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So a= c= d, in particular, a= cand b= d. 2. 0000010201 00000 n
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For our purposes, it will sufce to approach basic logical concepts informally. startxref
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Methods of Proof. ha;bi= ffag;fa;bgg Theorem 1.5. ha;bi= hc;dii a= cand b= d. Proof. >>
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14 Chapter 1 Sets and Probability Empty Set The empty set, written as /0or{}, is the set with no elements. 0000075927 00000 n
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Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. This proves that P.X/“X, and P.X/⁄Xby the Axiom of Extensionality. 0000047721 00000 n
Then ffag;fa;bgg= ffag;fa;agg= ffag;fagg= ffagg Since ffagg= ffcg;fc;dggwe must have fag= fcgand fag= fc;dg. 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 important properties of sets, and give examples. 0000038078 00000 n
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V. Naïve Set Theory. 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. 0000063750 00000 n
Let Xbe an arbitrary set; then there exists a set Y Df u2 W – g. Obviously, Y X, so 2P.X/by the Axiom of Power Set.If , then we have Y2 if and only if – [SeeExercise 3(a)]. In the second half of the last century, logic as pursued by mathematicians gradually branched into four main areas: model theory, computability theory (or recursion theory), set theory, and proof theory. 2 0 obj
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Formal Proof. Subsets A set A is a subset of a set B iff every element of A is also an element of B. in set theory, one that is important for both mathematical and philosophical reasons. Such a relation between sets is denoted by A ⊆ B. Multiple Quantifiers. These notes were prepared using notes from the course taught by Uri Avraham, Assaf Hasson, and of course, Matti Rubin. In 1874 Cantor had shown that there is a one-to-one correspondence between the natural numbers and the algebraic numbers. An appendix on second-order logic will give the reader an idea of the advantages and limitations of the systems of first-order logic used in Set Theory and Logic Supplementary Materials Math 103: Contemporary Mathematics with Applications A. Calini, E. Jurisich, S. Shields c 2008. 0000011497 00000 n
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A. Hajnal & P. Hamburger, ‘Set Theory’, CUP 1999 (for cardinals and ordinals) 4. Set Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos) A.1. We study two types of relations between statements, implication and equivalence. <<
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In Chapter 2, a section has been added on logic with empty domains, that is, on what happens when we allow interpretations with an empty domain. Chapter 1 Set Theory 1.1 Basic deﬁnitions and notation A set is a collection of objects. The Axiom of Pair, the Axiom of Union, and the Axiom of 0000072804 00000 n
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Although Elementary Set Theory is well-known and straightforward, the modern subject, Axiomatic Set Theory, is both conceptually more diﬃcult and more interesting. ����sɞ
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��Xe�e���� �81��c������ ˷�孇f�0h_mw. III. Any object which is in a set is called a member of the set. The major changes in this new edition are the following. Basics of Set Theory and Logic S. F. Ellermeyer August 18, 2000 Set Theory Membership A setis a well-defined collection of objects. 0000041887 00000 n
This proves that P.X/“X, and P.X/⁄Xby the Axiom of Extensionality. Set Theory and Logic is the result of a course of lectures for advanced undergraduates, developed at Oberlin College for the purpose of introducing students to the conceptual foundations of mathematics.Mathematics, specifically the real number system, is approached as a unity whose operations can be logically ordered through axioms. /Filter /FlateDecode
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Unique Existence. An Overview of Logic, Proofs, Set Theory, and Functions aBa Mbirika and Shanise Walker Contents 1 Numerical Sets and Other Preliminary Symbols3 2 Statements and Truth Tables5 3 Implications 9 4 Predicates and Quanti ers13 5 Writing Formal Proofs22 6 Mathematical Induction29 7 Quick Review of Set Theory & Set Theory Proofs33 0000056119 00000 n
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They are meta-statements about some propositions. (Caution: sometimes ⊂ is used the way we are using ⊆.) Primitive Concepts. If A ⊆ B and A ≠ B we call A a proper subset of B and write A ⊂ B. 0000065343 00000 n
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File Name: Logic And Set Theory With Applications 6th Edition Pdf.pdf Size: 6514 KB Type: PDF, ePub, eBook Category: Book Uploaded: 2020 Nov 20, 04:15 Rating: 4.6/5 from 914 votes. The empty set can be used to conveniently indicate that an equation has no solution. t IExercise 7 (1.3.7). Clearly if a= cand b= dthen ha;bi= ffag;fa;bgg= ffcg;fc;dgg= hc;di 1. 0000072849 00000 n
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SECTION 1.4 ELEMENTARY OPERATIONS ON SETS 3 Proof. P. T. Johnstone, ‘Notes on Logic & Set Theory’, CUP 1987 2. 0000021855 00000 n
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