Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. The proof that example 4 is a partial order is left to the reader. Set theory for computer science university of cambridge. 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 platonic level, this is intended to communicate something about proof, sets, and logic. In doing so, our examples to illustrate the techniques should not be very. For any set a, the empty set is an element of the power set of a. A set theory textbook can cover a vast amount of material depending on the mathematical background of the readers it was designed for. The overflow blog how the pandemic changed traffic trends from 400m visitors across 172 stack. Basically, the definition states it is a collection of elements. Basic concepts of set theory, functions and relations. Already in his famous \mathematical problems of 1900 hilbert, 1900 he raised, as the second.
Here is another set equality proof from class about set operations. To prove that p is true, assume that p is not true. 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. However, instead of establishing set equality using venn diagrams, we use logical symbols and. Browse other questions tagged elementarysettheory or ask your own question. Axiomatic set theory introduced to control these paradoxes. In maths, set theory was developed to explain about collections of objects. Discrete mathematicsset theoryexercises wikibooks, open. The empty set is a subset of a, hence it is an element of the power set of a.
Set theory has its own notations and symbols that can seem unusual for many. 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. Additional materials on logic contributed by eugenio omodeo and domenico cantone. Two sets are equal if and only if they have the same elements.
There are other conceptions of set, but although they have genuine mathematical interest they are not our concern here. Set theory tutorial problems, formulas, examples mba. The notation and symbols for sets are based on the operations performed on them. This text is for a course that is a students formal introduction to tools and methods of proof. The set t 2,3,1 is equal to s because they have the same members. 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. Proof by contradiction a proof by contradiction is a proof that works as follows. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. We also demonstrate how to work with families of sets.
Since this is a set theory text, we reverse the usual procedure by putting in the rst chapter some mathematics that will prove essential to the serious study of set theory. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. Pure set theory deals exclusively with sets, so the only sets under consideration are those whose members are also sets. The set of all books written about travel to chile. This is being written as a textbook for math 502, logic and set theory, and math 522, advanced set theory, at boise state university, on the practical level. Remember that our aim is not to learn more about the theory of numbers, most of the examples will be statements that you know are true, rather we are interested in the way that the proofs are.
Website devoted to highfalutin intellectual issues. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts. Vowels in the english alphabet v a, e, i, o, u first seven prime numbers. We apply the previous theorem with n2, noting that every rational number can be written as ba,whereband aare integers. Set theory basic theorems with many examples we discuss the basics of elementary set theory including set operations such as unions, intersections, complements, and cartesian products. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. How to write proofs in set theory with the math sorcerer. Aug, 2014 learn how to write proof of demorgans law. They are not guaranteed to be comprehensive of the material covered in the course. Mat231 transition to higher math proofs involving sets fall 2014 3 11. Since any element x in k is also in s, we know that every element x in k is also in s, thus k s. 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.
These elements could be numbers, alphabets, variables, etc. As we have seen from our examples, sets may contain a. The theory of the hereditarilyfinite sets, namely those finite sets whose. A book of set theory, first published by dover publications, inc. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student.
Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. We discuss the basics of elementary set theory including set operations such as unions, intersections, complements, and cartesian products. Corollary 19 the set of all rational numbers is countable. Basic set theory we will use the standard notation for containments. Selecting the material for presentation in this book often came down to deciding how much detail should be provided when explaining. Mar 12, 2014 the same sets as investigated in proof and problem solving sets example 04 are also examined here. A set may be defined by a membership rule formula or by listing its. Duplicates dont contribute anythi ng new to a set, so remove them. These objects are sometimes called elements or members of the set. Since the set of pairs b,a is countable, the set of quotients ba, and thus the set of rational numbers, is countable. The order of the elements in a set doesnt contribute anything new. In naive set theory, a set is a collection of objects called members or elements that is regarded as being a single object.
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. S contradiction but every element must either be in s or not in s. Proof theory was created early in the 20th century by david hilbert to prove the consistency of the ordinary methods of reasoning used in mathematics in arithmetic number theory, analysis and set theory. This book has been reprinted with the cooperation of kyung moon publishers, south korea. 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. In standard introductory classes in algebra, trigonometry, and calculus there is currently very little emphasis on the discipline of proof. This page covers set theory, common sets, venn diagrams, intersections and subsets. Set theory, branch of mathematics that deals with the properties of welldefined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions. Some examples of sets defined by describing the contents. Set theory is the mathematical theory of welldetermined collections, called sets, of objects that are called members, or elements, of the set. Set theory as a programming language, logic proof verifiers, topolgy. One may specify a set explicitly, that is by listing all the elements the set contains, or implicitly, using a predicate description as seen in predicate logic, of the form fx. A set can be defined by describing the contents, or by listing the elements of the set, enclosed in curly brackets.
Introduction to modern set theory judith roitman december 6, 2011 1. Proving equality between sets elementary set theory 6. A set is a collection of distinct objects, called elements of the set. To indicate that an object x is a member of a set a one writes x. That is, it is possible to determine if an object is to be included in the set or not. This alone assures the subject of a place prominent in human culture. The theory of numbers provides an excellent source for such examples. Set theory proof example proving b\a b intersect ac.
Sets are often also represented by letters, so this set might be. Instead, i am going to show you what can be done with sets. Mat231 transition to higher math proofs involving sets fall 2014 8 11. 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 always false. But even more, set theory is the milieu in which mathematics takes place today. When expressed in a mathematical context, the word statement is viewed in a. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. Let s all sets of type 2 set of all sets not containing themselves as elements s. In mathematics, the notion of a set is a primitive notion. While we usually list the members of a set in a standard order if one is available there is no. Based on the assumption that p is not true, conclude something impossible. Implicit descriptions tend to be preferred for in nite sets. Elementary set theory proofs mathematics stack exchange. Set theory \a set is a many that allows itself to be thought of as a one.