Mathematical Thinking: Exercises for the Mind

271 Pages·2020·10.354 MB·English

by  Stephen Fratini

Mathematical Thinking

1,1,2,3,5,8,13,21,?
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(cid:3041) (cid:3041)(cid:2879)(cid:2869) (cid:3041)(cid:2879)(cid:2870)

by Stephen Fratini
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Table of Contents
List of Figures ....................................................................................................................................... 10
List of Tables ........................................................................................................................................ 14
Preface ................................................................................................................................................. 16
Acknowledgments ............................................................................................................................... 17
1 Introduction.................................................................................................................................. 18
1.1 Purpose ................................................................................................................................. 18
1.2 Intended Audience ............................................................................................................... 18
1.3 Prerequisites ......................................................................................................................... 18
1.4 Terminology .......................................................................................................................... 18
1.5 Outline .................................................................................................................................. 18
2 What is Mathematics? ................................................................................................................. 21
3 Methods of Reasoning ................................................................................................................. 22
3.1 Deductive Reasoning ............................................................................................................ 22
3.2 Inductive Reasoning ............................................................................................................. 22
3.3 Comparison ........................................................................................................................... 23
4 Propositional Logic ....................................................................................................................... 24
4.1 Overview ............................................................................................................................... 24
4.1.1 Definition ...................................................................................................................... 24
4.1.2 Motivation ..................................................................................................................... 24
4.2 Basic Logical Operations and Definitions ............................................................................. 24
4.3 Truth Tables .......................................................................................................................... 25
4.4 Derived Operations............................................................................................................... 26
4.5 Conditional and Biconditional Propositions ......................................................................... 27
4.6 Redundancy .......................................................................................................................... 28
4.7 Logical Laws .......................................................................................................................... 29
4.8 Determining a Statement that Satisfies a Truth Table ......................................................... 30
4.9 Arguments ............................................................................................................................ 32
4.9.1 Definitions ..................................................................................................................... 32
4.9.2 Conversion from Prose to Logical Arguments .............................................................. 33
4.9.2.1 Modus Ponens Example ............................................................................................ 33
4.9.2.2 Conjunction and Modus Ponens Example ................................................................ 33
4.9.2.3 College Entrance Example ......................................................................................... 34

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4.9.3 Proofs ............................................................................................................................ 34
4.9.4 Equivalence Rules ......................................................................................................... 35
4.10 Exercises ............................................................................................................................... 36
5 First-Order (or Predicate) Logic .................................................................................................... 38
5.1 Overview ............................................................................................................................... 38
5.2 Provable Identities ................................................................................................................ 38
5.3 Scope of a Quantifier ............................................................................................................ 39
5.4 Arguments ............................................................................................................................ 40
5.4.1 Single Variable Example ................................................................................................ 40
5.4.2 Multi-variable Example ................................................................................................. 41
5.5 Paradoxes ............................................................................................................................. 42
5.5.1 Berry’s Paradox ............................................................................................................. 42
5.5.2 The Liar’s Paradox ......................................................................................................... 43
5.5.3 Drinking Paradox ........................................................................................................... 43
5.5.4 The Unexpected Hanging .............................................................................................. 44
5.6 Exercises ............................................................................................................................... 45
6 Sets ............................................................................................................................................... 47
6.1 Overview ............................................................................................................................... 47
6.2 Terminology .......................................................................................................................... 47
6.3 Venn Diagrams ...................................................................................................................... 49
6.4 Theorems .............................................................................................................................. 49
6.5 Equivalence Relationships and Partitions ............................................................................ 53
6.6 Paradoxes and Interesting Facts .......................................................................................... 56
6.6.1 Hilbert's Paradox of the Grand Hotel ........................................................................... 56
6.6.2 Cantor’s Diagonalization Argument .............................................................................. 58
6.6.3 The Cantor Set............................................................................................................... 60
6.6.4 Russell's Paradox ........................................................................................................... 62
6.6.5 Ross–Littlewood Paradox .............................................................................................. 65
6.6.6 Zeno’s Paradox of Motion ............................................................................................. 66
6.7 Exercises ............................................................................................................................... 67
7 Boolean Algebra ........................................................................................................................... 68
7.1 Definitions ............................................................................................................................. 68
7.2 Theorems .............................................................................................................................. 68

