Logic, Sets, and Functions

Title ......Page 1
Copyright ......Page 2
Contents ......Page 3
Preface ......Page 6
1 Basic Concepts of Logic ......Page 7
1.1 Arguments ......Page 8
1.2 Validity ......Page 14
1.3 Implication and Equivalence ......Page 20
1.4 Logical Properties of Sentences ......Page 24
1.5 Satisfiability ......Page 29
2.1 The Language of Sentential Logic ......Page 33
2.2 Truth Functions ......Page 36
2.3 A Sentential Language ......Page 39
2.4 Translation ......Page 42
2.5 Validity ......Page 44
2.6 Truth Tables ......Page 47
2.7 Truth Tables for Formulas ......Page 49
2.8 Examples ......Page 51
2.9 Truth Tables for Argument Forms ......Page 54
2.10 Implication, Equivalence and Satisfiability ......Page 57
3.1 Natural Deduction Systems ......Page 61
3.2 Rules for Negation and Conjunction ......Page 63
3.3 Rules for the Conditional and Biconditional ......Page 70
3.4 Rules for Disjunction ......Page 73
3.5 Derivable Rules ......Page 75
4 Quantifiers ......Page 83
4.1 Constants and Quantifiers ......Page 84
4.2 Categorical Sentence Forms ......Page 88
4.3 Polyadic Predicate ......Page 90
4.4 The Language Q ......Page 93
4.5 Translation ......Page 96
4.5.1 Noun Phrases ......Page 97
4.5.2 Verb Phrases ......Page 101
4.5.3 Connectives ......Page 103
4.6 Interpretations ......Page 106
5.2 Universal Proof ......Page 115
5.3 Derived Rules for Quantifiers ......Page 125
6.1 Identity ......Page 129
6.2 Deduction Rules for Identity ......Page 133
6.3 Function Symbols ......Page 135
7.1 Extensionality ......Page 143
7.2 Abstraction ......Page 148
7.3 Pair Sets, Unit Sets and Enumeration ......Page 151
7.4 The Null Set ......Page 153
7.5 Binary Unions, Intersections and Complements ......Page 155
7.6 Unions and Intersections of Single Sets ......Page 162
7.7 Power Sets ......Page 165
7.8 Complex Abstraction ......Page 167
8.1 Sequences and Ordered Pairs ......Page 169
8.2 Cartesian Products ......Page 171
8.3 Relations ......Page 176
8.4 Properties of Relations ......Page 181
8.5 Ordering Relation ......Page 186
8.6 Relations between Relations ......Page 191
8.7 Restrictions and Images ......Page 192
8.8 Equivalence Relations and Partitions ......Page 201
9.1 Functions and Relations ......Page 205
9.2 Properties of Functions ......Page 209
10.1 The Natural Numbers and Definition by Recursion ......Page 215
10.2 Weak Induction on the Natural Numbers ......Page 223
10.3 Strong Induction on Natural Numbers ......Page 227
10.4 Induction on Sets other than the Natural Numbers ......Page 230
10.5 Graphs, Trees and Lists ......Page 239
10.6 Formal Languages ......Page 247
A.1 Introduction ......Page 257
A.3.1 The Menus: An Overview ......Page 258
A.3.3 The Edit Menu ......Page 259
A.3.4 The Annotations Menu ......Page 260
A.4 Homework Mode the Windows Version ......Page 261
A.6.1 The Language of Sentential Logic ......Page 262
A.7.1 Proof Formats ......Page 263
A.7.2 Derivation Rules ......Page 265
A.7.3 The Proofs Menu ......Page 266
A.7.4 Imported Rules ......Page 270
A.7.5 The Quantification Rule Set ......Page 271
A.8 Macintosh Keyboard Shortcuts ......Page 272
A.8.1 Mackintosh Keyboard Equivalents for Logical Symbols ......Page 273
B Answers to Selected Problems ......Page 275
Index ......Page 290