They are not guaranteed to be comprehensive of the material covered in the course. The objects that make up a set also known as the sets elements or members can be anything. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. Set theory 1 set theory one of the main objectives of a statistician is to draw conclusions about a population of objects by conducting an experiment. The notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is, but we can give an informal description, describe. Set theory is the branch of mathematical logic that studies sets, which are collections of objects. Ideal as a supplement to a curriculum or for independent study. Designed for teachers, students, parents, homeschoolers, educators and all lifelong learners. 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. Department of pure mathematics and mathematical statistics.
Complex issues arise in set theory more than any other area of pure mathematics. Certainly there is room for more examples and more simple exercises to. Ifthereisaproofoffrom,say provesorsyntacticallyimplies,written. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Basic set theory a set is a many that allows itself to be thought of as a one. Only informal, or nave, set theory notation and results are needed in chapter i at the metalevel, i. In 1829, dirichlet had shown that a function always had a fourier series converging to itself, under certain sufficient conditions he considered periodic functions with finitely many extrema in every period and equal to the halfsum of. Everyday low prices and free delivery on eligible orders.
For those that take axiomatic set theory, you will learn about something. There you develop logic for finite or explicitly presented countable vocabularies. It will be suitable for all mathematics undergraduates coming to the subject for the first time. A set is a welldefined collection of distinct objects. Usually we denote sets with uppercase letters, elements with lowercase letters.
The strong tradition, universality and neutrality of set theory make it rm common ground on which to provide uni cation between seemingly disparate areas and notations of computer science. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. Notes on logic and set theory cambridge mathematical. A statement is a sentence that is either true or false. These notes are not endorsed by the lecturers, and i have modified them often significantly. Sentential logic we go into the mathematical theory of the simplest logical notions. 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. It only remains to define a, b in terms of set theory. Part ii logic and set theory based on lectures by i. Students need to be familiar with the following concepts.
A equals b,written ab,iff every element of a is also an element of b,and conversely,every element of b is also an element of a. Meant as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy, this twovolume work is written in a userfriendly conversational lecture style that makes it equally effective for selfstudy or class use. Cambridge mathematical text books, cambridge university press, 1987, paperback, 110pp. He includes many methodological remarks and explanations, and demonstrates how the basic concepts of mathematics can be reduced to set theory. Choi department of mathematical science kaist, daejeon, south korea fall semester, 2011 s. Nafisha tasnim zaman b1506150 shamin yeaser rahman b15061 sarker humaira mostareen b1506014 farjana islam mim b1506182 3. Then by the axiom schema of comprehension, there is a set bdf x2vw g. There the importance stems from the fact that set theory provides an incredibly versatile. Choi kaist logic and set theory september 23, 2011 1 38.
The introduction to nlp module assumes some basic knowledge of set theory and logic. Leader, lentterm 2005, 2010 chapter 1 propositional logic 1 chapter 2 wellorderings and ordinals 7 chapter 3 posets and zorns lemma 16 chapter 4 predicate logic 24 chapter 5 set theory 34 chapter 6 cardinals 43. The old theory of types is still discussed somewhat in the philosophy department version of mathematical logic, but the mathematics department principally remembers russells work only. A total order or linear order on a set is a relation notes on logic and set theory by johnstone, p. Volume ii, on formal zfc set theory, incorporates a selfcontained chapter 0 on proof techniques based on formal logic, in the. Studies in logic and the foundations of mathematics.
Set theory and logic supplementary materials math 103. Suitable for all introductory mathematics undergraduates, notes on logic and set theory covers the basic concepts of logic. Basic set theory and logic university of cambridge. 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.
