Contents
Chapter 0: Reasoning
§ 1.1 Inductive and Deductive Reasoning ...... ... 1
Exercises ... 6
Chapter 1: An Introduction to
Logic
§ 1.1 Statements, negations conjunctions, disjunctions and truth tables ... 9
Exercises ... 15
§ 1.2 Conditional and biconditional statements ... 16
Exercises .. 20
§ 1.3 The Laws of logic . 23
Exercises . 30
§ 1.4 Arguments, argument forms, proofs, and counterexamples . .. 31
Exercises . 42
§ 1.5 More on logic proofs .... . 43
Exercises . 46
§ 1.6 More on fallacies .. .... . 47
Chapter 2: An Introduction to
the Basic Concepts of Sets, Syllogistic Logic,
and Quantification
§ 2.1 Basic notation and concepts for sets .. .. 51
Exercises .. 58
§ 2.2 Venn diagrammes and other illustrations for sets .. . 61
Exercises .. . 72
§ 2.3 An introduction to syllogistic logic and basic quantification.. .. .. 74
Exercises .. . 82
§ 2.4 More on syllogistic logic and two place quantification .. .. .. . 85
Exercises 90
§ 2.5 Logic and Deduction .. .. .. ..93
Exercises 96
§ 2.6 A Treatise on Deductive Logic, Sets, and Mathematics .. .. 98
Chapter 3: An Introduction to
Axioms and Mathematical Systems,
Arithmetic, The
Peano Axioms, and Mathematical Induction
§ 3.1 Basic rational for axiom systems and an introduction to
mathematical systems .103
Exercises ..110
§ 3.2 Some fundamental axiom systems .. .. .....113
§ 3.3 A bit of formal Natural arithmetic .. . . .. ... 120
Exercises 122
§ 3.4 Another type of arithmetic . . .. ... 123
Exercises 126
§ 3.5 Modular arithmetic. .. ... .. 127
Exercises ....130
§ 3.5 The Peano axioms and mathematical induction ... .. 131
Exercises ....137
§ 3.7 On the foundation of pure, applied, and computational mathematics ... 138
Chapter 4: Combinatorics
§ 4.1 The axioms of counting theory ... .140
Exercises ..149
§ 4.2 Elementary counting procedures ... ... .. .. .....151
Exercises ..163
§ 4.3 Generalised Counting Principle and Permutations ... ..... ...166
Exercises 175
§ 4.4 Combinations ... .. . .. .....179
Exercises 189
§ 4.5 Ordered Partitions and More on Permutations and Combinations ..... .....191
Exercises ..197
§ 4.6 Elementary Binomial Expansions, Pascals Triangle,
Multinomial Expansions and More . ...199
Exercises 211
Chapter 5: An Introduction to
the Axioms of Probability,
Elementary
Probability Theory, and Stochastic
Processes
§ 5.1 The Axioms of Probability .. ... .217
Exercises ..227
§ 5.2 Elementary probability theory for finite sample spaces ... .. .. .....231
Exercises ..243
§ 5.3 Elementary probability theory for infinite sample spaces ... .. ...246
Exercises 261
§ 5.4 Probability measure ... ... .. . .. .....263
Exercises 269
§ 5.5 Conditional probability---------------------------------------... .. .. .....271
Exercises ..283
§ 5.6 Independent events ... ... .. ...286
Exercises 295
§ 5.7 Finite stochastic processes ... .. . .. .....298
Exercises 305
Chapter 6: More Results from
Probability Theory and an Introduction to Mathematical Statistics.