Set theory is indivisible from logic where computer science has its roots. In this case, we say that uis an upper bound for e. A set t of real numbers hollow and filled circles, a subset s of t filled circles, and the infimum of s. Basic set theory a set is a many that allows itself to be thought of as a one.
Rd is the density of body forces acting on the uid e. A, the set a is then bounded from below and inf a exists. They are extensively used in real analysis, including the axiomatic construction of the real numbers and the formal definition of the riemann integral. This alone assures the subject of a place prominent in human culture. 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. Mar 03, 2004 similarly, the inf of a set sizesmallsame legaleesesize is the minimum of that set, unless youre back in that heartless world where the set doesnt have a minimum. A fuzzy set is a class of objects with a continuum of grades of membership. Since a n is bounded, x nis a real number for each n 1. The boundary of the domain is only assumed to be lipschitz. Since the latter is always measurable, we conclude that ais a measurable set. Let u be a universe of discourse, and m a set included in u.
S is called bounded above if there is a number m so that any x. Since the set of pairs b,a is countable, the set of quotients ba, and thus the set of rational numbers, is countable. Let t, i, f be real standard or nonstandard subsets of the nonstandard unit interval. We say that eis bounded above if there exists a real number u such that x ufor all x2e. The partially ordered set x, is a lattice if for all pairs x,y. Every finite subset of r has both upper and lower bounds. For all complete lattice fand all set e, the set fe is a complete lattice, with for any a. The superior limit is the larger of the two, and the inferior limit is the smaller of the two. Dec 07, 2016 the definition of supremum and infimum of a set. For those of you new to abstract mathematics elementary does not mean simple though much of the material.
The infimum and supremum are concepts in mathematical analysis that generalize the notions of minimum and maximum of finite sets. A set a of real numbers blue circles, a set of upper bounds of a red diamond and circles, and the smallest such upper bound, that is, the supremum of a. But even more, set theory is the milieu in which mathematics takes place today. In mathematics, the notion of a set is a primitive notion. We will only say the supremum or infimum of a set exists if it is a finite. They are not guaranteed to be comprehensive of the material covered in the course. A is bounded if it is bounded by both a sup m and an inf m.
Classes of subsets semialgebras, algebras and sigmaalgebras, and set functions duration. Hence ais the union of an open set, inta, and a subset of the null set. In all of the examples considered above, the least upper bound. The limits of the infimum and supremum of parts of sequences of real numbers are used in some convergence tests. This inequalityis fundamentalin the study of the stokesequations. An introduction to set theory university of toronto. Suppose that aand b are subsets of r which are nonempty and bounded below. Corollary 19 the set of all rational numbers is countable. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. Measure theory is not needed unless we discuss the probability of two types of events. Set theory this chapter describes set theory, a mathematical theory that underlies all of modern mathematics.
The two red curves approach the limit superior and limit inferior of xn, shown as dashed black lines. The number m is called an upper bound for the set s. Consequently, the supremum is also referred to as the least upper bound or lub. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions.
In this sections, all the subsets of r are assumed to be nonempty. In this case, the sequence accumulates around the two limits. Since inf ais a lower bound of aand inf bis a lower bound of b, the smaller of these two is a lower bound of ab. Note that since w 2w h the supremum is taken in a much smaller set and the discrete infsup condition bnb2 h does not follow from the in nite dimensional case bnb2. We apply the previous theorem with n2, noting that every rational number can be written as ba,whereband aare integers. 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.
Thus f kg k 1 converges towards the in mum of its range. The supremum of a set is its least upper bound and the infimum is its greatest. Application to stokes equations references 1 1 2 4 5 8 1. Set theory \a set is a many that allows itself to be thought of as a one. Sets may be described by listing their elements between curly braces, for example 1,2,3 is the set containing the elements 1, 2, and 3. For each particular choice of the spaces v h and w. Likewise there is a largest number smaller than all numbers in a called infa the in. A series converges implies all the people who cant get another job eventually swallow their pride and become regulars too. Such a set is characterized by a membership characteristic function which assigns to each object a grade of membership ranging between zero and one. Set theory for computer science university of cambridge. Set theory, and set operations statistical science. An illustration of limit superior and limit inferior. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory.
It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. We say that eis bounded below if there exists a real number lso that x lfor all x2e. Ill use the symbol r to denote the set of real numbers. Note that since w 2w h the supremum is taken in a much smaller set and the discrete inf sup condition bnb2 h does not follow from the in nite dimensional case bnb2. Classes of subsets semialgebras, algebras and sigmaalgebras, and set functions. The lattice x, is a complete lattice if in addition, for all nonempty subsets s. Note that for finite, totally ordered sets the infimum and the minimum are equal. Inequalities and operations on functions are defined pointwise as usual. Let e be a subset of rn, and inte the set of all interior points of e. Find the max, min, sup, and inf of the following set and prove your answer.
Introduction one of the most important inequalities in the theory of incompressible uids is the socalled inf sup condition, cf. Introduction one of the most important inequalities in the theory of incompressible uids is the socalled infsup condition, cf. A set sequence an converges if and only if liman liman. The notions of inclusion, union, intersection, complement, relation, convexity, etc.
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