Mathematical Logic for Computer Science 3rd edition 2012 by Mordechai Ben-Ari

Appendix - Set Theory

p327 - For an elementary, but detailed, development of set theory, see Velleman "
How to Prove It" 2nd edition 2006

Elements of Set Theory 1st Edition by Herbert B. Enderton

Contents

Preface - XI

List of Symbols - XIII

Ch 1 - Introduction - 1

Ch 2 - Axioms and Operators - 17

Ch 3 - Relations and Functions - 35

Ch 4 - Natural Numbers - 66

Ch 5 - Construction of the Real Numbers - 90

Ch 6 - Cardinal Numbers and the Axiom of Choice - 128

Ch 7 - Orderings and Ordinals - 167

Ch 8 - Ordinals and Order Types - 209

Ch 9 - Special Topics - 241

Appendix - Notation, Logic, and Proofs - 263

Selected References for Further Study - 269

List of Axioms - 271

Index - 273-279 end of book

Symbols

\(\dashv\) - end of proof - \dashv - pXII p22

iff - if and only if - pXII p2

Preface

pXI - some set theory knowledge needed for math study - can be studied for its own interest - no end to what can be learned of set theory - axiomatic material marked by stripe in the margin

pXII - no specific background needed - book gives real proofs (first difficult proof end of ch 4) - ... - 300 exercises - appendix dealing with topics from logic (turntables, quantifiers) - examples how discover a proof - list of books - two stylistic matters: end of proof \(\dashv\)

Ch 1 - Introduction - Baby Set Theory

p1 - beginning math course with discussion of set theory has become widespread - set is collection of things, called its members or elements - collection is regarded as a single object - write "\(t \in A\)", say t is a member of A - \(t \notin A\) say t is not a member of A - example: prime numbers less than 10 are: {2, 3, 5, 7} --- call it set A ---

p2 - B is all solutions to polinomial x4 - 17x3 + 101x2 + 247x + 210 = 0 --- solutions are again 2, 3, 5, 7 --- therefore A and B same set, A = B --- no matter that they are defined in different ways, elements are exactly the same, equal

Principle of Extensionality: If two sets have exactly the same members, then they are equal.

more concisely by symbolic notation: If A and B are sets such that for every object t,

\(t \in {A}\) iff \(t \in {B}\),

then \({A = B}\).If \({A = B}\) then any object \(t, t \in A\) iff \t is

Vocabulary

ambiguously - zweideutig ... - p2

concisely - kurz und klar, prägnant - p2

denote - bezeichnen, verzeichnen, bedeuten, hindeuten, kennzeichnen, anzeigen ... - p13

Bill Evans - Alone (1968 Album)