Mathematics
Sets Math
In mathematics, a set is a collection of distinct objects, considered as an object in its own right. These objects can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are fundamental to various mathematical concepts, including functions, relations, and probability. They are often represented using curly braces and commas.
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11 Key excerpts on "Sets Math"
eBook - PDFModern Mathematics
Made Simple
- Patrick Murphy(Author)
- 2014(Publication Date)
- Made Simple(Publisher)
1 1 SETS The language of sets, together with the simplicity of combining sets according to elementary rules, gives us the opportunity to examine basic mathematical ideas and to illustrate some aspects of mathematical language which are common to all its branches. The study of sets enables us to attach mathemat-ical overtones to everyday terms and experiences which are not always appa-rent in normal conversation. This is not to turn us into bores but more to begin to make us appreciate the definitive nature of our subject, for while much may be discovered by comparisons with the real world the fact remains that the activity of mathematics is a constant search for, and identification of, its results. A subsequent classification of those elements from which the results are derived is the starting point of almost any mathematical thinking. Definition 1.1. A set is a collection of distinct objects. If it is always possible to decide whether or not any object whatsoever belongs to the collection then we say that the set is well defined. The objects which belong to the set are called the elements of the set. Two sets are considered to be the same if they contain exactly the same objects, in which case we also say that the sets are equal to each other. Notation Usually a set will be represented by capital letters like A, K, P, X, and the elements of a set by what are called lower-case letters, such as b, m, n, t. A set may be defined by enclosing a list of its elements, in any order, between two brackets { and }.
eBook - PDF- N. Ya. Vilenkin, D. Allan Bromley, Nicholas Declaris, W. Magnus(Authors)
- 2014(Publication Date)
- Academic Press(Publisher)
Indeed, this explains the extraordinary breadth of the theory of sets and its applicability to the most varied branches of knowledge (mathematics, mechanics, physics, biology, linguistics, etc.). For mathematics, of course, the sets composed of mathemat-ical objects'' play an especially important role; among these mathematical objects are geometric figures, algebraic equa-tions, functions, etc. Some of these sets are involved in elementary mathematics, but the word set is usually lacking there (this is explained simply if we recall that the most modern part of elementary mathematics came into being at the end of the 17th century, while the theory of sets is a child of the 19th century). Indeed, in elementary mathematics we encounter sets at every turn. Sets of numbers, i.e., sets composed of numbers, are met with especially frequently. As examples of such sets we can take: (a) the set of all natural numbers, (b) the set of all integers (positive, negative, and zero), (c) the set of all rational numbers, (d) the set of all real numbers, (e) the set of all complex numbers. Subsets 27 Two kinds of sets turn up in geometry. First of all, in geometry we ordinarily talk about the properties of some set of geometric figures. For example, the theorem stating that the diagonals of a parallelogram bisect each other relates to the set of all parallelograms. Secondly, the geometric figures are themselves sets composed of the points occurring within them. We can therefore speak of the set of all points contained within a given circle, of the set of all points within a given cone, etc. In algebra we meet such sets as the set of all polynomials in two variables, the set of all quadratic equations, the set of all roots of a given equation, etc. In other words, almost every part of elementary mathematics is connected with the theory of sets in some way or other. Subsets The concept of set turns out to be very useful in mathe-matics.
eBook - PDF- Valentin Deaconu, Donald C. Pfaff(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
3 Elementary theory of sets The notion of a set is a relatively recent development in mathematical history. Created at the end of 19th century, the idea of a set has become the cornerstone for virtually all contemporary mathematics. In its simplest form, a set is simply a bunch of things gathered together to form a new entity which is considered to be a single object. The English language is rich in words that could be regarded as synonymous with the word “set”; among them are collection, group, family, class, club, flock, herd, or team. Note that when we refer to a club, for example, we are not generally thinking of the individual members, but of the totality of all members, presumed to be a single entity. In fact, a club (or team, family, class, etc.) is often referred to in the singular. We try to find a balance between a naive point of view of set theory and an axiomatic point of view. We only discuss the axiom of extent, which tells us when two sets are the same, and the axiom of separation, which enables us to form new sets from old ones. Later we will add the axiom of choice. To avoid paradoxes, like Russell’s paradox, we will assume that our sets are subsets of a fixed big set named universe. We will talk about subsets of a given set, then we will define the power set and operations with sets like union, intersection, complement, difference and symmetric difference. Finally, we introduce ordered pairs and define the Cartesian product of sets. 3.1 Axioms for set theory We will usually denote the sets by capital letters and we will mostly be inter- ested in sets of mathematical entities like numbers, functions, etc. The fact that an element a belongs to a set A is written a ∈ A. The negation of a ∈ A is written a / ∈ A. The sets can be specified by the roster notation using braces, like A = {a, b, c}, an enumeration of its elements, or by the set-builder no- tation, which will be explained below after the Axiom of Separation.
