49 is a perfect square.Ans. F (49 is a perfect square.)

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Let p and q be propositions. If both p and q are true, then the compound proposition p and q (denoted by p q) is true. Otherwise p q is false. 84

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pqp qTTTFFTFF

TFFFTruth Table for a Conjunction85

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Give the truth value of each statement.1. 2 is prime and 2 is even. Ans. T2. 3 is even and I am Darna. Ans. F

FT

T

(Whatever!)86

Or is inclusive when it means either or both.Or is exclusive when it means either but not both. In logic, or is always inclusive.87

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Let p and q be propositions. If at least one of p or q is true, then the compound proposition p or q (denoted by p q) is true. Otherwise, p q is false.88

pqp qTTTFFTFF

FTTTTruth Table for a Disjunction89

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Give the truth value of each statement.1. 2 is odd or 3 is even. Ans. F2. 4 is even or I am Darna. Ans. T

TF

F

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A conditional proposition is a proposition joined by the words if and then.Form: If p then q.p is called the antecedent, while q is called the consequent.

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The conditional proposition if p then q (denoted by p q) is true unless p is true and q is false.92

pqp qTTTFFTFF

TTTFTruth Table for a Conditional

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Examples:94If Makati is not an agricultural region, then the fertilizer funds must have gone somewhere.If a function is differentiable at a number, then it is continuous at that number. Differentiability implies continuity.3. If you take Caltrate plus, then hindi ka liliiit.

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Give the truth value of each statement.1. If 2 is odd then 3 is even. Ans. T2. If 4 is even then 5 is even. Ans. FTF

F

F

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A biconditional proposition has the formp if and only if q.

The proposition p if and only if q (denoted by p q) is true if p and q have the same truth values. It is false otherwise.

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A number is even if and only if it is divisible by 2. A number n is even if and only if there exists an integer k such that n = 2k.

2. ab = 0 if and only if a = 0 or b = 0.

3. The sum of two numbers is even iff both numbers are odd or both are even.Examples:97

Truth Table for a Biconditional

pqp qTTTFFTFF

TFTF

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Give the truth value of each statement.Ans. TAns. FTF

F

F

1. 2 is odd iff 3 is even. 2. 4 is even iff 5 is even. 99

A compound proposition is said to be a tautology if it is true for all truth value assignments of the involved simple propositions.Illustration:For each proposition p, p ~p is a tautology.

pTF~ pFTTT

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For all propositions p and q, [(p q) p] q is a tautology.pTT

qpqTTFFTFFTTFTTFFTFFFTFTFTTTT

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A compound proposition is said to be a contradiction if it is false for all truth value assignments of the involved simple propositions.Illustration:For each proposition p, p ~p is a contradiction.

pTF~ pFTFF

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A compound proposition is said to be a contingent if it is sometimes true and sometimes false, depending on the truth value assignments of the involved simple propositions.103

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Illustration:For all propositions p and q, [(p q) q] p is a contingent.pTT

qqpTTFFTFFTTFTFTFTFTFTTFFTTFT

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Exercise 1.2 Truth Table AnalysisDetermine if each proposition is a tautology, a contradiction or a contingent.Support your answer with a truth table.1. [(p q) q] p 2. (p q) (q p)3. [(p q) (q r)] (p r) 105