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 important properties of sets, and give examples. Laws in set theory know set theory laws with solved examples. Some examples of sets defined by listing the elements of the set. Orderedpairs after the concepts of set and membership, the next most important concept of set theory is the concept of orderedpair. A complement, or not a is the set of all elements not in a. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. In cases where one wishes to not to have this, the term proper subset is meant not to have this possibility. This alone assures the subject of a place prominent in human culture. 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. When expressed in a mathematical context, the word statement is viewed in a. Implicit descriptions tend to be preferred for in nite sets. Introduction to logic and set theory 202014 bgu math. Set theory provides both general descriptions that entail all members of a set and the representations for specific elements of the set. For example, suppose that committee a, consisting of the 5 members jones, blanshard, nelson, smith, and hixon.
Let a, b, c be three sets as shown in the following venn diagram. If you get a number larger than 11 or smaller than 0, add or subtract 12 to get a valid pitch class number. The selfconsidering object in the set theory was existing too, an example numbers. Learn the basic concepts and formulas of set theory with the help of solved examples. The set of all books written about travel to chile. Set theory for computer science university of cambridge. The algebra of sets defines the properties and the set theoretic operations of union, laws in sets theory, intersection, and relations and the complementation of set inclusion and set equality. For example, one can construct sets over the domain of natural numbers, or over the domain of persons, or whatever.
While we usually list the members of a set in a standard order if one is available there is no. Set theory basic concepts of set theory hitbullseye. From this example, it is clear that a set is a subset of itself. Thus, three features are the full integration into the text of the study of models of set theory, the use of illustrative examples both in the text and and in the exercises, and the integration of. To transpose a set up by n half steps, add n to each pitch class in the set. Set operations and venn diagrams university of hawaii. This text contains all information relevant for the exams. These objects are sometimes called elements or members of the set. We define uniform notations and convention for such abbreviation terms. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. An inefficient method would be to test each value from 2 to 165 330 2 to see if each is a divisor of 330. Set symbols of set theory and probability with name and definition. 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.
For example, if we regard the collection of all sets as a set, we will get into theoretical troubles. Inversional symmetry this is a property of a set which is able to map onto itself through inversion at some value of n. Axioms and set theory mathematics university of waterloo. 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. A set is a collection of objects, nothing more and nothing less. Set theory some basics and a glimpse of some advanced techniques.
From this simple beginning, an increasingly complex and useful. Basic concepts of set theory, functions and relations. For example, tall people is not a set, because people tend to disagree. The set difference a minus b is the set of elements that are in a, with those that are in b subtracted out. Bagaria and others published set theory find, read and cite all. This means that our description of the elements of a set is clear and unambiguous. Discrete mathematicsset theory wikibooks, open books. Some examples of sets defined by describing the contents. Set theory tutorial problems, formulas, examples mba. Set theory considers a universe of sets which is ordered by the membership or element relation all other mathematical objects are coded into this universe and studied within this framework. Compare this to the concept of modes of limited transposition found in messiaens theory. Any learner with insufficiency in hisher knowledge of set theory may struggle to distinguish the descriptions of general forms from the specific member representations. In this method a set is described by listing elements, separated by commas, within braces. Drawing venn diagrams can help you to visualise, but they do not count as formal proofs.
In more general framework, such a collection is named a proper class. We want to see the notation x, y as an example of a class term. Laws in set theory is essential for solving certain problems in cat quantitative aptitude. Of sole concern are the properties assumed about sets and the membership relation. Let us see the different types of symbols used in mathematics set theory with its meaning and examples. Set theory is the axiomatization of mathematics, and without ac noone seriously doubts its truth, or at least consistency.
We have already dealt with the notion of unorderedpair, or doubleton. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. Set theory is indivisible from logic where computer science has its roots. Set theory some basics and a glimpse of some advanced. In mathematics, the notion of a set is a primitive notion. In contrast to naive set theory, the attitude adopted in an axiomatic development of set theory is that it is not necessary to know what the things are that are called sets or what the relation of membership means. The big questions cannot be dodged, and students will not brook a flippant or easy answer. 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. Set of natural numbers 1,2,3, set of whole numbers 0,1,2,3, each object is called an element of the set. The set that contains all the elements of a given collection is called the universal set and is represented by the symbol, pronounced as mu. These will be the only primitive concepts in our system. They are not guaranteed to be comprehensive of the material covered in the course. Set theory is also the most philosophical of all disciplines in mathematics. In this way, set theory is one of the foundations of mathematics.
Complex issues arise in set theory more than any other area of pure mathematics. It sounds simple, but set theory is one of the basic building blocks for higher mathematics, so it helps to understand the basics well. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. The set t 2,3,1 is equal to s because they have the same members. For exam ple, the set t described in example a is in s since t is not an element of itself. Vowels in the english alphabet v a, e, i, o, u first seven prime numbers. Formal set notation description informal english description. A union b is the set of all elements that are in a, or b, or both.
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. For each of the following sets, draw a venn diagram and shade the area representing the given set. But even more, set theory is the milieu in which mathematics takes place today. We say that two sets a and b are mutually exclusive if, that is, the sets have no elements in common. Describe the following sets in both formal and informal ways. Know about the important set theory rules and check the set theory examples based on concepts set theory. Examples of discrete structures built with the help of sets. Example example simplify ab c \a a\bc using the laws of set algebra. Questions are bound to come up in any set theory course that cannot be answered mathematically, for example with a formal proof. We study two types of relations between statements, implication and equivalence. Here is another set equality proof from class about set operations.
To transpose a set down by n half steps, subtract n from each pitch class in the set. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. This page sets out the principles of sets, and the elements within them. For example, if we say that a and b are exhaustive with respect to c. For example, tall people is not a set, because people tend to disagree about what tall means. Informal set theory begins with an existing domain of objects, presumed not to be sets, and constructs all sets over that domain. Set theory and relations set theory a set is well defined class or collection of objects. As long as we consider collections of subsets of a set a, its. By contrast, pure set theory assumes no preexisting domain, but. For example, the set s of all teams in a particular hockey league. Math1081 revision set theory, number theory and graph. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. Set theory and relations studypivot free download dpp.
A set is often described in the following two ways. Bread a union b or the union of a and bis defined as the set that consists of all elements belonging to either set a or set b or both. Let us substitute v2 for all free occurrences of v1 in the formula. 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. The set membership symbol is often used in defining operations that manipulate sets.