eBook - PDF- Ed Wheeler, Jim Brawner(Authors)
- 2013(Publication Date)
- Information Age Publishing(Publisher)
Sets, Functions, and Sequences Chapter 2.1 Sets 2.2 Relations 2.3 Functions 2.4 Sequences 2.5 Recursive Sequences 2 4 6 C H A P T E R 2 : S E T S , F U N C T I O N S , A N D S E Q U E N C E S 2.1 Sets Over a hundred years ago, the German mathematician Georg Cantor borrowed the everyday word set to describe a specific mathematical concept. The language of sets that he introduced was particularly useful in finding a precise way to talk about infinite quantities. His work provided much of the grammar and vocabulary of modern mathematics. The “new math” that was introduced in the 1960s emphasized the language of sets to such a degree that many critics feared children and adults beginning their study of mathematics would be lost in a sea of abstract symbols. Over time, however, it has become clear that the language of sets can be a very useful mode of representation that clarifies ideas across a broad spectrum of introductory mathematics. Perhaps much of the backlash in response to the “new math” could have been avoided if more emphasis had been placed on showing how set language and other abstractions can be useful tools for clarifying or drawing connections among more “traditional” mathematical ideas. The basic vocabulary of sets explored in this section, probably familiar already to many students, should not be considered as a topic in isolation. Rather, it will be used throughout the rest of the book to provide a clearer picture of the topics that follow. D EFINING S ETS What is a set? We define a set to be a collection of objects; the objects themselves are called the elements (or members ) of the set. We typically use a capital letter such as A , B , or C to represent a set and use a lower-case letter such as a , b , or c to represent an element of the set. We use the notation ∈ a A to mean that the object a belongs to (or is an element of ) the set A . If a is not an element of the set A , we write ∉ a A .
eBook - PDFProofs and Ideas
A Prelude to Advanced Mathematics
- B. Sethuraman(Author)
- 2021(Publication Date)
- American Mathematical Society(Publisher)
5 Sets and Functions In Chapter 3 we studied the first key ingredient of mathematical vocabulary, namely statements. Here we will study the second and third key ingredients: sets, and the functions between them. Together, these three form the bedrock of mathematics. In fact, according to one viewpoint, mathematics can be described most broadly as the discovery of theorems (true statements) about sets and the functions between them! As in Chapter 3, our goal here is to understand sets and functions in the intuitive way a mathematician understands them. Any deeper study (particularly of sets) will involve venturing into the field of mathematical logic. At this introductory level, we will not go there. Accordingly, we will use what may be termed a naive, as opposed to axiomatic, approach to sets. But the term naive does not mean that our approach is somehow de-ficient for the purpose of doing mathematics. It is not! It is just that in certain technical situations that we will not confront at this beginning level, our intuitive approach can lead to internal contradictions. At that point we will need to adjust our approach with a firm set of axioms. Once we firm up our axioms and revisit what we do here, we will find that everything we do is correct after all! It is harmless therefore to proceed along more intuitive lines at this stage. 5.1 Sets Definition 5.1. A set is a collection of objects. The members of the collection are called elements of the set. When an object ? is an element of a set ? , we write ? ∈ ? . As with statements (see Example 3.2 (3) of Chapter 3), objects must be unambiguous and well-defined. Example 5.2. Let us consider some examples: (1) We have considered the set of integers several times already, traditionally denoted ℤ . We write this as ℤ = {0, ±1, ±2, ±3, ... } . Note the curly braces at the beginning 69 70 Sets and Functions and the end. The dots indicate that the set is built up by continuing the obvious pattern.
eBook - PDF- Charles Roberts(Author)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
Using set notation in mathematics promotes precision and clarity in communicating mathematical ideas. The theory of sets also provides a means of simplifying and unifying a large number of subdisciplines of mathematics. In the late nineteenth century, Cantor defined the term “set.” This defini-tion led to a paradox known as Russell’s paradox. Giving a formal definition of a set ultimately leads to circularity of definition, since a set probably would be defined as a “collection,” a collection would be defined as an “aggregate,” and so forth until we eventually circle back to the word “collection” or “ag-gregate.” The modern method of developing mathematical theories such as the theory of sets is the axiomatic approach. In this approach to set theory, the terms “set” and “is an element of” are undefined terms—just as “lines,” “points,” and “intersects” are undefined terms in geometry. Intuitively, a set consists of objects called elements or members . Furthermore, a set is well-defined—that is, given a specific object, it is possible to determine if the object belongs to a given set or not. We will use the notation “ x ∈ A ” to denote that “ x is an element of the set A ,” “ x is a member of A ,” or, sim-ply, “ x is in A .” To indicate “ x is not an element of the set A ,” we write symbolically “ x / ∈ A .” Usually sets are described in roster notation , in which the elements of the set are enclosed in curly braces, { } , and separated by commas, or in set-builder notation , in which the set is specified in the form { x | P ( x ) } , which is read “the set of all x such that P ( x ) is true.” For example, “the set of all natural numbers less than 6” is specified in roster notation as { 1 , 2 , 3 , 4 , 5 } and in set-builder notation as { x | x ∈ N and x < 6 } . There is one and only one set which has no elements. It is called the empty set or null set . The empty set is denoted by the symbol ∅ . The following is a formal definition of the empty set.
