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A Transition to Advanced Mathematics


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Table of Contents

1. LOGIC AND PROOFS. Propositions and Connectives. Conditionals and Biconditionals. Quantifiers. Basic Proof Methods I. Basic Proof Methods II. Proofs Involving Quantifiers. Additional Examples of Proofs (optional). 2. SET THEORY. Basic Notions of Set Theory. Set Operations. Extended Set Operations and Indexed Families of Sets. Induction. Equivalent Forms of Induction. Principles of Counting (optional). 3. RELATIONS. Cartesian Products and Relations. Equivalence Relations. Partitions. Ordering Relations (optional). Graphs of Relations (optional). 4. FUNCTIONS. Functions as Relations. Constructions of Functions. Functions That Are Onto; One-to-One Functions. Induced Set Functions. Sequences as Functions (optional). 5. CARDINALITY. Equivalent Sets; Finite Sets. Infinite Sets. Countable Sets. The Ordering of Cardinal Numbers. Comparability of Cardinal Numbers and the Axiom of Choice (optional). 6. CONCEPTS OF ALGEBRA: GROUPS (optional). Algebraic Structures. Groups. Examples of Groups. Subgroups. 7. CONCEPTS OF ANALYSIS: COMPLETENESS OF THE REAL NUMBERS (optional). Ordered Field Properties of the Real Numbers. The Heine-Borel Theorem. The Bolzano-Weierstrass Theorem. The Bounded Monotone Sequence Theorem. Equivalents of Completeness. Answers to Exercises. Index. List of Symbols.

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