We typically use the bracket notation fgto refer to a set. Itis perhaps best to say that an extensional deﬁnition of a set is one that is givenbyanenumera-tion (listing) of all its elements. �=n�?�C �mY�m��ߣ7���'w8��uQӠ?g����rv���)����TuWmʰ��G�1a��>���~%��ؕ- ��k��I��� �����۟��b��,SF���S\�@���OX0�=�(�r��݂�h�x��q�. Leader Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modi ed them (often signi cantly) after lectures. The material is mostly elementary. Set Theory is the true study of inﬁnity. Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 3 Set Theory Basics.doc Predicate notation. The notion of set is taken as “undefined”, “primitive”, or “basic”, so we don’t try to define what a set is, but we can give an informal description, describe In mathematics, the notion of a set is a primitive notion. Introduction to Logic and Set Theory-2013-2014 General Course Notes December 2, 2013 These notes were prepared as an aid to the student. Part II | Logic and Set Theory Based on lectures by I. 1.1 Sets Mathematicians over the last two centuries have been used to the idea of considering a collection of objects/numbers as a single entity. That is, we admit, as a starting point, the existence of certain objects (which we call sets), which we won’t deﬁne, but which we assume satisfy some basic properties, which we express as axioms. Course Notes Page 1. These notes were prepared using notes from the course taught by Uri Avraham, Assaf Hasson, and of course, Matti Rubin. This alone assures the subject of a place prominent in human culture. Quite simply, Denition 1 A set is a collection of distinct objects. Implication and equivalence 10 4. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. These entities are what are typically called sets. They are not guaran-teed to be comprehensive of the material covered in the course. Example: {x x is a natural number and x < 8} Reading: “the set of all x such that x is a natural number and is less than 8” So the second part of this notation is a prope rty the members of the set share (a condition As in any axiomatic theory, they are not deﬁned (they are feature–less objects; in the context of the theory there is nothing to them apart from what the theory says). %���� Contents Chapter 1. Itis perhaps best to say that an extensional deﬁnition of a set is one that is givenbyanenumera-tion (listing) of all its elements. Basic Set Theory LX 502 - Semantics I September 11, 2008 1. Elementary logic 5 Some history 5 Objectives 5 1. ������9���UGK���Y96`��7�����6��+>�'�؃���Hb9^��5�"cy�r\bY��Ť��c��7�b5Z�y!�mR+�0>4w�O٘� :���_�{����0=Q��O��S]��]���r�R�z��q�3��� This book has been reprinted with the cooperation of Kyung Moon Publishers, South Korea. Elements of Set Theory eleven; all oxygen molecules in the atmosphere; etc. Before the 19th century it was uncommon to think of sets as completed objects in their own right. Typesetter’s Introduction Thesenotesprovideagreatintroductiontoaxiomaticsettheoryandtopicsthereinappropriate for a ﬁrst … Bibliographical Note A Book of Set Theory, first published by Dover Publications, Inc., in 2014, is a revised and corrected republication of Set Theory, originally published in 1971 by Addison-Wesley Publishing Company, Reading, Massachusetts. Statements 5 2. Although this is not sufﬁciently well appreciated, it is difﬁcult to give a general characterization of extensional deﬁnitions. The number of such objects can be nite or innite. Basic Set Theory A set is a Many that allows itself to be thought of as a One. This chapter will be devoted to understanding set theory, relations, functions. Sets and elements Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. As such, it is expected to provide a ﬁrm foundation for the rest of mathematics. %PDF-1.4 NOTES ON SET THEORY The purpose of these notes is to cover some set theory terminology not included in Solow’s book. - Georg Cantor This chapter introduces set theory, mathematical in- duction, and formalizes the notion of mathematical functions. Axiomatic set theory: ZFC Z is for Ernst Zermelo, F is for Abraham Fraenkel, C is for the Axiom of Choice. Exercices 13 Chapter 2. Proofs 11 5. Quanti ers 12 6. The objects of set theory are sets. xڽZ[o��~ϯ0�R��p�\&�Cr��(�h\m�F�%�H�KJu��;��\^��� 틸\.ggf���P�_��A��Y������B�T�c�R_\o.��|�t����JY�|���,����&�����_.s����_����cr+.SYv���n�����\�L'��K����+a��s�*in��qW�Sus,;��:�:6t�n.E��J\$F�]s�[t�ѿ�+��ޓ.�CU7�~�l���w�a~�?m�M� �@'\%˕�s�w����&_A�,i���\ic@7V\$͉Vt���и-?z{{��)�M��O1N`�]�No&w�ذ�x��iˁvUo�� S�ϔ�ɜT�T:�'������i�����K�9�E���+�����7d_�ó}y�W��g>��)��\����F},PF��v��I��~���� B. But even more, Set Theory is the milieu in which mathematics takes place today.

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