Contents 1 Introduction 1.1 Rules of logic 1.2 Taxonomy of Proofs 1.3 Bibliography for Chapter 1 2 Set Theory 2.1 Set Operations 2.1.1 Algebraic properties of set operations 2.2 Cartesian Products 2.3 Relations 2.3. 1 Equivalence relations 2.3.2 Order relations 2.4 Correspondences and Functions 2.4.1 Restrictions and extension 2.4.2 Composition of functions 2.4.3 Injections and inverses 2.4.4 Surjections and bijections 2.5 Finite and Infinite Sets 2.6 Algebras of Sets 2.7 Bibliography for Chapter 2 2.8 End of Chapter Problems 3 The Space of Real Numbers 3.1 The Field Axioms 46 3.2 The Order Axioms 3.3 The Completeness Axiom 3.4 Open and Closed Sets 3.5 Borel setsContents 1 Introduction 13 1.1 Rules of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2 Taxonomy of Proofs . . . . . . . . . . . . . . . . . . . . . . . 17 1.3 Bibliography for Chapter 1 . . . . . . . . . . . . . . . . . . . . 19 2 Set Theory 21 2.1 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.1 Algebraic properties of set operations . . . . . . . . . . 24 2.2 Cartesian Products . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1 Equivalence relations . . . . . . . . . . . . . . . . . . . 25 2.3.2 Order relations . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Correspondences and Functions . . . . . . . . . . . . . . . . . 30 2.4.1 Restrictions and extensions . . . . . . . . . . . . . . . 32 2.4.2 Composition of functions . . . . . . . . . . . . . . . . . 32 2.4.3 Injections and inverses . . . . . . . . . . . . . . . . . . 33 2.4.4 Surjections and bijections . . . . . . . . . . . . . . . . 33 2.5 Finite and Infinite Sets . . . . . . . . . . . . . . . . . . . . . . 34 2.6 Algebras of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.7 Bibliography for Chapter 2 . . . . . . . . . . . . . . . . . . . . 43 2.8 End of Chapter Problems. . . . . . . . . . . . . . . . . . . . . 44 3 The Space of Real Numbers 45 3.1 The Field Axioms . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 The Order Axioms . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 The Completeness Axiom . . . . . . . . . . . . . . . . . . . . 50 3.4 Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . 53 3.5 Borel Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3