Basic Concepts of Mathematics and Logic

by Gemignani, Michael C.

• ISBN13: 9780486435060
• ISBN10: 0486435067
• Format: Paperback
• Copyright: 2004-04-22
• Publisher: Dover Publications

Summary

This text emphasizes logic and the theory of sets. Students who take no further courses in the field will find it an excellent resource for developing an appreciation for the nature of mathematics. Others will discover the foundations for future studies set theory, logic, counting, numbers, functions, and more. 1968 edition. 43 figures. 25 tables.

Table of Contents

Chapter 1 Introduction
1.1 Abstraction and Mathematics	1	(1)
1.2 Mathematics and "Reality"	2	(4)
1.3 An Example	6	(3)
1.4 Mathematical Statements	9	(4)
1.5 Notation	13	(2)
1.6 On Rigor	15	(2)
Chapter 2 Introduction to Logic
2.1 Necessary and Sufficient Conditions	17	(4)
2.2 Negations	21	(2)
2.3 Mathematical Disproofs	23	(4)
2.4 Conjunction and Disjunction	27	(3)
2.5 Truth Tables	30	(5)
2.6 Logical Equivalence. Tautologies	35	(4)
Chapter 3 More about Logic
3.1 Converses and Contrapositives	39	(3)
3.2 On the Validity of a Proof	42	(3)
3.3 Some Elementary Arguments	45	(5)
3.4 More Elementary Arguments	50	(5)
3.5 Proofs by Contradiction	55	(3)
3.6 Summary	58	(4)
Chapter 4 Sets
4.1 The Notion of a Set	62	(4)
4.2 Set-Building Notation	66	(2)
4.3 Subsets. Equality of Sets	68	(4)
4.4 Union and Intersection	72	(5)
4.5 The Difference of Two Sets. Complements	77	(5)
Chapter 5 Set Theory and Logic
5.1 Set Diagrams	82	(5)
5.2 Sets and Language. Venn Diagrams	87	(6)
5.3 Applying Set Theory to Arguments	93	(4)
Chapter 6 Counting
6.1 The Notion of Counting	97	(5)
6.2 More About Infinity	102	(4)
6.3 More About Countable Sets	106	(5)
6.4 The Notion of Number	111	(5)
6.5 The Addition of Cardinal Numbers	116	(7)
Chapter 7 The Cartesian Product. Functions
7.1 The Cartesian Product of Two Sets	123	(4)
7.2 The Multiplication of Cardinal Numbers	127	(4)
7.3 Functions	131	(6)
7.4 More About Functions	137	(5)
7.5 Composition of Functions	142	(5)
Chapter 8 Relations
8.1 The Notion of a Relation	147	(5)
8.2 Equivalence Relations	152	(5)
8.3 A Definition of the Integers	157	(6)
8.4 Orderings	163	(8)
Chapter 9 More about Total Ordering
9.1 The Ordering of the Integers, an Intuitive View	171	(6)
9.2 The Ordering of the Integers, a Formal View	177	(3)
9.3 Finite Induction	180	(5)
9.4 Another Definition of the Integers	185	(5)
Chapter 10 Probability
10.1 The Nature of Probability	190	(4)
10.2 The Basic Rules of Probability	194	(6)
10.3 Methods of Assigning Probabilities	200	(6)
10.4 Sampling	206	(7)
10.5 Some Special Probabilities	213	(6)
Chapter 11 An Elementary Geometry
11.1 The Projective Plane	219	(4)
11.2 Basic Properties of the Projective Plane	223	(6)
11.3 The Afline Plane	229	(6)
Chapter 12 Conclusion
12.1 Where shall We Go From Here?	235	(1)
12.2 Mathematics as a Science	236	(1)
12.3 Mathematics as an Art	237	(1)
12.4 Conclusion	238	(1)
Appendix	239	(4)
Answers to Odd-Numbered Exercises	243	(34)
Index of Symbols	277	(1)
Index	278