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- Author: Douglas Smith, Maurice Eggen, Richard St. Andre
- Title: A Transition to Advanced Mathematics
- Amazon Link: https://www.amazon.com/dp/0495562025/?tag=pfamazon01-20
- Level: Undergrad
Table of Contents:
- Preface
- Preface to the Student
- 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
- Set Theory
- Basic Concepts of Set Theory
- Set Operations
- Extended Set Operations and Indexed Families of Sets
- Mathematical Induction
- Equivalent Forms of Induction
- Principles of Counting
- Relations and Partitions
- Cartesian Products and Relations
- Equivalence Relations
- Partitions
- Ordering Relations
- Graphs
- Functions
- Functions as Relations
- Constructions of Functions
- Functions That Are Onto; One-to-One Functions
- One-to-One Correspondences and Inverse Functions
- Images of Sets
- Sequences
- Cardinality
- Equivalent Sets; Finite Sets
- Infinite Sets
- Countable Sets
- The Ordering of Cardinal Numbers
- Comparability of Cardinal Numbers and the Axiom of Choice
- Concepts of Algebra
- Algebraic Structures
- Groups
- Subgroups
- Operation Preserving Maps
- Rings and Fields
- Concepts of Analysis
- Completeness of the Real Numbers
- The Heine–Borel Theorem
- The Bolzano–Weierstrass Theorem
- The Bounded Monotone Sequence Theorem
- Equivalents of Completeness
- Answers to Selected Exercises
- Index
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