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Preface Since more than 30 years, I taught several courses of basic mathematics for beginners at the university. This concerns students of engineering sciences, students of natural sciences and students of economical sciences. In particular, since almost 20 years I taught a course ‘Mathematics of Economics and Management’ at the Otto-von-Guericke University Magdeburg. The latter class included all relevant subjects from calculus and algebra. Roughly 10 years ago, I jointly with my co-author Yuri Sotskov from Minsk wrote a book ‘Mathematics of Economics and Business’ following exactly the structure of this lecture. This book appeared 2006 at Routledge, and I used it as the first item on my reading list for this class. However, this book explicitly includes only some refreshments from school in a short form in Chapter 4, namely how to work with real numbers. The foundations of calculus discussed in this book are, of course, also already a subject of school education in the upper classes so that there is a larger overlapping with mathematical subjects from secondary school education. I noticed that at the beginning of their study, the majority of students has some partial knowledge about the basic mathematical subjects from school, but not at the required extent. It seems so that this tendency is even increasing currently. In any case, one can observe that beginners of a university study enormously range in their mathematical skills and aptitudes. I know that for many beginners at the university, mathematical subjects appear to be rather difficult. However, if these gaps are not filled at the beginning of the study, this will definitively cause subsequent difficulties in other courses. Without any doubt, nowadays a solid mathematical knowledge is the base for most (almost all) study courses. So, I felt that there is a need to present some necessary foundations from the mathematical education at school in more detail and also some additional supplementary material, where I wish that university beginners are familiar with. When writing this booklet, I also used the experience collected in several classes of extra-occupational study courses, among them also a bridge course, which refreshes the main subjects from mathematics in school. Typically, the latter students have even more difficulties with mathematical subjects because their school education finished already some years ago. Summarizing, I found that there is a need to write such a booklet from my personal point of view, using the experience collected over the past decades. The goal was roughly not to exceed 250 pages. Sure, the content of the mathematical education in secondary school varies from country to country a bit. So, I tried to cover a broad range of subjects which might be useful for a university study from an overall point of view. The booklet consists of 12 chapters. Chapters 1 - 2 discuss some basics: some mathematical foundations as well as real numbers and arithmetic operations. Chapters 3 - 5 deal with equations and inequalities. Chapter 6 surveys some basics from analytic geometry in the plane. This is nowadays not so intensively taught as at the time 8when I attended school, but nevertheless it addresses some useful subjects. Chapters CONTENTS 7 - 10 treat classical subjects from calculus. Chapter 11 presents some aspects of vectors. Chapter 12 discusses some foundations from combinatorics, probability theory and statistics. 7 possible. So, it is not necessary to read all I tried to write the chapters as independent as chapters beginning from the first one. Instead, the student can go immediately to a particular subject. Sure, the chapters are not completely independent since in mathematics, there are often specific relationships between different subjects. Nevertheless, there is no necessity to study the chapters systematically one by one in the given sequence for the understanding of the book. Moreover, since it is an repetition and summary of elementary material of mathematics, I avoided the formal use of theorems and definitions. Instead of, the major notions are shaded in grey and in addition, important formulas and properties are given in boxes. Download free eBooks at bookboon.com Each chapter gives the learning objectives at the beginning. Moreover, every chapter finishes with a number of exercises. The solutions to the7 exercises (i.e., the concrete results) are given on my homepage so that the reader can verify whether to be able or not to solve typical problems
8
CONTENTS
A REFRESHER COURSE IN MATHEMATICS
Preface
