An Introduction to Mathematical Proofs

Description

Cover Half Title Series Page Title Page Copyright Page Dedication Contents Preface 1. Logic 1.1 Propositions, Logical Connectives, and Truth Tables 1.2 Logical Equivalences and IF-Statements 1.3 IF, IFF, Tautologies, and Contradictions 1.4 Tautologies, Quanti ers, and Universes 1.5 Quantifier Properties and Useful Denials 1.6 Denial Practice and Uniqueness Statements 2. Proofs 2.1 Definitions, Axioms, Theorems, and Proofs 2.2 Proving Existence Statements and IF Statements 2.3 Contrapositive Proofs and IFF Proofs 2.4 Proofs by Contradiction and Proofs of OR-Statements 2.5 Proofs by Cases and Disproofs 2.6 Proving Quantified Statements 2.7 More Quantifier Properties and Proofs (Optional) Review of Logic and Proofs 3. Sets 3.1 Set Operations and Subset Proofs 3.2 Subset Proofs and Set Equality Proofs 3.3 Set Equality Proofs, Circle Proofs, and Chain Proofs 3.4 Small Sets and Power Sets 3.5 Ordered Pairs and Product Sets 3.6 General Unions and Intersections 3.7 Axiomatic Set Theory (Optional) 4. Integers 4.1 Recursive Definitions and Proofs by Induction 4.2 Induction Starting Anywhere and Backwards Induction 4.3 Strong Induction 4.4 Prime Numbers and Integer Division 4.5 Greatest Common Divisors 4.6 GCDs and Uniqueness of Prime Factorizations 4.7 Consequences of Prime Factorization (Optional) Review of Set Theory and Integers 5. Relations and Functions 5.1 Relations 5.2 Inverses, Identity, and Composition of Relations 5.3 Properties of Relations 5.4 Definition of Functions 5.5 Examples of Functions and Function Equality 5.6 Composition, Restriction, and Gluing 5.7 Direct Images and Preimages 5.8 Injective, Surjective, and Bijective Functions 5.9 Inverse Functions 6. Equivalence Relations and Partial Orders 6.1 Reflexive, Symmetric, and Transitive Relations 6.2 Equivalence Relations 6.3 Equivalence Classes 6.4 Set Partitions 6.5 Partially Ordered Sets 6.6 Equivalence Relations and Algebraic Structures (Optional) 7. Cardinality 7.1 Finite Sets 7.2 Countably Infinite Sets 7.3 Countable Sets 7.4 Uncountable Sets Review of Functions, Relations, and Cardinality 8. Real Numbers (Optional) 8.1 Axioms for R and Properties of Addition 8.2 Algebraic Properties of Real Numbers 8.3 Natural Numbers, Integers, and Rational Numbers 8.4 Ordering, Absolute Value, and Distance 8.5 Greatest Elements, Least Upper Bounds, and Completeness Suggestions for Further Reading Index