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7.3 Examples ............................................................................................................................... 72
7.4 Switching Circuits .................................................................................................................. 72
7.5 Exercises ............................................................................................................................... 76
8 Functions ...................................................................................................................................... 77
8.1 Terminology and Examples .................................................................................................. 77
8.2 Composition of Functions ..................................................................................................... 79
8.3 Visual Representations ......................................................................................................... 80
8.4 Absolute Value ...................................................................................................................... 81
8.5 Polynomials ........................................................................................................................... 82
8.6 Exponential and Logarithmic Functions ............................................................................... 84
8.7 Transforming the Graph of a Function ................................................................................. 86
8.7.1 Up and Down ................................................................................................................ 86
8.7.2 Left and Right ................................................................................................................ 87
8.7.3 Reflections ..................................................................................................................... 88
8.7.4 Stretching a Graph ........................................................................................................ 89
8.7.5 Compressing a Graph .................................................................................................... 90
8.7.6 Summary of Transformations ....................................................................................... 91
8.8 Exercises ............................................................................................................................... 92
9 Number Theory ............................................................................................................................ 93
9.1 Background ........................................................................................................................... 93
9.1.1 Well-ordering Principle ................................................................................................. 93
9.1.2 Principle of Finite Induction .......................................................................................... 93
9.1.3 Binomial Theorem ......................................................................................................... 95
9.2 Divisibility .............................................................................................................................. 97
9.2.1 Greatest Common Divisor (GCD) .................................................................................. 97
9.2.2 Euclid’s Algorithm ......................................................................................................... 99
9.2.3 Least Common Multiple (LCM) ................................................................................... 100
9.2.4 Divisibility Tests ........................................................................................................... 101
9.3 Diophantine Equations ....................................................................................................... 102
9.4 Prime Numbers ................................................................................................................... 104
9.5 Exercises ............................................................................................................................. 106
10 Combinatorics......................................................................................................................... 108
10.1 Overview ............................................................................................................................. 108

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10.2 Fundamentals ..................................................................................................................... 108
10.3 The Labeling Principle ......................................................................................................... 111
10.3.1 Distinguishable Objects............................................................................................... 111
10.3.2 Indistinguishable Objects ............................................................................................ 111
10.4 Problems Involving the Product Rule and Labeling Principle ............................................. 113
10.5 Inclusion-Exclusion Principle .............................................................................................. 114
10.5.1 Divisibility .................................................................................................................... 116
10.5.2 Cryptography via Letter Substitutions ........................................................................ 117
10.5.3 Hat-check Problem ...................................................................................................... 118
10.5.4 Surjective Mapping from One Finite Set to Another .................................................. 118
10.6 Exercises ............................................................................................................................. 119
11 Calculus ................................................................................................................................... 121
11.1 Limits ................................................................................................................................... 121
11.1.1 Example 1: f(x) = x/(x+10) ........................................................................................... 121
11.1.2 Example 2: Missing Point in Straight Line ................................................................... 122
11.1.3 Example 3: Continuous Interest and Euler’s Number ................................................ 123
11.2 Differential calculus ............................................................................................................ 124
11.3 Integral calculus .................................................................................................................. 126
11.3.1 Example: Integral of the Square Root of x .................................................................. 127
11.3.2 Example: Area Between Two Curves .......................................................................... 128
11.4 Exercises ............................................................................................................................. 129
12 Probability .............................................................................................................................. 131
12.1 Definitions and Axioms ....................................................................................................... 131
12.2 Approaches for Computing Probabilities ........................................................................... 132
12.3 Alternate Terminology for Likelihoods ............................................................................... 133
12.3.1 Fractional Odds ........................................................................................................... 133
12.3.2 Decimal Odds .............................................................................................................. 134
12.3.3 Moneyline Odds .......................................................................................................... 135
12.4 Basic Theorems ................................................................................................................... 136
12.5 Independent Events ............................................................................................................ 138
12.5.1 Example: Pairwise but Not Mutually Independent .................................................... 138
12.5.2 Example: Independent Card Selections ...................................................................... 139
12.5.3 Example: Independent Basketball Free-Throw .......................................................... 139

