Table of Contents Last updated Sep 16, 2022 Save as PDF InfoPage Licensing Page ID82453 ( \newcommand{\kernel}{\mathrm{null}\,}\) Table of Contents Licensing 1: Introduction and Notation 1.1: Basic Sets 1.2: Definitions - Prime Numbers 1.3: More Scary Notation 1.4: Definitions of Elementary Number Theory 1.5: Some Algorithms of Elementary Number Theory 1.6: Rational And Irrational Numbers 1.7: Relations 2: Logic and Quantifiers 2.1: Predicates and Logical Connectives 2.2: Implication 2.3: Logical Equivalences 2.4: Two-Column Proofs 2.5: Quantified Statements 2.6: Deductive Reasoning and Argument Forms 2.7: Validity of Arguments and Common Errors 3: Proof Techniques I 3.1: Direct Proofs of Universal Statements 3.2: More Direct Proofs 3.3: Indirect Proofs- Contradiction and Contraposition 3.4: Disproofs 3.5: Even More Direct Proofs- By Cases and By Exhaustion 3.6: Proofs and Disproofs of Existential Statements 4: Sets 4.1: Basic Notions of Set Theory 4.2: Containment 4.3: Set Operations 4.4: Venn Diagrams 4.5: 4.5 Russell’s Paradox 5: Proof Techniques II - Induction 5.1: The Principle of Mathematical Induction 5.2: Formulas for Sums and Products 5.3: Divisibility Statements and Other Proofs Using PMI 5.4: The Strong Form of Mathematical Induction 6: Relations and Functions 6.1: Relations 6.2: Properties of Relations 6.3: Equivalence Relations 6.4: Ordering Relations 6.5: Functions 6.6: Special Functions 7: Proof Techniques III - Combinatorics 7.1: Counting 7.2: Parity and Counting Arguments 7.3: The Pigeonhole Principle 7.4: The Algebra of Combinations 8: Cardinality 8.1: Equivalent Sets 8.2: Examples of Set Equivalence 8.3: Cantor’s Theorem 8.4: Dominance 8.5: The Continuum Hypothesis and The Generalized Continuum Hypothesis 9: Proof Techniques IV - Magic 9.1: Morley’s Miracle 9.2: Five Steps Into the Void 9.3: Monge’s Circle Theorem References Index Glossary Detailed Licensing