A Transition to Mathematics with Proofs

by Cullinane, Michael J

• ISBN13: 9781449627782
• ISBN10: 1449627781
• Edition: 1st
• Format: Hardcover
• Copyright: 2012-01-13
• Publisher: Jones & Bartlett Learning

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

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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