Such a set is characterized by a membership characteristic function which assigns to each object a grade of membership ranging between zero and one. A fuzzy set is a class of objects with a continuum of grades of membership. Every finite subset of r has both upper and lower bounds. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. Likewise there is a largest number smaller than all numbers in a called infa the in. This alone assures the subject of a place prominent in human culture. 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. Find the max, min, sup, and inf of the following set and prove your answer. 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. 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. Introduction one of the most important inequalities in the theory of incompressible uids is the socalled infsup condition, cf.
Dec 07, 2016 the definition of supremum and infimum of a set. The partially ordered set x, is a lattice if for all pairs x,y. They are not guaranteed to be comprehensive of the material covered in the course. We say that eis bounded above if there exists a real number u such that x ufor all x2e. The number m is called an upper bound for the set s.
Hence ais the union of an open set, inta, and a subset of the null set. For all complete lattice fand all set e, the set fe is a complete lattice, with for any a. An introduction to set theory university of toronto. In mathematics, the notion of a set is a primitive notion. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. 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. Consequently, the supremum is also referred to as the least upper bound or lub. Since a n is bounded, x nis a real number for each n 1. A series converges implies all the people who cant get another job eventually swallow their pride and become regulars too.
Introduction one of the most important inequalities in the theory of incompressible uids is the socalled inf sup condition, cf. Let u be a universe of discourse, and m a set included in u. The two red curves approach the limit superior and limit inferior of xn, shown as dashed black lines. The notions of inclusion, union, intersection, complement, relation, convexity, etc. But even more, set theory is the milieu in which mathematics takes place today. Let e be a subset of rn, and inte the set of all interior points of e. 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. Application to stokes equations references 1 1 2 4 5 8 1. The superior limit is the larger of the two, and the inferior limit is the smaller of the two. Let t, i, f be real standard or nonstandard subsets of the nonstandard unit interval. We will only say the supremum or infimum of a set exists if it is a finite. Set theory for computer science university of cambridge.
Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. Set theory this chapter describes set theory, a mathematical theory that underlies all of modern mathematics. In this sections, all the subsets of r are assumed to be nonempty. The lattice x, is a complete lattice if in addition, for all nonempty subsets s. Basic set theory a set is a many that allows itself to be thought of as a one. Since the set of pairs b,a is countable, the set of quotients ba, and thus the set of rational numbers, is countable. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. They are extensively used in real analysis, including the axiomatic construction of the real numbers and the formal definition of the riemann integral. A set t of real numbers hollow and filled circles, a subset s of t filled circles, and the infimum of s. 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. For those of you new to abstract mathematics elementary does not mean simple though much of the material. We say that eis bounded below if there exists a real number lso that x lfor all x2e. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student.
Set theory is indivisible from logic where computer science has its roots. Set theory, and set operations statistical science. The infimum and supremum are concepts in mathematical analysis that generalize the notions of minimum and maximum of finite sets. An illustration of limit superior and limit inferior. The boundary of the domain is only assumed to be lipschitz. A is bounded if it is bounded by both a sup m and an inf m. Corollary 19 the set of all rational numbers is countable. Then the inf is the number that would be the minimum if it could. 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. Set theory \a set is a many that allows itself to be thought of as a one. The supremum of a set is its least upper bound and the infimum is its greatest. Ill use the symbol r to denote the set of real numbers. In all of the examples considered above, the least upper bound.
Rd is the density of body forces acting on the uid e. Since the latter is always measurable, we conclude that ais a measurable set. Classes of subsets semialgebras, algebras and sigmaalgebras, and set functions duration. Thus f kg k 1 converges towards the in mum of its range. 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 this case, we say that uis an upper bound for e. Measure theory is not needed unless we discuss the probability of two types of events. S is called bounded above if there is a number m so that any x. For each particular choice of the spaces v h and w. We apply the previous theorem with n2, noting that every rational number can be written as ba,whereband aare integers. Inequalities and operations on functions are defined pointwise as usual. Suppose that aand b are subsets of r which are nonempty and bounded below. In this case, the sequence accumulates around the two limits. 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. A set sequence an converges if and only if liman liman. Note that for finite, totally ordered sets the infimum and the minimum are equal. Classes of subsets semialgebras, algebras and sigmaalgebras, and set functions.
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