If you apply t he axiom of replacement to a functionclass f, you must explain clearly why f is indeed a functionclass. The whiteheadrussell way of doing logic and set theory has long been relegated to the history books, while zf set theory and its spawn have dominated the scene. These notes were prepared using notes from the course taught by uri avraham, assaf. Cambridge core logic, categories and sets notes on logic and set theory by p. Boolean algebra is given by a set s and two operations. Leader, lentterm 2005, 2010 chapter 1 propositional logic 1 chapter 2 wellorderings and ordinals 7 chapter 3 posets and zorns lemma 16 chapter 4 predicate logic 24 chapter 5 set theory 34 chapter 6 cardinals 43 bonus lecture incompleteness examples sheets prerequisites. Though propositional logic will prove a useful tool to describe certain aspects of meaning, like the reasoning in 1, it is a blunt. Set theory is likely to be around long after most presentday programming languages have faded from memory.
In pure mathematics, it is the place where questions about in. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. In this introduction to set theory and logic, the author discusses first order logic, and gives a rigorous axiomatic presentation of zermelofraenkel set theory. Lecture notes on elementary logic and set theory by jeanmarc. The basic idea of a set, set intersection, set union etc. There are many other details and plenty of terminology that i will skip. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Logic and set theory virginia commonwealth university. Basic set theory lx 502 semantics i september 11, 2008 1. State the axiom of foundation and the principle of induction, and show that they are equivalent in the presence of the other axioms of zfc.
Oleary, the structure of proof with logic and set theory, 2002. An elementary introduction to logic and set theory. Introduction to logic and set theory 202014 bgu math. Although this is a fascinating study of permanent interest, it does not account for the importance of set theory in applied areas. In mathematics, the notion of a set is a primitive notion. Sentential logic we go into the mathematical theory of the simplest. Curiously, set theory arose in the context of sets of real numbers related to the convergence of fourier series. These notes are for students of math 251 as a revision workbook and are not meant to. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Tens of thousands of excellent, educational videos in a huge, intuitive directory. B for example, if s is the set of all juices in the supermarket, and t is the set of all. Set theory if we are interested in elements of a set a that are not contained in a set b, we can write this set as a. Mathematics, specifically the real number system, is approached as a unity whose operations can be logically ordered through axioms. Examples of structures the language of first order logic is interpreted in mathematical structures, like the following.
Like logic, the subject of sets is rich and interesting for its own sake. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. Perhaps landau hoped to induce the student to read the preface for the teacher, but not to worry about digesting. Set theory a mathematical model that we will use often is that of. Some familiarity with logic and set theory is required for studying calculus also, but as far as a systematic study of these topics are concerned, you should do it if you are really interested in it. Logic and set theory educational videos watchknowlearn. Download notes on logic and set theory 97805235027. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. Logic and set theory these notes correspond to mathematical appendix 1 in the text. For example, a deck of cards, every student enrolled in. Propositional logic, interpretation with respect to a model expressed in terms of sets.
Department of pure mathematics and mathematical statistics, university of cambridge. Set theory and logic dover books on mathematics reprint. Using this, you can develop formal set theory, and inside set theory you can redevelop logic, and go on to model theory, etc, now without any size or definability restrictions. In this lecture, we will be more rigorous than in htp and use axioms to establish facts. We will need only a few facts about sets and techniques for dealing with them, which we set out in this section and the next. The book is based on lectures given at the university of cambridge and covers the basic concepts of logic. The following notation is used to show set membership means that x is a member of the set a means that x is not a member of the set a. You wont be needing very advanced knowledge of logic for studying topics like abstract algebra, linear algebra, analysis etc. We will return to sets as an object of study in chapters 4 and 5. No student is expected to really benefit from these notes unless they have regularly attended the lectures.
Mathematical logic is a branch of mathematics which is mainly concerned with the relationship between semantic concepts i. Choi kaist logic and set theory november 20, 2012 4 14. Basic concepts of set theory, functions and relations. This short textbook provides a succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. For those of you new to abstract mathematics elementary does not mean simple though much of the material. Propositional logic, interpretation with respect to a. It will be suitable for all mathematics undergradua. The notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is.
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