eBook - PDF- Peter Grossman(Author)
- 2017(Publication Date)
- Red Globe Press(Publisher)
Sets and relations 5.1 Sets In this chapter, we will present some basic mathematical ideas about sets and relations. Some of the material on sets may be familiar to you already, in which case you may wish to scan over those sections fairly briefly. The main reason for introducing sets is to provide some useful terminology and notation for the work that follows; we will not be studying the mathematical theory of sets as such. Relations arise in computing in the theory of relational databases, and we will need them in Chapter 12 when we study congruences. The word set is used in mathematics to mean any well defined collection of items. The items in a set are called the elements of the set. For example, we can refer to the set of all the employees of a particular company, the set of all ASCII characters 1 , or the set of all the integers that are divisible by 5. A specific set can be defined in two ways. If there are only a few elements, they can be listed individually, by writing them between braces (‘curly’ brackets) and placing commas in between. For example, the set of positive odd numbers less than 10 can be written in the following way: {1, 3, 5, 7, 9} If there is a clear pattern to the elements, an ellipsis (three dots) can be used. For example, the set of odd numbers between 0 and 50 can be written: {1, 3, 5, 7, ..., 49} Some infinite sets can also be written in this way; for example, the set of all positive odd numbers can be written: {1, 3, 5, 7, ...} 74 CHAPTER 5 set element 1 The ASCII character set is a standard set of 128 characters, including letters, digits, punctuation marks, mathematical symbols and non-printing (control) characters. Each character has a unique numeric code from 0 to 127. The 8-bit binary equivalent of the numeric code is used in many computers for the internal representation of character data. (ASCII stands for American Standard Code for Information Interchange.)
eBook - PDFSets, Functions, and Logic
An Introduction to Abstract Mathematics, Third Edition
- Keith Devlin(Author)
- 2018(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 3 Set Theory 3.1 Sets The concept of a set is extremely basic and pervades the whole of present-day mathematical thought. Any well-defined collection of objects is a set. For instance we have: • the set of all students in your class • the set of all prime numbers • the set whose members are you and my left ear All it takes to determine a set is some way of specifying the collection. If A is a set, then the objects in the collection A are called either the members of A or the elements of A . (With a concept as simple as a set, there is no way to avoid such circular definitions: but it should be clear what is meant.) We write x ∈ A to denote that x is an element of A . Some sets occur frequently in mathematics, and it is convenient to adopt a standard notation for them: N : the set of all natural numbers (i.e., the numbers 1, 2, 3, etc.) Z : the set of all integers (i.e., 0 and all positive and negative whole numbers) Q : the set of all rational numbers (i.e., fractions) R : the set of all real numbers R + : the set of all non-negative real numbers Thus, for example, x ∈ R means that x is a real number. And (x ∈ Q ) ∧ (x > 0 ) means that x is a positive rational number. 57 58 Set Theory There are several ways of specifying a set. If it has a small number of elements we can list them. In this case we denote the set by enclosing the list of the elements in curly brackets; thus, for example, { 1 , 2 , 3 , 4 , 5 } denotes the set consisting of the natural numbers 1, 2, 3, 4, and 5. By use of “dots” we can extend this notation to any finite set, e.g., { 1 , 2 , 3 , . . ., n } denotes the set of the first n natural numbers. Again { 2 , 3 , 5 , 7 , 11 , 13 , 17 , . . ., 53 } could (given the right context) be used to denote the set of all primes up to 53. Certain infinite sets can also be described by the use of dots (only now the dots have no end), e.g., { 2 , 4 , 6 , 8 , . . ., 2 n, . . . } denotes the set of all even natural numbers.
