A Transition To Advanced Mathematics

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

• ISBN13: 9780534399009
• ISBN10: 0534399002
• Edition: 6th
• Format: Hardcover
• Copyright: 2005-06-14
• 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

Preface	vi
Preface to the Student	x
Logic and Proofs	1	(68)
Propositions and Connectives	1	(10)
Conditionals and Biconditionals	11	(9)
Quantifiers	20	(8)
Basic Proof Methods I	28	(11)
Basic Proof Methods II	39	(8)
Proofs Involving Quantifiers	47	(9)
Additional Examples of Proofs	56	(13)
Set Theory	69	(62)
Basic Concepts of Set Theory	69	(9)
Set Operations	78	(8)
Extended Set Operations and Indexed Families of Sets	86	(10)
Induction	96	(14)
Equivalent Forms of Induction	110	(8)
Principles of Counting	118	(13)
Relations	131	(48)
Cartesian Products and Relations	131	(14)
Equivalence Relations	145	(9)
Partitions	154	(6)
Ordering Relations	160	(9)
Graphs	169	(10)
Functions	179	(42)
Functions as Relations	179	(10)
Constructions of Functions	189	(9)
Functions That Are Onto; One-to-One Functions	198	(10)
Images of Sets	208	(6)
Sequences	214	(7)
Cardinality	221	(40)
Equivalent Sets; Finite Sets	221	(9)
Infinite Sets	230	(7)
Countable Sets	237	(10)
The Ordering of Cardinal Numbers	247	(7)
Comparability of Cardinal Numbers and the Axiom of Choice	254	(7)
Concepts of Algebra	261	(38)
Algebraic Structures	261	(8)
Groups	269	(7)
Subgroups	276	(6)
Operation Preserving Maps	282	(8)
Rings and Fields	290	(9)
Concepts of Analysis	299	(32)
Ordered Field Properties of the Real Numbers	300	(7)
The Heine--Borel Theorem	307	(10)
The Bolzano--Weierstrass Theorem	317	(5)
The Bounded Monotone Sequence Theorem	322	(4)
Equivalents of Completeness	326	(5)
Answers to Selected Exercises	331	(35)
Index	366