P.R. Halmos. Naive Set Theory. Springer Verlag, 1974.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....r # and its usual properties follow immediately. 2.6. Application: The Schr oder Bernstein Theorem The Schroder Bernstein Theorem plays a vital role in the theory of cardinal numbers. If there are two injections f : X X, then the Theorem states that there is a bijection h : X Y . Halmos [11] gives a direct but complicated proof. Simpler is to use the Knaster Tarski Theorem to prove a key lemma, Banach s Decomposition Theorem [7] Recall from 1.2 the image and converse operators. These apply to functions also, because functions are relations in set theory. If f is an injection then ....

Halmos, P. R., Naive Set Theory,Van Nostrand, 1960

....which we can define concisely: split(p, x y . x) split(p, x y . y) Like other destructors, split is defined using a description: split(p, f) #z . #xy.p= #x, y##z = f(x, y) 4.4 Cartesian products The set A B consists of all pairs b# such that a B. Many authors [8, 22] define the Cartesian product in a cumbersome manner. If a A and B then a , a, b B) so they define A using Separation: B # z # B) x#A . y#B .z= There is a historical and pedagogical case for this definition, which postpones the introduction of Replacement. ....

Paul R. Halmos. Naive Set Theory. Van Nostrand, 1960.

....relation R can be composed of all the R s. The paper [36] provided a major motivation for this work. The early results have been presented in [22] The paper is a revised version of [23] We assume that the reader is familiar with such concepts as function, relation, Cartesian product, etc. [12, 40]. The standard mathematical notation is used throughout the paper. In Section 2, we introduce tabular expressions of relations, present six topologically di erent types of cell connection graphs, and give the de nition of tabular expression (or table) as 6 tuple. In section 3, we elaborate on ....

....nodes. Let 7 be a relation 7 Components Components satisfying: 8A; B 2 Components A 7 B ) A = G B = G) A 6= B) 1) In other words, each arc that represents 7 must either start from or end at the grid G. The relation 7 , transitive and re exive closure of 7 , is a partial order [12], so we can talk about both maximal and minimal elements w.r.t. 7 . The relation 7 is a cell connection graph if 1. A is maximal w.r.t. 7 = A 2 V alues, 2. A is minimal w.r.t. 7 = A 2 Guards, 3. 8A 2 Guards(T ) 8B 2 V alues(T ) A 7 B. The cell connection graph 7 ....

P. R. Halmos, Naive Set Theory, Springer 1960.

....cation, and nal, comments are in Chapter 10. The paper [24] provided a major motivation for this work. The early results have been presented in [14] The paper is a revised version of [15] We assume that the reader is familiar with such concepts as function, relation, Cartesian product, etc. [7, 28]. The standard mathematical notation is used throughout the paper. 2 Raw Table Skeleton Intuitively, a table is an organized collection of sets of cells, each cell contains an appropriate expression. Such an organized collection of empty cells, without expressions, will be called a (raw or ....

.... is an asymmetric relation 7 Components(T ) Components(T ) satisfying: 8A; B 2 Components(T ) A 7 B ) A = G B = G) A 6= B) 1) plus a decomposition of Components(T ) into Guards(T ) and V alues(T ) The relation 7 , transitive and re exive closure of 7 , is a partial order [7]. A component A 2 Components(T ) is maximal if A 7 B implies B = A for every B 2 Components(T ) Similarly A 2 Components(T ) is minimal if B 7 A implies B = A for every B 2 Components(T ) A component A 2 Components(T ) is neutral if it is neither minimal nor maximal. The relation 7 ....

P. R. Halmos, Naive Set Theory, Springer 1960.

....cell connection graphs. 3 Relations The tables were designed to represent relations, with functions as a special case. Formally, a relation R is a subset of Cartesian product of the set X and the set Y . The concept of Cartesian product has two equivalent but different meanings in mathematics [6, 16], and we will make use of this differentiation in this paper. The relations functions that are represented by tables are defined on IN Theta OUT, where IN and OUT are the sets of input and output values respectively. However the sets IN and OUT are frequently themselves the products of different ....