eBook - PDF- Karl Smith(Author)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Every language has an abundance of words that mean “a collection” or “a grouping.” For example, we speak of a herd of cattle, a flock of birds, school of fish, a track team , a stamp collection , and a set of dishes. All these grouping words serve the same purpose, and in mathematics we use the word set to refer to any collection of objects. The study of sets is sometimes called set theory, and the first person to formally study sets was Georg Cantor (1845–1918). In this chapter, we introduce many of Cantor’s ideas. Denoting Sets In Section 1.2, we introduced the idea of undefined term. The word set is a perfect exam-ple of why there must be some undefined terms. Every definition requires other terms, so some undefined terms are necessary to get us started. To illustrate this idea, let’s try to define the word set by using dictionary definitions: “Set : a collection of objects.” What is a collection? “ Collection : an accumulation. ” What is an accumulation? “Accumulation : a collection , a pile, or a heap .” We see that the word collection gives us a circular definition. What is a pile ? “ Pile : a heap .” What is a heap? “Heap : a pile .” Do you see that a dictionary leads us in circles? In mathematics, we do not allow circular definitions, and this forces us to accept some words without definition. The term set is undefined. Remember, the fact that we do not define set does not prevent us from having an intuitive grasp of how to use the word. Sets are usually specified in one of two ways. The first is by description , and the other is by the roster method. In the description method, we specify the set by describing it in such a way that we know exactly which elements belong to it. An example is the set of 50 states in the United States of America. We say that this set is well defined, since there is no doubt that the state of California belongs to it and that the state of Germany does not; neither does the state of confusion.
- Michael L. O'Leary(Author)
- 2015(Publication Date)
- Wiley(Publisher)
CHAPTER 3 SET THEORY 3.1 SETS AND ELEMENTS The development of logic that resulted in the work of Chapters 1 and 2 went through many stages and benefited from the work of various mathematicians and logicians through the centuries. Although modern logic can trace its roots to Descartes with his mathesis universalis and Gottfried Leibniz’s De Arte Combinatoria (1666), the beginnings of modern symbolic logic is generally attributed to Augustus De Morgan [Formal Logic (1847)], George Boole [Mathematical Analysis of Logic (1847) and An Investigation of the Laws of Thought (1847)], and Frege [Begriffsschrift (1879), Die Grundlagen der Arithmetik (1884), and Grundgesetze der Arithmetik (1893)]. How- ever, when it comes to set theory, it was Georg Cantor who, with his first paper, “Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen” (1874), and over a decade of research, is the founder of the subject. For the next four chapters, Cantor’s set theory will be our focus. A set is a collection of objects known as elements. An element can be almost any- thing, such as numbers, functions, or lines. A set is a single object that can contain many elements. Think of it as a box with things inside. The box is the set, and the things are the elements. We use uppercase letters to label sets, and elements will usu- A First Course in Mathematical Logic and Set Theory, First Edition. Michael L. O’Leary. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc. 117 118 Chapter 3 SET THEORY ally be represented by lowercase letters. The symbol ∈ (fashioned after the Greek letter epsilon) is used to mean “element of,” so if A is a set and a is an element of A, write ∈ aA or, the more standard, a ∈ A. The notation a, b ∈ A means a ∈ A and b ∈ A. If c is not an element of A, write c ∉ A. If A contains no elements, it is the empty set. It is represented by the symbol ∅. Think of the empty set as a box with no things inside.
- John P. Mayberry(Author)
- 2001(Publication Date)
- Cambridge University Press(Publisher)
This semantic requirement is the minimum necessary if our discourse is to have the rigour and precision needed for definition and proof. But what things, or what kinds of things, if any, satisfy this require- ment? That is the ontological problem, and it is most unlikely that we shall ever find a definitive solution to it. Fortunately, we do not need a definitive solution. The modern concept of set derives from the ancient concept of number (arithmos). A set is a plurality, finite in multitude, composed of determi- nate, definite things, that is to say, of objects, in the technical, Fregean sense of "object" I have adopted here. The simplest, and most straight- forward examples of sets are those composed of two or more individuals, 70 Semantics, Ontology, and Logic that IS to say, objects that are not themselves sets 3 . In such sets, the individuals play the role of Euclid's units (Elements VII, Definition 1). But Euclid makes no attempt to catalogue the various kinds of things that make suitable units, nor should we attempt to catalogue the kinds of things that are suitable to play the role of individuals. For the purposes of mathematics this is not necessary. Indeed to do so would be most ill-advised, for mathematics ought to be onto logically neutral, insofar as that is possible. The mathematician has absolutely nothing to gain, in his capacity as mathematician, by committing himself to the existence of individuals of certain particular kinds. As Aristotle says, he need only posit the unit, or the individual 4 . "Positing" doesn't mean "inventing" or "conjuring out of thin air" here: it means rather "accepting as given without further specification or analysis". The mathematician should regard individuals, not as un- specifiable, or abstract in the manner of the "pure units" that make up Plato's "mathematical numbers", but as unspecified.
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