A Transition to Advanced Mathematics

by Smith, Douglas; Eggen, Maurice; St. Andre, Richard

• ISBN13: 9780534382148
• ISBN10: 0534382142
• Format: Hardcover
• Copyright: 2001-01-12
• Publisher: Brooks Cole

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