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Tables of Contents for Mathematical Proofs
Chapter/Section Title
Page #
Page Count
Communicating Mathematics
01
12
Learning Mathematics
01
2
What Others Have Said About Writing
03
1
Mathematical Writing
04
1
Using Symbols
05
2
Writing Mathematical Expressions
07
1
Common Words and Phrases in Mathematics
08
3
Some Closing Comments About Writing
11
2
Sets
13
16
Describing a Set
13
2
Special Sets
15
1
Subsets
16
2
Set Operations
18
3
Indexed Collections of Sets
21
2
Partitions of Sets
23
1
Cartesian Products of Sets
24
5
Exercises for Chapter 1
24
5
Logic
29
22
Statements
29
2
The Negation of a Statement
31
1
The Disjunction and Conjunction of Statements
32
1
The Implication
33
2
More On Implications
35
1
The Biconditional
36
2
Tautologies and Contradictions
38
1
Logical Equivalence
39
2
Some Fundamental Properties of Logical Equivalence
41
1
Characterizations of Statements
42
2
Quantified Statements and Their Negatiors
44
7
Exercises for Chapter 2
46
5
Direct Proof and Proof By Contrapositive
51
16
Trivial and Vacuous Proofs
51
2
Direct Proofs
53
3
Proof by Contrapositive
56
4
Proof by Cases
60
3
Proof Evaluations
63
4
Exercises for Chapter 3
64
3
More on Direct Proof and Proof by Contrapositive
67
16
Proofs Involving Divisibility of Integers
67
3
Proofs Involving Congruence of Integers
70
3
Proofs Involving Real Numbers
73
1
Proofs Involving Sets
74
3
Fundamental Properties of Set Operations
77
2
Proofs Involving Cartesian Products of Sets
79
4
Exercises for Chapter 4
80
3
Proof by Contradiction
83
10
Proof by Contradiction
83
1
Examples of Proof by Contradiction
84
1
The Three Prisoners Problem
85
2
Other Examples of Proof by Contradiction
87
1
The Irrationality of √2
87
1
A Review of the Three Proof Techniques
88
5
Exercises for Chapter S
90
3
Prove or Disprove
93
20
Conjectures in Mathematics
93
3
A Review of Quantifiers
96
2
Existence Proofs
98
2
A Review of Negations of Quantified Statements
100
1
Counterexamples
101
2
Disproving Statements
103
2
Testing Statements
105
2
A Quiz of ``Prove or Disprove'' Problems
107
6
Exercises for Chapter 6
108
5
Equivalence Relations
113
22
Relations
113
1
Reflexive, Symmetric, and Transitive Relations
114
2
Equivalence Relations
116
3
Properties of Equivalence Classes
119
4
Congruence Modulo n
123
4
The Integers Modulo n
127
8
Exercises for Chapter 7
130
5
Functions
135
18
The Definition of Function
135
3
The Set of All Functions From A to B
138
1
One-to-one and Onto Functions
138
2
Bijective Functions
140
3
Composition of Functions
143
3
Inverse Functions
146
3
Permutations
149
4
Exercises for Chapter 8
150
3
Mathematical Induction
153
22
The Well-Ordering Principle
153
2
The Principle of Mathematical Induction
155
3
Mathematical Induction and Sums of Numbers
158
4
Mathematical Induction and Inequalities
162
1
Mathematical Induction and Divisibility
163
2
Other Examples of Induction Proofs
165
1
Proof By Minimum Counterexample
166
2
The Strong Form of Induction
168
7
Exercises for Chapter 9
171
4
Cardinalities of Sets
175
22
Numerically Equivalent Sets
176
1
Denumerable Sets
177
6
Uncountable Sets
183
5
Comparing Cardinalities of Sets
188
3
The Schroder-Bernstein Theorem
191
6
Exercises for Chapter 10
194
3
Proofs in Number Theory
197
18
Divisibility Properties of Integers
197
1
The Division Algorithm
198
4
Greatest Common Divisors
202
2
The Euclidean Algorithm
204
2
Relatively Prime Integers
206
2
The Fundamental Theorem of Arithmetic
208
2
Concepts Involving Sums of Divisors
210
5
Exercises for Chapter 11
211
4
Proofs in Calculus
215
28
Limits of Sequences
215
5
Infinite Series
220
4
Limits of Functions
224
6
Fundamental Properties of Limits of Functions
230
5
Continuity
235
2
Differentiability
237
6
Exercises for Chapter 12
239
4
Proofs in Group Theory
243
26
Binary Operations
243
4
Groups
247
5
Permutation Groups
252
3
Fundamental Properties of Groups
255
2
Subgroups
257
3
Isomorphic Groups
260
9
Exercises for Chapter 13
263
6
Answers and Hints to Selected Odd-Numbered Exercises
269
12
References
281
2
Index of Symbols
283
2
Index of Mathematical Terms
285