A Transition to Advanced Mathematics,9780534382148

A Transition to Advanced Mathematics

by ; ;
ISBN13: 9780534382148
ISBN10: 0534382142
Format: Hardcover
Pub. Date: 2001-01-12
Publisher(s): Brooks Cole
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Summary

Bridge the gap between calculus and advanced math with TRANSITION TO ADVANCED MATHEMATICS! This mathematics text will improve your ability to think and write in a mature mathematical fashion and provide you with a solid understanding of the material most useful for advanced courses. With a readable, concise style, discussions of sequences and real analysis concepts found throughout the text are tied to your experience in elementary calculus in order to clarify material. Worked examples and exercises throughout the text, ranging from the routine to the challenging, reinforce the concepts.

Table of Contents

Logic and Proofs
1(64)
Propositions and Connectives
1(8)
Conditionals and Biconditionals
9(9)
Quantifiers
18(8)
Basic Proof Methods I
26(11)
Basic Proof Methods II
37(7)
Proofs Involving Quantifiers
44(9)
Additional Examples of Proofs
53(12)
Set Theory
65(58)
Basic Notions of Set Theory
65(8)
Set Operations
73(7)
Extended Set Operations and Indexed Families of Sets
80(11)
Induction
91(12)
Equivalent Forms of Induction
103(8)
Principles of Counting
111(12)
Relations
123(48)
Cartesian Products and Relations
123(15)
Equivalence Relations
138(8)
Partitions
146(5)
Ordering Relations
151(10)
Graphs of Relations
161(10)
Functions
171(36)
Functions as Relations
171(10)
Constructions of Functions
181(9)
Functions That Are Onto; One-to-One Functions
190(10)
Induced Set Functions
200(7)
Cardinality
207(38)
Equivalent Sets; Finite Sets
207(8)
Infinite Sets
215(6)
Countable Sets
221(9)
The Ordering of Cardinal Numbers
230(8)
Comparability of Cardinal Numbers and the Axiom of Choice
238(7)
Concepts of Algebra: Groups
245(40)
Algebaric Structures
245(8)
Groups
253(5)
Examples of Groups
258(5)
Subgroups
263(7)
Cosets and Lagrange's Theorem
270(4)
Quotient Groups
274(4)
Isomorphism; The Fundamental Theorem of Group Homomorphisms
278(7)
Concepts of Analysis: Completeness of the Real Numbers
285(36)
Ordered Field Properties of the Real Numbers
286(7)
The Heine--Borel Theorem
293(10)
The Bolzano--Weierstrass Theorem
303(4)
The Bounded Monotone Sequence Theorem
307(9)
Equivalents of Completeness
316(5)
Answers to Selected Exercises 321(32)
Index 353(8)
List of Symbols 361

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