Table of contents for Sets, functions, and logic : an introduction to abstract mathematics / Keith Devlin.


Bibliographic record and links to related information available from the Library of Congress catalog. Note: Electronic data is machine generated. May be incomplete or contain other coding.


Counter






Preface ......      ...................     ........................... vii

Students Start Here .........        ................................. ix

1  What Is Mathematics and What Does It Do for Us? ..................    1
   1.1  It's Not Just Numbers ....................................................  1
   1.2  M athematical Notation  ....................................................  4
   1.3  Making the Invisible Visible ............................................  7
   1.4  This Is Where You Come In ................................       9
   1.5  The Stuff of Modern Mathematics .............. .....................  11

2  Math Speak ......................................................... 13
   2.1  The Language of Mathematics: Part 1 ...................................  13
   2.2  Properties of the Language .............  ........................  20
   2.3  The Language of Mathematics: Part 2 ........    ................  29
   2.4  Properties of Quantification  ..............................................  33
   2.5  Proofs in Mathematics ........................................ 40
   2.6  The Integers .................  ................................ ..  50
   2.7  Mathematical Truth .......................................................  54

3  Set Theory       .......................................................... 57
   3.1  Sets .................................. ..................... .  57
   3.2  Operations on Sets  .......   .............................   61
   3.3  Real Intervals .................. .......... .......................  68
   3.4  Absolute Values ......................................................  69
   3.5  Inequalities .................  ..........    ..............  71
   3.6  Arbitrary Unions and Intersections ......................................  75
   3.7  Cartesian Products .............. ..........................................  78
   3.8  The Historical Development of Set Theory ........................... ..  81

4  Functions            ......................................................... 87
   4.1  The Function Concept ....................................................  87
   4.2  Examples of Functions ..............  ............................  89
   4.3  History of the Modem Function Concept .............................. .  93
   4.4  One-One and Onto Functions ........................................  95
   4.5  Composition and Inverse Functions ..................................... 100





   4.6  Denumerability ................................   .............. 104
   4.7  Uncountability ...........................               ....... 108

5  Relations ......................................                    113
   5.1  Binary Relations ........................     .............. 113
   5.2  Properties of Relations ......................................  115
   5.3  Relations as Sets of Ordered Pairs  .......................................  118
   5.4  Relations as Graphs ........................................ 121
   5.5  Equivalence Relations ........................................ 122
   5.6  Functions as Relations ........................................ 127
   5.7   An Example: The Reals ........................................ 129
   5.8   Upper Bounds. Completeness ........................................ 131
   5.9   Sequences ................. ........ .....    .............  133

No Answers to the Exercises ................................ 137

List of Symbols .......   ...................................... 139

Index            ........................................              141





Library of Congress Subject Headings for this publication: Mathematics