Set Theory and Logic Supplementary Materials Math 103: Contemporary Mathematics with Applications A. Calini, E. Jurisich, S. Shields c 2008. Proof by Counter Example. g Yet the density of squares goes down as we go up. Negation of Quantified Predicates. Methods of Proof. TABLE OF CONTENTS LOGIC 1. Mathematical Induction. Sentential logic.....1 2. Unique Existence. Predicates. Notes on Set Theory and Logic August 29, 2013. 2. Indirect Proof. The axioms of set theory.....67 6. Predicate Logic and Quantifiers. For example, a deck of cards, every student enrolled in Math 103, the collection of all even integers, these are all examples of sets of things. ii. These entities are … Universal and Existential Quantifiers. . – Ian Stewart Does God play dice? 2 CHAPTER 1. Also, their activity led to the view that logic + set theory can serve as a basis for 1. This chapter will be devoted to understanding set theory, relations, functions. This era did not produce theorems in mathematical logic of any real depth, 1 but it did bring crucial progress of a conceptual nature, and the recognition that logic as used in mathematics obeys mathematical rules that can be made fully explicit. academic career, you may wish to study set theory and logic in greater detail. PRELIMINARIES all of mathematics. . Chapter 1 Set Theory 1.1 Basic deﬁnitions and notation A set is a collection of objects. (Georg Cantor) In the previous chapters, we have often encountered "sets", for example, prime numbers form a set, domains in predicate logic form sets as well. Our main purpose here is to learn how to state mathematical results clearly and how to prove them. 1.1 Statements A proof in mathematics demonstrates the truth of certain statement . 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. 8 In the Two New Sciences book, his ﬁnal masterpiece, Galileo observed that there is a one-to-one correspondence between natural numbers and squares: n !n2 f1,2,3,. Abstraction is what makes mathematics work. Multiple Quantifiers. 8. Formal Proof. The proof that p = t in Chapter 34 is based upon notes of Fremlin and a thesis of Roccasalvo. Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Conditional Proof. First-order logic.....12 3. . We start with the basic set theory. The completeness theorem .....42 ELEMENTARY SET THEORY 5. III. . A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. IV. V. Naïve Set Theory. De ning a set formally is a pretty delicate matter, for now, we will be happy to consider an intuitive de nition, namely: De nition 24. Proofs .....24 4. Informal Proof. g !f1,4,9,. The consistency proofs in Chapter 35 are partly from Kunen and partly from the author. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. It is therefore natural to begin with a brief discussion of statements. Set Theory \A set is a Many that allows itself to be thought of as a One."
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