SetTheory - Foundations of Mathematics I Set Theory (only a...

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Unformatted text preview: Foundations of Mathematics I Set Theory (only a draft) Ali Nesin Mathematics Department Istanbul Bilgi University Ku¸ stepe S ¸i¸ sli Istanbul Turkey [email protected] February 12, 2004 2 Contents I Naive Set Theory 7 1 Basic Concepts and Examples 11 1.1 Sets, Subsets and Emptyset . . . . . . . . . . . . . . . . . . . . . 11 1.2 Notes on Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 Number Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Subsets Defined By a Property . . . . . . . . . . . . . . . . . . . 20 1.5 Sets of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.6 Parametrized Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2 Operations with Sets 25 2.1 Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Cartesian Product of Two Sets . . . . . . . . . . . . . . . . . . . 29 3 Functions 31 3.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 More On Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Binary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 Operations with Functions . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Injections, Surjections, Bijections . . . . . . . . . . . . . . . . . . 38 3.5.1 Injections . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5.2 Surjections . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5.3 Bijections . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.6 Sym( X ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.7 Families, Sequences and Cartesian Products . . . . . . . . . . . . 44 4 Relations 47 4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Partial Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.4 Total Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5 Induction 57 3 4 CONTENTS 6 Bijections, revisited 61 6.1 Schr¨ oder-Bernstein’s Theorem . . . . . . . . . . . . . . . . . . . . 61 6.2 Examples of Sets in Bijection . . . . . . . . . . . . . . . . . . . . 61 7 Some Automorphism Groups 63 7.1 Binary Unirelational Structures . . . . . . . . . . . . . . . . . . . 63 7.2 Automorphism Groups of Graphs . . . . . . . . . . . . . . . . . . 64 7.3 Geometric Automorphism Groups . . . . . . . . . . . . . . . . . 65 7.4 Back and Forth Argument . . . . . . . . . . . . . . . . . . . . . . 67 7.4.1 Dense Total Orderings . . . . . . . . . . . . . . . . . . . . 67 7.4.2 Random Graphs . . . . . . . . . . . . . . . . . . . . . . . 68 8 Formulae 69 9 Miscellaneous Exercises 71 II Axiomatic Set Theory 73 10 Basics 75 10.1 Russell’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . .10....
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SetTheory - Foundations of Mathematics I Set Theory (only a...

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