....k g (X 1 Theta : Theta Xm ) X i 1 Theta : Theta X i k , if fi 1 ; i k g f1; 2; mg, a i 2 X i , for i = 1; m. We shall write t (Y ) instead of ftg (Y ) for t 2 T , and then t (Y ) is a projection of Y on the t th coordinate. All the above definitions are classical [6, 16]. The new concepts are below. t2T X t , and B be a set. We write B v A ( 9S T : B S (A) Clearly B A ) B v A. For example, if A X 1 Theta X 2 Theta X 3 Theta X 4 and B X 2 Theta X 4 , then B v A ( 8(x 2 ; x 4 ) 2 B:9x 1 2 X 1 ; x 3 2 X 3 : x 1 ; x 2 ; x 3 ; x 4 ) 2 A: Let ....

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P. R. Halmos, Naive Set Theory, Springer 1960.

....in Zermelo Fr ankel set theory. Therefore, previous formal semantics to OCL ( 14, 15, 4] has been based on a parameterized semantics approach, i.e. the semantic function is parameterized by an arbitrary, but fixed diagram C, and its definition is based on naive set theoretic reasoning [18]. Even if one is not too much concerned about the foundational problems with this approach, we argue that parametric semantics does not cover the most important aspect of object orientation: reuseability. In principle, if one extends a class diagram C by a diagram extension E (containing new ....

Halmos, P.R.: Naive Set Theory. van Nostrand (1979)

....is conservative to add a bijection of the ACL2 universe with the natural numbers. 2.3 Total Orders From a Set Theoretic Perspective In this section we review some well know facts about total orders in a purely set theoretic setting. Good references include the books by Kunen, Devlin, and Halmos [11, 3, 4] and Part B of the Handbook of Mathematical Logic [1] especially chapter B.2 on the Axiom of Choice [5] This section can be skipped without impacting readability of the rest of this paper. Recall the Axiom of Choice, which (among many equivalent formulations) can be stated as follows: every set ....

P. R. Halmos. Naive Set Theory. Van Nostrand, 1960.

.... a set X is just the usual identity map x 7 x, and the composition operator is the ordinary composition of mappings (so if (ff; fi) are composable, fi ffi ff is defined by mapping first by ff and then by fi) The collection of all sets is too big to itself be a set; it is a proper class (see [10]) Categories whose objects form a set (rather than a proper class) are called small categories. In many categories, the objects are sets with structure and the arrows are structure preserving mappings between objects . This structure is often algebraic in nature. Such categories are called ....

....T = Set we recover the familiar fact the power set of a set is a lattice (under union and intersection) and that B is a lattice. However, in the latter case we know that in fact both powersets and B are in fact Boolean algebras. This means that we can model the elementary calculus of propositions [10][23] implication relation together with the operations of disjunction, conjunction, and negation) using the calculus of subsets (inclusion relation together with the operations of union, intersection and complementation) Let us recall very briefly how this works. 1. The elements 0; 1 of B may be ....

Paul R. Halmos, Naive Set Theory. New-York, Springer-Verlag, 1974.

.... induction principle, then we can 8 use the equations in the constructor destructor principle to define functions Label and Children as required and we can deduce the recursion principle by adapting a well known argument for justifying definition by recursion over the natural numbers (e.g. see [3]) So it suffices to define T (X) and Node and show monotonicity and the induction principle. We describe the construction of T (A) for a concrete set A, leaving it to the reader to check that the construction can be captured in a typable operator T (X) To define T (A) we first construct the ....

Paul R. Halmos. Naive Set Theory. Springer-Verlag, 1974.

