Author(s): Guido Osimo
Edition: 2
Year: 2023
Language: English
Pages: 180
Contents
Introduction
Part I - Relations, functions, operations
Chapter 1 Relations
1.1 Prerequisites
1.2 Cartesian product
1.3 Relations
1.4 Equivalence relations
1.5 Order relations
1.6 Exercises and suggestions for further study
Chapter 2 Functions
2.1 Functions
2.2 Surjective, injective, bijective functions
2.3 Exercises
Chapter 3 Operations
3.1 Internal operations, fields
3.2 External operations, vector spaces
3.3 Suggestions for further study
Part II Real functions of one real variable
Chapter 4 Definitions, first examples
4.1 Prerequisites
4.2 Real functions of one real variable
4.3 First examples
4.4 Exercises
Chapter 5 Composite function, inverse function
5.1 Composite function
5.2 Invertible function, inverse function
5.3 Exercises
Chapter 6 Fundamental examples
6.1 Power functions
6.2 Exponential functions
6.3 Logarithmic functions
6.4 Trigonometric functions
6.5 Final remarks
6.6 Exercises
Chapter 7 More examples
7.1 Piecewise defined functions
7.2 Transformations of functions
7.3 Positive part and negative part
7.4 Graphical solution of equations
7.5 Exercises
Chapter 8 Some properties
8.1 Bounded functions
8.2 Monotonic functions
8.3 Global maxima and global minima
8.4 Exercises
Part III Theorems and proofs
Chapter 9 Mathematical knowledge
9.1 Introduction
9.2 Theorems and axioms
9.3 Definitions and primitive notions
9.4 An axiomatic-deductive system
9.5 Suggestions for further study
Chapter 10 First examples of proofs
10.1 The parts of a theorem
10.2 Example 1
10.3 Example 2
10.4 Example 3
10.5 Example 4
10.6 Example 5
10.7 Example 6
10.8 Conjectures
10.9 Exercises
Chapter 11 Implications and equivalences
11.1 Implications
11.2 Converse of an implication
11.3 Equivalences
11.4 Example 1
11.5 Example 2
11.6 Exercises
Chapter 12 Proofs by contradiction
12.1 How to negate a property
12.2 Proofs by contradiction
12.3 Example 1
12.4 Example 2
12.5 Example 3
12.6 Example 4
12.7 Exercises
Chapter 13 Proofs by contrapositive
13.1 Contrapositive of an implication
13.2 Proofs by contrapositive
13.3 Example 1
13.4 Example 2
13.5 Exercises
Chapter 14 Proofs by induction
14.1 The principle of induction
14.2 Example 1
14.3 Example 2
14.4 Example 3
14.5 Bernoulli inequality
14.6 Elements of the power set
14.7 A fake proof by induction
14.8 Peano’s axioms for natural numbers
14.9 Exercises
Part IV Combinatorics
Chapter 15 Permutations, combinations
15.1 Permutations
15.2 Combinations
15.3 Exercises
Chapter 16 The binomial theorem
16.1 The statement and two proofs
16.2 Exercises
Solutions of exercises
