In algebra, we know about the distributive property of multiplication over sums and. Additional materials on logic contributed by eugenio omodeo and domenico cantone. 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. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. A set gwith a associative binary operation is called a semigroup. A few elementary examples are the set of natural numbers. Halmos, naive set theory undergraduate texts in mathematics. 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. Set theory for computer science university of cambridge. This alone assures the subject of a place prominent in human culture. Complex issues arise in set theory more than any other area of pure mathematics. To indicate that an object x is a member of a set a one writes x. This frequency of occurrence of an outcome can be thought of as a probability.
Hauskrecht basic discrete structures discrete math study of the discrete structures used to represent discrete objects many discrete structures are built using sets sets collection of objects examples of discrete structures built with the help of sets. Therefore continuity is usually defined in terms of open sets. I offer no definition of what a set is beyond the intuitive notion described above. Bagaria and others published set theory find, read and cite all the research you need on.
Let us see the different types of symbols used in mathematics set theory with its meaning and examples. Probably the most important single example is the wikipedia. Set theory is also the most philosophical of all disciplines in mathematics. We can use these sets understand relationships between groups, and to analyze survey data. The most straightforward way to define a set is to say that it is an unordered collection of objects. Some examples of sets defined by listing the elements of the set.
But even more, set theory is the milieu in which mathematics takes place today. Many difficulties of cognition in linear algebra furthermore can be explained by the lack of the presence of the mastery of set theory concepts. In naive set theory, a set is a collection of objects called members or elements that is regarded as being a single object. 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. Algebraic set theory uses the methods of category theory to study elementary set theory. A set can be defined by describing the contents, or by listing the elements of the set, enclosed in curly brackets. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.
In this case, the order in which elements are combined does not matter. While we usually list the members of a set in a standard order if one is available there is no. This page covers set theory, common sets, venn diagrams, intersections and subsets. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory.
Duplicates dont contribute anythi ng new to a set, so remove them. Some common examples of sets include r \mathbbr r, the set of all real numbers, z \mathbbz z, the set of all integers, and the cantor set. Basic set theory a set is a many that allows itself to be thought of as a one. Also called boolean algebra or field of sets by some authors. Chapter 4 set theory \a set is a many that allows itself to be thought of as a one. Feb 11, 2014 modern set theory is it a religious belief system. The problems with defining sets led mathematicians to develop various systems for formalizing set theory. However, the open set definition is more general, and applies to many different structures in abstract algebra where distances such as. The set of all objects x which can be described by the sentence x is a man loosely speaking, the set of all men. This series of lessons cover the essential concepts of math set theory the basic ways of describing sets, use of set notation, finite sets, infinite sets, empty sets, subsets, universal sets, complement of a set, basic set operations including intersection and union of sets, using venn diagrams and simple applications of sets. Mathematics 144 set theory fall 2012 version ucr math.
Math 103, the collection of all even integers, these are all examples of sets of things. Two sets are equal if and only if they have the same elements. We will denote sets using capital letters a,b and the elements of the set using curly braces fg. Sets and set operations cs 441 discrete mathematics for cs m. Subsets, proper subsets, number of subsets, subsets of real numbers, examples and step by step solutions, notation or symbols used for subsets and proper subsets, how to determine the number of possible subsets for a given set, distinguish between elements, subsets and proper subsets. The purpose of this website is to link together current research in algebraic set theory and make it easily available. When expressed in a mathematical context, the word statement is viewed in a. Basics of probability theory when an experiment is performed, the realization of the experiment is an outcome in the sample space. Databases can be found in almost all software applications.
By countable additivity, any countable set in rn has measure zero. Linear algebra is one of the most applicable areas of mathematics. They are not guaranteed to be comprehensive of the material covered in the course. Hopefully this encourages you to believe that abstract algebra may be of serious use both inside and outside mathematics, and indeed, it is so, in addition to being a fascinating and beautiful theory in its own right for those so inclined.
These will be the only primitive concepts in our system. The most popular of these is zfc, where the z and f stand for zermelo and frankel, two mathematicians who developed these axioms, and the c stands for the axiom of choice, one of the axioms involved. The intersection of the set with members 1, 2, 3 together with the set with members 3, 4, 5 is the set that has only the 3. Complete set of operations some operators can be expressed in terms of others e. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. It is hoped that this will encourage and facilitate further development of the subject. Other examples of associative binary operations are matrix multiplication and function composition. A set may be defined by a membership rule formula or by listing its. It is used by the pure mathematician and by the mathematically trained scientists of all disciplines. Abstract algebra and discrete mathematics, set theory. A set is a collection of distinct objects, called elements of the set. Abstract algebra is the study of algebraic systems in an abstract way. The two definitions agree, and you can use which ever one is convenient. Practice exercises stitzzeager prerequisites pages 14.
This chapter introduces set theory, mathematical in duction. Such as, sets could be a collection of odd numbers, even numbers, natural numbers, whole numbers, real or complex numbers and all the set of numbers which comes in the number line. As we have seen from our examples, sets may contain a. These objects are called the elements of the set, and they can be organized into smaller collections called subsets. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. Set theory is indivisible from logic where computer science has its roots. 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. Instead, i am going to show you what can be done with sets. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely.
Hauskrecht basic discrete structures discrete math study of the discrete structures used to represent discrete objects many discrete structures are built using sets sets collection of objects examples of. Our approach is based on a widely used strategy of mathematicians. The set definition above is spoken the set of twice. 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. In mathematics, the notion of a set is a primitive notion. Introduction basic probability general ani probability space. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. A group is a set of elements together with an operation that combines any two of its elements to form a third element also in the set while satisfying four conditions called the group axioms, namely closure, associativity, identity and invertibility. The order of the elements in a set doesnt contribute. If the experiment is performed a number of times, di.
Rs chapter 1 random variables 6142019 1 chapter 1 probability theory. A set theory textbook can cover a vast amount of material depending on the. Set theory as a programming language, logic proof verifiers, topolgy. Basic concepts of set theory, functions and relations. 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. Axioms and set theory mathematics university of waterloo. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. Note that in the second identity, we show the number of elements in each set by the corresponding shaded area. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. The set t 2,3,1 is equal to s because they have the same members.
Dedekind is mostly known for his research in algebra and set theory. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. If you were to let the set with 1,2,3 be set a, and the set with 3,4,5 be set b, then you could use venn diagrams to illustrate the situation see figure 2. Set theory in cognition of linear algebra concepts clearly set theory language is an essential prerequisite for the cognition of linear algebra concepts. The singleton points in rn, and lines and curves in rn, n. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Examples the set of all objects x which satisfy the condition x is an irrational number and 0 x 1 loosely speaking, the set of all irrational numbers between 0 and 1. Website devoted to highfalutin intellectual issues. Set theory and algebra in computer science a gentle. For those of you new to abstract mathematics elementary does not mean simple though much of the material. It is natural for us to classify items into groups, or sets, and consider how those sets overlap with each other.