# A Transition To Advanced Mathematics

• ISBN13:

• ISBN10:

## 0534399002

• Edition: 6th
• Format: Hardcover
• Publisher: 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.

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)
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)