Studying Mathematics

402 Pages·2018·6.403 MB·English

by  Marco Bramanti, Giancarlo Travaglini

Marco Bramanti · Giancarlo Travaglini
Studying
Mathematics
The Beauty, the Toil and the Method
Studying Mathematics
Marco Bramanti • Giancarlo Travaglini
Studying Mathematics
The Beauty, the Toil and the Method
123
MarcoBramanti GiancarloTravaglini
DipartimentodiMatematica DipartimentodiMatematicaeApplicazioni
PolitecnicodiMilano Universita`diMilano-Bicocca,EdificioU5
ViaBonardi9,Milano,Italy ViaCozzi55,Milano,Italy
Translation from the Italian language edition: Matematica. Questione di metodo by Marco Bramanti and
GiancarloTravaglini.Copyright©ZanichellieditoreS.p.A.2009.Allrightsreserved.
ISBN978-3-319-91354-4 ISBN978-3-319-91355-1 (eBook)
https://doi.org/10.1007/978-3-319-91355-1
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to our friends
Contents
Why This Book XI
Instructions for Use XV
About the Authors XVII
I The Language of Mathematics 1
1 An Introductory Test (Level A) 5
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Solutions of the Test . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Quantifying (Level A) 13
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Using the Sets (Level A) 21
3.1 The Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 “Or” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Logic and Operations Between Sets . . . . . . . . . . . . . 27
3.4 “All” and “Only” . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Numerical Sets . . . . . . . . . . . . . . . . . . . . . . . . 32
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Propositions and Properties (Level B) 37
4.1 Propositions and Properties . . . . . . . . . . . . . . . . . 37
4.2 More on Sets Defined by Properties . . . . . . . . . . . . . 41
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
VII
VIII CONTENTS
5 Proofs, Implications, and Counterexamples 45
5.1 Implications and Counterexamples
(Level A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2 Inclusions Between Sets and Implications (Level B) . . . . 51
5.3 Implications Between Propositions
(Level B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6 Negations and Indirect Proofs 61
6.1 Negation of a Proposition (Level A) . . . . . . . . . . . . . 62
6.2 Negation and Complement of a Set
(Level B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.3 Proofs by Contradiction (Level A) . . . . . . . . . . . . . . 66
6.4 More on Proofs by Contradiction: Nonconstructive Proofs
(Level B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7 Formulae and Indices (Level A) 75
7.1 To Understand a Formula . . . . . . . . . . . . . . . . . . 76
7.2 Use of Indices . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.3 Summations and Products . . . . . . . . . . . . . . . . . . 80
7.4 Functions and Their Compositions . . . . . . . . . . . . . . 85
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
8 Saturation of Indices and Syntactic Consistency of
a Formula 91
8.1 Saturation of Indices in Sums and Products . . . . . . . . 91
8.2 Syntactic Consistency . . . . . . . . . . . . . . . . . . . . . 93
8.3 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.4 Union and Intersection of Families of Sets . . . . . . . . . . 97
8.5 Operators that Saturate Continuous Variables . . . . . . . 98
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
9 Induction and Natural Numbers (Level B) 103
9.1 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
9.2 The Axioms of the Natural Numbers . . . . . . . . . . . . 109
Solutions of the Exercises of Part I 113
CONTENTS IX
II Studying a Math Book 135
10 To Read a Definition 139
10.1 Trailer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Solutions of the Trailer . . . . . . . . . . . . . . . . . . . . . . . . 142
10.2 Exercises of Level A . . . . . . . . . . . . . . . . . . . . . . 146
10.3 Exercises of Level B . . . . . . . . . . . . . . . . . . . . . . 148
10.4 Exercises of Level C . . . . . . . . . . . . . . . . . . . . . . 150
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
11 To Understand, i.e., to Know How to Apply 167
11.1 Trailer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Solutions of the Trailer . . . . . . . . . . . . . . . . . . . . . . . . 174
11.2 Exercises of Level A . . . . . . . . . . . . . . . . . . . . . . 179
11.3 Exercises of Level B . . . . . . . . . . . . . . . . . . . . . . 183
11.4 Exercises of Level C . . . . . . . . . . . . . . . . . . . . . . 186
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
12 Learning to Correct 201
12.1 Trailer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Solutions of the Trailer . . . . . . . . . . . . . . . . . . . . . . . . 205