discusses some foundations from combinatorics, probability theory and statistics. I tried to write the chapters as independent as possible. So, it is not necessary to read all chapters beginning from the first one. Instead, the student can go immediately to a particular subject. Sure, the chapters are not completely independent since in mathematics, there are often specific relationships between different subjects. Nevertheless, there is no necessity to study the chapters systematically one by one in the given sequence for the understanding of the book. Moreover, since it is an repetition and summary of elementary material of mathematics, I avoided the formal use of theorems and definitions. Instead of, the major notions are shaded in grey and in addition, important formulas and properties are given in boxes. Each chapter gives the learning objectives at the beginning. Moreover, every chapter finishes with a number of exercises. The solutions to the exercises (i.e., the concrete results) are given on my homepage so that the reader can verify whether to be able or not to solve typical problems from a particular topic. They can be downloaded as a pdf file under: http://www.math.uni-magdeburg.de/∼werner/solutions-refresher-course.pdf The author is grateful to many people for suggestions and comments. In particular, I would like to thank Dr. Michael H¨ oding and my Ph.D. student Ms. Julia Lange from the Institute of Mathematical Optimization of the Faculty of Mathematics at the Otto-von-Guericke-University Magdeburg for their many useful hints during the preparation of this booklet and the support in the preparation of the figures, respectively. I would also like to use this opportunity to thank both for their long-term support in the teaching process at the Otto-von-Guericke-University Magdeburg. I hope that this small booklet will help the students to overcome their initial difficulties when studying the required mathematical foundations at the university. Typically, the first term at the university is the hardest one due to many changes compared with secondary school. I want to finish this preface with a hint: Long time ago, I was told that learning mathematics is somehow like learning swimming. Nobody learns it by looking how other people do it! One does not learn mathematics by exclusively listening to the lecturer or tutor and copying notes from the blackboard or slides. Only own practice contributes to a significant progress. The time necessary for getting a sufficient progress varies for the individual students significantly, so everybody has to find this out by solving a sufficient number of exercises. So, students in their first year at the university should not be afraid of mathematics but should take into account that some (or even a lot of) time is needed to get a sufficiently wide experience in applying mathematical tools. The author is also grateful for all hints that improve further the content and the presentation of this edition. All suggestions should be addressed preferably to the email address given below. It is my pleasure to thank the publisher Bookboon for the delightful cooperation during the preparation of this booklet. Frank Werner Otto-von-Guericke-University Magdeburg Faculty of Mathematics PSF 4120 39016 Magdeburg / Germany Internet: http://www.math.uni-magdeburg.de/∼werner email: [email protected]
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Magdeburg, March 2016
A REFRESHER COURSE IN MATHEMATICS
Some Mathematical Foundations
Chapter 1
Some Mathematical Foundations This chapter intends to refresh some basic mathematical foundations which are necessary to understand the elementary mathematics in first- or second-year classes on mathematically oriented subjects at universities. We deal with • sets and operations on sets, • the use of the sum and product notations and • mathematical proofs by induction. Proofs by induction can be used e.g. for verifying certain sum and product formulas as well as specific inequalities. They can also be used for particular combinatorial and geometrical problems.
1.1
Sets
In this section, we introduce the basic notion of a set and discuss operations on sets. A set is a fundamental notion of mathematics, and so there is no definition of a set by other basic mathematical notions for a simplification. A set may be considered as a collection of distinct objects which are called the elements of the set. For each object, it can be uniquely decided whether it is an element of the set or not. We write: a ∈ A: b∈ / A:
a is an element of the set A; b is not an element of the set A.
A set can be given either by enumeration or by description. In the first case, we give explicitly the elements of the set, e.g. A = {1, 3, 5, 7, 9, 11, 13, 15}, i.e., A is the set that contains the eight elements 1, 3, 5, 7, 9, 11, 13, 15. Alternatively, we can describe a set in the form B = {b | b has property P }.
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A REFRESHER COURSE IN MATHEMATICS
10
Some Mathematical Foundations
CHAPTER 1. SOME MATHEMATICAL FOUNDATIONS
The set B is the set of all elements b which have the property P . In this way, we can describe the above set A e.g. as follows: A = {a | a is an odd integer and 1 ≤ a ≤ 15}. For special subsets of real numbers, one often uses also the interval notation. In particular, we have [a, b] = {x ∈ R | a ≤ x ≤ b}; (a, b) = {x ∈ R | a < x < b}; [a, b) = {x ∈ R | a ≤ x < b};
(a, b] = {x ∈ R | a < x ≤ b},
where R is the set of all real numbers. The interval [a, b] is called a closed interval whereas (a, b) is called an open interval. Accordingly, the intervals (a, b] and [a, b) are called half-open (leftopen and right-open, respectively) intervals. The set R of real numbers can also be described by the interval (−∞, ∞). Moreover, we use the following abbreviations: R≥a = [a, ∞),
360° thinking
R>a = (a, ∞),
R≤a = (−∞, a]
Thus, R≥0 denotes the set of all non-negative real numbers.
Autonomous University of Baja California, Mexicali, Mexico. Download free eBooks at bookboon.com. Page 3 of 284. a-refresher-course-in-mathematics.pdf.