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12.6 Conditional Probability ....................................................................................................... 140
12.6.1 Example: School Subject Preference .......................................................................... 141
12.6.2 Example: Rolling an Octahedron Die .......................................................................... 142
12.6.3 Example: Life Expectancy ............................................................................................ 142
12.6.4 Pólya’s Urn Model ....................................................................................................... 144
12.7 Law of Total Probability ...................................................................................................... 144
12.8 Bayes’ Theorem .................................................................................................................. 145
12.8.1 Pólya’s Urn Model ....................................................................................................... 146
12.8.2 Colored Marbles in Three Containers ......................................................................... 147
12.8.3 Example: Sensitivity and Specificity ............................................................................ 147
12.9 Random Variables ............................................................................................................... 148
12.9.1 Overview ..................................................................................................................... 148
12.9.2 Examples ..................................................................................................................... 149
12.9.2.1 Example 1: Rolling Two Dice ............................................................................... 149
12.9.2.2 Example 2: Poker Hands ...................................................................................... 150
12.9.2.3 Example 3: Continuous Random Variable concerning Temperature ................. 151
12.9.3 Discrete Random Variables ......................................................................................... 152
12.9.3.1 Definitions and Some Examples .......................................................................... 152
12.9.3.2 Binomial Distribution .......................................................................................... 155
12.9.3.3 Geometric Distribution ........................................................................................ 156
12.9.3.4 Poisson Distribution ............................................................................................ 157
12.9.4 Continuous Random Variables.................................................................................... 159
12.9.4.1 Definitions and Some Examples .......................................................................... 159
12.9.4.2 Normal Distribution ............................................................................................. 161
12.9.4.3 Exponential Distribution and the Memoryless Property .................................... 163
12.9.5 Standardized Random Variables ................................................................................. 165
12.9.6 Z-Score ........................................................................................................................ 166
12.9.7 Law of Large Numbers ................................................................................................ 167
12.9.8 Central Limit Theorem ................................................................................................ 169
12.10 Exercises ............................................................................................................................. 171
13 Statistics .................................................................................................................................. 174
13.1 Overview ............................................................................................................................. 174
13.2 Descriptive Statistics ........................................................................................................... 174

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13.2.1 Central Tendency ........................................................................................................ 174
13.2.2 Dispersion ................................................................................................................... 176
13.2.3 Shape of a Probability Distribution Function .............................................................. 177
13.3 Inferential Statistics ............................................................................................................ 178
13.3.1 Point Estimators .......................................................................................................... 178
13.3.2 Confidence Intervals ................................................................................................... 179
13.3.2.1 Using Normal Distribution to Compute Confidence Interval for the Mean ....... 180
13.3.2.2 Using Student’s T-distribution to Compute Confidence Interval for the Mean . 181
13.3.2.3 Confidence Intervals for Proportions .................................................................. 183
13.3.3 Hypothesis Testing ...................................................................................................... 184
13.3.3.1 Overview.............................................................................................................. 184
13.3.3.2 Steps in an Hypothesis Test ................................................................................. 185
13.3.3.3 P-value ................................................................................................................. 186
13.3.3.4 P-value versus Level of Significance .................................................................... 187
13.3.3.5 Example – Large Vat of Ping-Pong Balls .............................................................. 188
13.3.3.6 Example – Comparing Battery Type Lifetimes .................................................... 189
13.3.3.7 Example – Type II Errors ...................................................................................... 191
13.3.4 Relationship between Confidence Intervals and Hypothesis Testing ........................ 193
13.3.5 Regression Analysis ..................................................................................................... 194
13.3.5.1 Simple Linear Regression .................................................................................... 194
13.3.5.2 Multiple Linear Regression .................................................................................. 196
13.3.5.3 Nonlinear Regression .......................................................................................... 196
13.3.5.4 Autoregression .................................................................................................... 197
13.4 Statistical Paradoxes ........................................................................................................... 198
13.4.1 Berkson's Paradox ....................................................................................................... 198
13.4.1.1 Conditional Dependence ..................................................................................... 198
13.4.1.2 Restaurant Example ............................................................................................ 199
13.4.1.3 Stamp Example .................................................................................................... 201
13.4.2 False Positive Paradox................................................................................................. 201
13.4.3 Simpson’s Paradox ...................................................................................................... 202
13.5 Exercises ............................................................................................................................. 203
14 Graph Theory .......................................................................................................................... 204
14.1 Basic Concepts .................................................................................................................... 204

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14.2 Classification ....................................................................................................................... 206
14.2.1 Almost Irregular Graphs .............................................................................................. 206
14.2.2 Regular Graphs ............................................................................................................ 207
14.2.3 Bipartite Graphs .......................................................................................................... 210
14.2.4 Trees ............................................................................................................................ 213
14.2.5 Subgraphs .................................................................................................................... 214
14.2.6 Isomorphic Graphs ...................................................................................................... 215
14.3 Connectivity ........................................................................................................................ 215
14.3.1 Trails, Path and Cycles ................................................................................................ 215
14.3.2 Cut-vertices and Bridges ............................................................................................. 217
14.4 Minimum Spanning Trees ................................................................................................... 218
14.5 Traversing Graphs ............................................................................................................... 220
14.6 Exercises ............................................................................................................................. 222
15 Linear Algebra ......................................................................................................................... 224
15.1 Matrices and Vectors .......................................................................................................... 224
15.2 Vectors – Geometric Approach .......................................................................................... 227
15.3 Systems of Linear Equations ............................................................................................... 230
15.3.1 Geometric Considerations .......................................................................................... 230
15.3.2 Gaussian Elimination ................................................................................................... 232
15.4 Vector Spaces ..................................................................................................................... 235
15.4.1 Basic Definitions and Theorems ................................................................................. 235
15.4.2 Examples of Vector Spaces ......................................................................................... 236
15.4.2.1 Matrices ............................................................................................................... 236
15.4.2.2 Functions ............................................................................................................. 236
15.4.2.3 Infinite Sequences of Real Numbers ................................................................... 237
15.5 Linear Independence, Linear Transformations and Bases ................................................. 237
15.6 Exercises ............................................................................................................................. 241
16 Proofs ...................................................................................................................................... 243
16.1 Direct Proof ......................................................................................................................... 243
16.2 Mathematical Induction ..................................................................................................... 243
16.3 Constructive Proof .............................................................................................................. 243
16.4 Existence Proof ................................................................................................................... 244
16.5 Proof by contraposition ...................................................................................................... 244

