This alone assures the subject of a place prominent in human culture. Elementary logic 5 Some history 5 Objectives 5 1. We typically use the bracket notation fgto refer to a set. 1. First, the book de nes the notion of the complement, denoted by Ac, of a set A in some universal set U. Primitive Concepts. Introduction to Logic and Set Theory-2013-2014 General Course Notes December 2, 2013 These notes were prepared as an aid to the student. Contents Chapter 1. - Georg Cantor This chapter introduces set theory, mathematical in- duction, and formalizes the notion of mathematical functions. �=n�?�C �mY�m��ߣ7���'w8��uQӠ?g����rv���)����TuWmʰ��G�1a��>���~%��ؕ-
��k��I��� �����۟��b��,SF���S\�@���OX0�=�(�r��݂�h�x��q�. Elements of Set Theory eleven; all oxygen molecules in the atmosphere; etc. In mathematics, the notion of a set is a primitive notion. This book has been reprinted with the cooperation of Kyung Moon Publishers, South Korea. Operations on statements 6 3. The objects can be real, physical things, or abstract, mathematical things. NOTES ON SET THEORY The purpose of these notes is to cover some set theory terminology not included in Solow’s book. 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. Although this is not sufﬁciently well appreciated, it is difﬁcult to give a general characterization of extensional deﬁnitions. The technique of using the concept of a set to answer questions is hardly new. Itis perhaps best to say that an extensional deﬁnition of a set is one that is givenbyanenumera-tion (listing) of all its elements. Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 3 Set Theory Basics.doc Predicate notation. 3 0 obj << >> 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��~���� They are not guaran-teed to be comprehensive of the material covered in the course. Part II | Logic and Set Theory Based on lectures by I. Basic Set Theory LX 502 - Semantics I September 11, 2008 1. These entities are what are typically called sets. /Filter /FlateDecode 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. 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). 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 These notes were prepared using notes from the course taught by Uri Avraham, Assaf Hasson, and of course, Matti Rubin. Basic Concepts of Set Theory. 2. i A brief history of sets A set is an unordered collection of objects, and as such a set is determined by the objects it contains. NB (Note Bene) - It is almost never necessary in a mathematical proof to remember that a function is literally a set of ordered pairs. Implication and equivalence 10 4. Motivation When you start reading these notes, the first thing you should be asking yourselves is “What is Set Theory and why is it relevant?” Though Propositional Logic will prove a useful tool to describe certain aspects of meaning, like the reasoning in (1), it is a blunt instrument. /Length 2763 Set Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos) A.1. Basic Set Theory A set is a Many that allows itself to be thought of as a One. 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. As such, it is expected to provide a ﬁrm foundation for the rest of mathematics. We start with the basic set theory. No speci c prerequisites. SET THEORY AND FORCING 1 0. A set is one of those fundamental mathematical ideas whose nature we understand without direct reference to other mathematical ideas. 1.1. %PDF-1.4 Quite simply, Denition 1 A set is a collection of distinct objects. Set Theory is the true study of inﬁnity. Statements 5 2. Notes on Set Theory for Computer Science by Prof Glynn Winskel c Glynn Winskel. ������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��� Although this is not sufﬁciently well appreciated, it is difﬁcult to give a general characterization of extensional deﬁnitions. Lecture notes on elementary logic and set theory by Jean-Marc Schlenker1 Mathematical Research Unit, University of Luxembourg 1Translated into English by Sergei Merkulov. The number of such objects can be nite or innite. De nition 1.8 (Injection). Quanti ers 12 6. %���� Axiomatic set theory: ZFC Z is for Ernst Zermelo, F is for Abraham Fraenkel, C is for the Axiom of Choice. 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.

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