The book first tackles the foundations of set theory and infinitary combinatorics. Discussions focus on the suslin problem, martins axiom. An introduction to elementary set theory guram bezhanishvili and eachan landreth 1 introduction in this project we will learn elementary set theory from the original historical sources by two key gures in the development of set theory, georg cantor 18451918 and richard dedekind 18311916. The objects that belong to vare called the elements of vor the members of v. Click download or read online button to get elementary set theory proof techniques book now. Proof is, how ever, the central tool of mathematics. These will be the only primitive concepts in our system. That is, it is possible to determine if an object is to be included in the set or not. Sets we discussed in class how to formally show that one set is a subset of another and how to show two sets are equal. For any set a, the empty set is an element of the power set of a.
An introduction to proofs, logic, sets, and numbers introduces key concepts from logic and set theory as well as the fundamental definitions of algebra to prepare readers for further study in the. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Proofs of settheoretic relations and equalities among sets are among the simplest types of proofs and therefore present an excellent opportunity to familiarize. A book of set theory, first published by dover publications, inc. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. This book has been reprinted with the cooperation of.
Set theory proof, revolving around the number of elements in a set 0 given an infinite set, can we give an infinite chain of finite subsets of that set, with the chain ordered by the proper subset operator. Find materials for this course in the pages linked along the left. A mathematical proof of a statement strongly depends on who the proof is written for. The empty set is a subset of a, hence it is an element of the power set of a.
Other words roughly synonymous with the word set are class. 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. Proofs formal proofs follow simple welldefined rules and should be easy for a machine to check as assembly language is easy for a machine to execute english proofs correspond to those rules but are designed to be easier for humans to read also easy to check with practice almost all actual math and theory in cs is done this way. This proof will be omitted, though the theorem is equivalent to the axiom of choice. Set theory wikibooks, open books for an open world. This lecture is based on but not limited to to chapter 6 in discrete mathema. About elementwise proofs in set theory an elementwise proof is a method for showing that one set is a subset of another set.
Set theory is indivisible from logic where computer science has its roots. Discrete mathematicsset theorypage 2 wikibooks, open. This lead to the formal study of set theory, where set theory is given a formal foundation based on some axioms of set theory. Home courses electrical engineering and computer science mathematics for computer science unit 1. A set v divides the mathematical universe into two parts. This handout is meant primarily for those students who are already familiar with most of the subject matter contained within that is. Two sets are equal if and only if they have the same elements. Basic set theory a set is a many that allows itself to be thought of as a one. Set theory has its own notations and symbols that can seem unusual for many. Axioms and set theory mathematics university of waterloo. But even more, set theory is the milieu in which mathematics takes place today.
Addressing the importance of constructing and understanding mathematical proofs, fundamentals of mathematics. Are the natural numbers larger than the even numbers. Let us consider a few examples of entities we may consider to be sets. This is similar to euclids axioms of geometry, and, in some sense, the group axioms. Proof, sets, and logic department of mathematics boise state. An introduction to independence proofs offers an introduction to relative consistency proofs in axiomatic set theory, including combinatorics, sets, trees, and forcing. Contradiction proofs this proof method is based on the law of the excluded middle. For example, the set e above is the set of all values the expression 2 nthat satisfy the rule 2 z. Halmos, naive set theory 1974 karel hrbacek, thomas j. Common types of proofs disproof by counterexample statement must be of the form every x satisfies fx disprove it by finding some x that does not satisfy fx application of quantifier negation. Review of set theory michael williams last updated. Empty setsubset properties theorem s empty set is a subset of any set.
Is it meaningful to say one infnite set is larger than another. Duplicates dont contribute anythi ng new to a set, so remove them. Continued since each set is a subset of the other, we have established the equality of the two sets so a b c a b \a c. Krzysztof ciesielski, set theory for the working mathematician 1997 p. Reviewed by david miller, professor, west virginia university on 41819. Proofs from group theory december 8, 2009 let g be a group such that a. Here are some basic subset proofs about set operations. Proofs for a research audience are quite different from those found in textbooks. Jech, set theory 3rd edition 2006 kenneth kunen, set theory. Basic concepts of set theory, functions and relations. Learn to prove distributive laws of set theory in writing. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. We must show the following implication holds for any s x x x s since the empty set does not contain any element, x is. There are other conceptions of set, but although they have genuine mathematical interest they are not our concern here.
Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Elementary set theory proof techniques download ebook. Set theory an introduction to independence proofs studies in. Proof methods mathematical and statistical sciences. Proof theory is concerned almost exclusively with the study of formal proofs. Set theory for computer science university of cambridge. 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. A first running example in the upcoming sections, were going to see how to reason rigorously about sets and set theory. The order of the elements in a set doesnt contribute. The foundational rules of set theory the laws listed below can be described as the foundational rules of set theory.
For more videos on set theory and many more other interesting topics subscribe or visit to. Georg cantor this chapter introduces set theory, mathematical in duction, and formalizes the notion of mathematical functions. Studies in logic and the foundations of mathematics, volume 102. Set theory an introduction to independence proofs studies. An accessible introduction to abstract mathematics with an emphasis on proof writing. The axioms of set theory of my title are the axioms of zermelofraenkel set theory, usually thought ofas arisingfromthe endeavourtoaxiomatise the cumulative hierarchy concept of set. Writing mathematical proofs is therefore an art form the art of convincing and a social process since it is directed at people the readers. On the test i will have to do proofs involving all of these subset, unions, intersections, relative complements, cartesian products, different kinds of relations, partial orderstrict order, inverse, composition, fuctions, onetoone functions, onto funtions, etc. This alone assures the subject of a place prominent in human culture.
Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. An overview of logic, proofs, set theory, and functions. They are not guaranteed to be comprehensive of the material covered in the course. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Basic set theory daniel murfet may 16, 2006 throughout we work with the foundation provided by standard zfc fct,section 3. Mat231 transition to higher math proofs involving sets fall 2014 8 11. This handout is meant primarily for those students who are already familiar with most of the subject matter contained within that is, those who have taken a proofs class before. This is the most convincing technique to use for proving subset inclusion. We derive them by going back to the definitions of intersection, union, universal set and empty set, and by considering whether a given element is in, or not in, one or more sets. Set theory proof, revolving around the number of elements in a set. By the associative property of groups, a b b 1a 1 abb 1a. March 30, 2009 basic set theory we will use the standard notation for containments. Set theory tutorial problems, formulas, examples mba.
The theory of numbers provides an excellent source for such examples. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. In this tutorial, we look at some solved examples to understand how set theory works and the kind of problems it can be used to solve. I have a test coming up on set theory, so the basic definitions, set operations, relations and functions. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. In particular we do not assume we are working inside any. The notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is. Set theory \a set is a many that allows itself to be thought of as a one. Basic laws of set theory it is quite clear that most of these laws resemble or, in fact, are analogues of laws in basic algebra and the algebra of propositions. On the test i will have to do proofs involving all of these subset, unions, intersections, relative complements, cartesian products, different kinds of relations, partial orderstrict order, inverse, composition, fuctions, onetoone.
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