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16.6 Proof by contradiction (Reductio ad absurdum) ................................................................ 244
16.7 Decomposition and Levels .................................................................................................. 245
16.8 Theorem, Lemma and Corollary ......................................................................................... 245
17 Algorithms .............................................................................................................................. 247
17.1 Overview ............................................................................................................................. 247
17.2 Classification ....................................................................................................................... 247
17.2.1 Recursive and Iterative Algorithms ............................................................................. 247
17.2.1.1 Fibonacci Sequence ............................................................................................. 248
17.2.1.2 Towers of Hanoi .................................................................................................. 249
17.2.2 Serial and Parallel Algorithms ..................................................................................... 251
17.3 Graphical Representation ................................................................................................... 251
17.3.1 Flow Charts ................................................................................................................. 251
17.3.2 Other Design Approaches for Algorithms ................................................................... 253
18 Universal Laws of Mathematics ............................................................................................. 254
18.1 Overview ............................................................................................................................. 254
18.2 Law of Large Numbers ........................................................................................................ 254
18.3 Central Limit Theorem ........................................................................................................ 255
18.4 Benford’s Law ..................................................................................................................... 255
18.5 Power Laws ......................................................................................................................... 256
18.5.1 Overview ..................................................................................................................... 256
18.5.2 Zipf’s Law..................................................................................................................... 257
18.5.3 Pareto Principle ........................................................................................................... 258
19 Conclusion .............................................................................................................................. 260
Acronyms and Symbols ...................................................................................................................... 261
References ......................................................................................................................................... 263
Index of Terms ................................................................................................................................... 269

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List of Figures
Figure 1. Black Box with Known Relationship between Inputs and Output ........................................ 30
Figure 2. Venn diagram showing the intersection of three sets ......................................................... 49
Figure 3. Venn diagram of A ∩ (B∪C) = (A∩B) ∪ (A∩C) ...................................................................... 52
Figure 4. First four iterations of the Cantor Set................................................................................... 60
Figure 5. Series and Parallel Circuits .................................................................................................... 73
Figure 6. Synchronized Switches ......................................................................................................... 73
Figure 7. Circuit Simplification Example .............................................................................................. 74
Figure 8. Simplified Circuit ................................................................................................................... 74
Figure 9. Bridge Circuit ......................................................................................................................... 75
Figure 10. Breaks Approach ................................................................................................................. 75
Figure 11. Series-Parallel Circuit .......................................................................................................... 76
Figure 12. Example Functions and Non-functions ............................................................................... 77
Figure 13. Injective and Surjective Functions ...................................................................................... 79
Figure 14. Bijective Function................................................................................................................ 79
Figure 15. Example of Composition of Functions ................................................................................ 80
Figure 16. Bar Chart for f(n)=3n+1 ....................................................................................................... 80
Figure 17. Graph of f(x) ........................................................................................................................ 81
Figure 18. Graph of the Absolute Value of x ....................................................................................... 82
Figure 19. Graph of a Cubic Polynomial .............................................................................................. 83
Figure 20. Graph of the Absolute Value of a Cubic Polynomial .......................................................... 84
Figure 21. Graph of Exponential Function with 𝑎 = 3 and its Inverse ................................................ 85
Figure 22. Vertical Transformation ...................................................................................................... 87
Figure 23. Horizontal Transformation ................................................................................................. 88
Figure 24. Reflections of an Exponential Function .............................................................................. 89
Figure 25. Stretching a Graph .............................................................................................................. 90
Figure 26. Compressing a Graph .......................................................................................................... 91
Figure 27. Pascal’s Triangle .................................................................................................................. 95
Figure 28. Pigeonhole Principle – Generalization #1 – Example 2 .................................................... 109
Figure 29. Pigeonhole Principle – Generalization #1 – Example 3 .................................................... 110
Figure 30. Pigeonhole Principle – Generalization #2 ......................................................................... 110
Figure 31. Inclusion-Exclusion for Two Sets ...................................................................................... 114
Figure 32. Inclusion-Exclusion for Three Sets .................................................................................... 115

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