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A Transition to Advanced Mathematics

ISBN: 9780534382148 | 0534382142
Edition: 5th
Format: Paperback
Publisher: Brooks Cole
Pub. Date: 1/12/2001

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SummaryTable of Contents
Successfully addressing the frustration many students feel as they make the transition from beginning calculus to a more rigorous level of mathematics, A Transition to Advanced Mathematics provides a firm foundation in the major ideas needed for continued work in the discipline. The authors guide students to think and to express themselves mathematically--to analyze a situation, extract pertinent facts, and draw appropriate conclusions. With their proven approach, Smith, Eggen, and St. Andre introduce students to rigorous thinking about sets, r... MORE
Logic and Proofs
1(64)
Propositions and Connectives
1(8)
Conditionals and Biconditionals
... MORE9(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 Exercises321(32)
Index353(8)
List of Symbols361

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