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Tables of Contents for A Transition to Advanced Mathematics
Chapter/Section Title
Page #
Page Count
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