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Preface | p. xi |
Mathematics and Mathematical Activity | p. 1 |
What Is Mathematics? | p. 1 |
Mathematical Research and Problem Solving | p. 2 |
An Example of a Mathematical Research Situation | p. 3 |
Conjectures and Theorems | p. 5 |
Methods of Reasoning | p. 5 |
Why Do We Need Proofs? | p. 7 |
Mathematical Writing | p. 8 |
Reading a Mathematics Textbook | p. 9 |
Problems | p. 11 |
Sets, Numbers, and Axioms | p. 15 |
Sets and Numbers from an Intuitive Perspective | p. 15 |
Set Equality and Set Inclusion | p. 23 |
Venn Diagrams and Set Operations | p. 31 |
Undefined Notions and Axioms of Set Theory | p. 45 |
Axioms for the Real Numbers | p. 50 |
Problems | p. 56 |
Elementary Logic | p. 69 |
Statements and Truth | p. 69 |
Truth Tables and Statement Forms | p. 80 |
Logical Equivalence | p. 86 |
Arguments and Validity | p. 91 |
Statements Involving Quantifiers | p. 98 |
Problems | p. 106 |
Planning and Writing Proofs | p. 117 |
The Proof-Writing Context | p. 117 |
Proving an If… Then Statement | p. 122 |
Proving a For All Statement | p. 129 |
The Know/Show Approach to Developing Proofs | p. 136 |
Existence and Uniqueness | p. 144 |
The Role of Definitions in Creating Proofs | p. 153 |
Proving and Expressing a Mathematical Equivalence | p. 160 |
Indirect Methods of Proof | p. 168 |
Proofs Involving Or. | p. 173 |
A Mathematical Research Situation | p. 177 |
Problems | p. 183 |
Relations and Functions | p. 199 |
Relations | p. 199 |
Equivalence Relations and Partitions | p. 207 |
Functions | p. 217 |
One-to-One Functions, Onto Functions, and Bijections | p. 225 |
Inverse Relations and Inverse Functions | p. 235 |
Problems | p. 239 |
The Natural Numbers, Induction, and Counting | p. 255 |
Axioms for the Natural Numbers | p. 255 |
Proof by Induction | p. 258 |
Recursive Definition and Strong Induction | p. 270 |
Elementary Number Theory | p. 276 |
Some Elementary Counting Methods | p. 286 |
Problems | p. 298 |
Further Mathematical Explorations | p. 311 |
Exploring Graph Theory | p. 311 |
Exploring Groups | p. 323 |
Exploring Set Cardinality | p. 337 |
Index | p. 347 |
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