ISBN10: 0534399002
ISBN13: 9780534399009
Edition/Copyright: 6TH 06

Publisher: Brooks/Cole Publishing Co.
Cover: Hardback
Year Published: 2006
Weight: 1.7lbs.

Transition to Advanced Mathematics

by Doug Smith, Maurice Eggen and Richard St. Andre


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Summary

Table of Contents

TRANSITION TO ADVANCED MATHEMATICS bridges the gap between calculus and advanced math in at least three ways. First, it guides students to think precisely and to express themselves mathematically--to analyze a situation, extract pertinent facts, and draw appropriate conclusions. Second, it provides a firm foundation of the basic concepts and methods needed for continued work. Finally, it provides introductions to concepts of modern algebra and analysis in sufficient depth to capture some of their spirit and characteristics. The text will improve the student's ability to think and write in a mature mathematical fashion and provide a solid understanding of the material most useful for advanced courses.

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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Transition to Advanced Mathematics

by Doug Smith, Maurice Eggen and Richard St. Andre


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