# Set Theory

## Set Theory

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

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