9781449627782

A Transition to Mathematics with Proofs

by
  • ISBN13:

    9781449627782

  • ISBN10:

    1449627781

  • Edition: 1st
  • Format: Hardcover
  • Copyright: 12/30/2011
  • Publisher: Jones & Bartlett Learning

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Summary

Developed for the "transition" course for mathematics majors moving beyond the primarily procedural methods of their calculus courses toward a more abstract and conceptual environment found in more advanced courses, A Transition to Mathematical Proofs emphasizes mathematical rigor and helps students learn how to develop and write mathematical proofs. The author takes great care to develop a text that is accessible and readable for students at all levels. It addresses standard topics such as set theory, number system, logic, relations, functions, and induction in at a pace appropriate for a wide range of readers. Throughout early chapters students gradually become aware of the need for rigor, proof, and precision, and mathematical ideas are motivated through examples.

Table of Contents

Prefacep. xi
Mathematics and Mathematical Activityp. 1
What Is Mathematics?p. 1
Mathematical Research and Problem Solvingp. 2
An Example of a Mathematical Research Situationp. 3
Conjectures and Theoremsp. 5
Methods of Reasoningp. 5
Why Do We Need Proofs?p. 7
Mathematical Writingp. 8
Reading a Mathematics Textbookp. 9
Problemsp. 11
Sets, Numbers, and Axiomsp. 15
Sets and Numbers from an Intuitive Perspectivep. 15
Set Equality and Set Inclusionp. 23
Venn Diagrams and Set Operationsp. 31
Undefined Notions and Axioms of Set Theoryp. 45
Axioms for the Real Numbersp. 50
Problemsp. 56
Elementary Logicp. 69
Statements and Truthp. 69
Truth Tables and Statement Formsp. 80
Logical Equivalencep. 86
Arguments and Validityp. 91
Statements Involving Quantifiersp. 98
Problemsp. 106
Planning and Writing Proofsp. 117
The Proof-Writing Contextp. 117
Proving an If… Then Statementp. 122
Proving a For All Statementp. 129
The Know/Show Approach to Developing Proofsp. 136
Existence and Uniquenessp. 144
The Role of Definitions in Creating Proofsp. 153
Proving and Expressing a Mathematical Equivalencep. 160
Indirect Methods of Proofp. 168
Proofs Involving Or.p. 173
A Mathematical Research Situationp. 177
Problemsp. 183
Relations and Functionsp. 199
Relationsp. 199
Equivalence Relations and Partitionsp. 207
Functionsp. 217
One-to-One Functions, Onto Functions, and Bijectionsp. 225
Inverse Relations and Inverse Functionsp. 235
Problemsp. 239
The Natural Numbers, Induction, and Countingp. 255
Axioms for the Natural Numbersp. 255
Proof by Inductionp. 258
Recursive Definition and Strong Inductionp. 270
Elementary Number Theoryp. 276
Some Elementary Counting Methodsp. 286
Problemsp. 298
Further Mathematical Explorationsp. 311
Exploring Graph Theoryp. 311
Exploring Groupsp. 323
Exploring Set Cardinalityp. 337
Indexp. 347
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