12.2 Exercises of Level A . . . . . . . . . . . . . . . . . . . . . . 208
12.3 Exercises of Level B . . . . . . . . . . . . . . . . . . . . . . 213
12.4 Exercises of Level C . . . . . . . . . . . . . . . . . . . . . . 215
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
13 To Discuss the Ideas 233
13.1 Trailer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Solutions of the Trailer . . . . . . . . . . . . . . . . . . . . . . . . 235
13.2 Exercises of Level A . . . . . . . . . . . . . . . . . . . . . . 237
13.3 Exercises of Level B . . . . . . . . . . . . . . . . . . . . . . 240
13.4 Exercises of Level C . . . . . . . . . . . . . . . . . . . . . . 242
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
14 To Understand, i.e., to Know How to Explain 265
14.1 Trailer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
Solutions of the Trailer . . . . . . . . . . . . . . . . . . . . . . . . 270
14.2 Exercises of Level A . . . . . . . . . . . . . . . . . . . . . . 273
14.3 Exercises of Level B . . . . . . . . . . . . . . . . . . . . . . 278
14.4 Exercises of Level C . . . . . . . . . . . . . . . . . . . . . . 281
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
14.5 Want to Know More? . . . . . . . . . . . . . . . . . . . . . 310
X CONTENTS
III Pages and Ideas 333
15 Majorizations 337
What We Are Talking About . . . . . . . . . . . . . . . . . . . . 337
15.1 Absolute Values (Level A) . . . . . . . . . . . . . . . . . . 338
15.2 Majorizing Term by Term (Level A) . . . . . . . . . . . . . 342
15.3 Using the Positivity of the Square
(Level B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
16 Uniqueness Proofs (Level B) 355
What We Are Talking About . . . . . . . . . . . . . . . . . . . . 355
16.1 The Distance of a Point from a Convex Set . . . . . . . . . 355
16.2 Uniqueness of the Solution for Triangular Systems . . . . . 357
17 Functions and Set Theoretic Arguments 363
What We Are Talking About . . . . . . . . . . . . . . . . . . . . 363
17.1 Topology of a Discrete Set (Level B) . . . . . . . . . . . . 364
17.2 Harmonic Functions on a Discrete Set (Level C) . . . . . . 369
17.3 Harmonic Functions and Electrical Networks (Level C) . . 373
18 Tiles, Polyhedra, and Characterizations 377
What We Are Talking About . . . . . . . . . . . . . . . . . . . . 377
18.1 Tiling the Plane (Level B) . . . . . . . . . . . . . . . . . . 377
18.2 Wisdom of the Bees and Isoperimetric Problems (Level B) 380
18.3 Regular Polyhedra (Level C) . . . . . . . . . . . . . . . . . 382
18.4 Euler’s Theorem (Level C) . . . . . . . . . . . . . . . . . . 386
Index 393
Why This Book
Mathematics is the science that yields the best opportunity to
observe the working of the mind. Its study is the best training of
ourabilitiesasitdevelopsboththepowerandtheprecisionofour
thinking. Mathematics is valuable on account of the number and
variety of its applications. And it is equally valuable in another
respect: by cultivating it, we acquire the habit of a method of
reasoning which can be applied afterwards to the study of any
subject and can guide us in life’s great and little problems [3].
It is always true that many students encounter some difficulties during
the study of mathematics, both in school and in college. According to us this
problem is often due to two deficiencies and it is useful to distinguish them.
1. Prerequisites. On the one hand the study of mathematics is orga-
nized into a logical path which, from primary school to university, proceeds
sequentially (in a more marked way than in other disciplines). Then a possi-
ble “difficult time” in the study of mathematics, at any stage of the student’s
school life, may cause a “gap in the prerequisites” which may be hard to fill
later. Among freshmen this problem is quite common and whoever is in this
situation needs to spend some time and effort acquiring the missing argu-
ments. In order to do this one can go through her/his high school math
textbooks, or look for specific texts which offer a reasonable and sufficiently
condensed presentation of the subject.
2. Method. On the other hand the difficulties encountered by many
students while studying math courses are also (and sometimes mainly) due
to the particular method of study that math requires: one has to understand
the need of a precise language and learn how to use it when it is necessary;
he/she has to familiarize with logical tools and mathematical symbols, and
understand the justifications of results (exercises, theorems, computational
techniques), checking every line, exemplifying the arguments and learning
how to apply them in similar situations. Briefly speaking: one needs to
XI