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| General Preface Preface for Instructors PART 1 Chapter 1 Proof Techniques Part A Elements of Logic and Induction A.1 Definitions A.2 Propositions and Predicates A.3 Conjunction and Negation A.4 Implication A.5 Proofs by Induction for Natural Numbers Part B Methods of Proof B.1 Chains of Implications B.2 Proof by Contradiction B.3 Instantiation B.4 Constructive and Non-Constructive Proofs B.5 Disproofs Chapter 2 Predicate Calculus 2.1 Logical Operators 2.2 Propositional Forms 2.3 Parentheses 2.4 Standard Logical Identities 2.5 Standard Rules of Inference (*)2.6 Aristotelean Logic 2.7 Logical Equivalence (*)2.8 Commutativity and Associativity 2.9 Proving Propositional Forms 2.10 Predicate Forms and Quantification 2.11 Standard Valid Predicate Forms 2.12 Proving Predicate Forms 2.13 Disproofs Chapter 3 Set Theory 3.1 Axioms and the Primitives of Set Theory 3.2 The Basics of Set Theory (*)3.3 ZFC Set Theory 3.4 Binary Operations on Sets 3.5 Cartesian Products 3.6 Absolute Complements and DeMorgan's Laws (*)3.7 The Set of Non-Negative Integers 3.8 Inductive Definitions 3.9 Sets of Numbers Chapter 4 Elementary Number Theory 4.1 Common Multiples 4.2 The Division Algorithm 4.3 Common Divisors 4.4 Relatively Prime Pairs 4.5 Linear Diophantine Equations 4.6 The Fundamental Theorem of Arithmetic 4.7 The Euclidean Algorithm (*)4.8 Continued Fractions Chapter 5 Relations 5.1 N-ary Relations 5.2 Binary Relations and Digraphs 5.3 Properties of Relations 5.4 Set-Theoretic Operations 5.5 Inversion 5.6 Composition (*)5.7 Iterations 5.8 Posets 5.9 Linear and Well Orders (*)5.10 Bounds in Posets (*)5.11 Axiom of Choice Revisited 5.12 Equivalence Relations 5.13 Partitions (*)5.14 Closure Operations (*)5.15 Meets and Joins of Partitions (*)5.16 Lattices Chapter 6 Functions 6.1 Definitions and Examples 6.2 Composition 6.3 Restriction and Extension 6.4 Injectivity, Surjeetivity, and Bijectivity 6.5 Inverses 6.6 Images and Pre-Images …… PART 2 |
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