....end can give di erent results, in the sense of having di erent properties as orders, not all the arithmetic identities for natural numbers still hold for ordinals (commutativity of addition fails for example) and we have to be more careful about the arguments used for recursive de nitions. Halmos [12] gives a good account of the ideas involved. In brief, ordinal arithmetic presents itself as an area where induction applies but one which contains higher order aspects. Not all theorems in standard arithmetic hold for ordinal numbers making it genuinely distinct from the nite case. 3 Proof ....

P. Halmos. Naive Set Theory. Van Nostrand, Princeton, NJ, 1960.

.... prove the following assertion (with generic parameter Y ) Y ] 8 e : PY Y ffl 9 1 h : FT Y ffl 8 x : F FT ffl h(k(x ) e(h(j x j) 6) The proof of (6) is similar to the proof of definition by induction for the natural numbers that one can find in elementary texts on set theory (e.g. [5]) Given Y and e as in the statement of the theorem, we consider partial approximations to the desired total function h, That is to say we consider functions g : FT Y which satisfy g(k(x ) e(g(j x j) whenever both sides of that equation are defined. We show that any two such approximations g ....

Paul R. Halmos. Naive Set Theory. Springer-Verlag, 1974.

....the k term Cartesian product S Theta S Theta Delta Delta Delta Theta S. When no ambiguity arises, fAg may be written as A, and id denotes the identity mapping on any set. As usual a partial ordering v on a set S is a binary relation on S which is reflexive, antisymmetric and transitive [5]. When v is a linear ordering, we denote it by . When each element is only comparable with itself, S is unordered, i.e. v is just the equality predicate = The structure hS; vi is called a partially ordered set. We use the term ordered to mean partially ordered unless stated to the contrary) ....

P. Halmos. Naive Set Theory, Springer-Verlag, New York, (1974).

....given the crucial importance of successfulness. At a level of methodology, it may be worth making explicit that perhaps until issues of linguistic context and anaphora are done justice referring expressions generation can, in large part, be viewed as an exercise in naive set theory (e.g. Halmos 1960): it is about nding combinations of sets whose intersection equals some given target set. Dale and Reiter showed that this analogy is a productive one (see, e.g. their remarks about set covers, Dale and Reiter 1995, section 2.2) Our own work (especially sections 3.2, 3.3, 4.1, and 4.2) suggests ....

Halmos 1960. P. Halmos. Naive Set Theory. Van Nostrand, Princeton, N.J.

....C. Paulson For expressing specifications, constructive type theories are no more powerful than the classical systems we have examined above. The # and # constructions can also be defined in both untyped set theory and in the typed set theory of higher order logic. Classic ZF texts define # [Halmos 1960, p. 36] and # has a simple definition. PVS s predicate subtypes provide the e#ect of # and # at the level of types. The main virtue of these type theories is precisely that they are constructive. A constructive proof that two arbitrary numbers always have a greatest common divisor provides an ....

Halmos, P. R. 1960. Naive Set Theory. Van Nostrand.

....a subscript they refer to the set R of real numbers. Note that R is a lattice such that for x; y 2 R it follows x y = minfx; yg and x y = maxfx; yg. Furthermore xky is always false in R hence the elements of R are conventionally called totally ordered and R is called a chain (lattice) 2] [31]. The concept fuzzy lattice has been introduced by the authors of this work in order to extend the lattice ordering relation to all pairs (x; y) 2 L L of a crisp lattice L. Such an extended relation may 5 be regarded as a fuzzy set on the universe of discourse LL [67] Note that in this work a ....

....U is the product of N identical constituent lattices, these are the chains I = 0; 1] where the implied lattice ordering relation is the conventional less than or equal to relation between real numbers. Recall that a chain is a lattice characterized by a total ordering of its elements [2] [31]. Moreover each of the N (lattice) chains I, is a complete one with least element O I = 0:0 and greatest element I I = 1:0. Fig.1 illustrates some notions of the FL framework with reference to lattice U in the case N = 2, that is the unit square on the plane. Note that a rectangle (a box) ....

Halmos, P.R. (1961). Naive Set Theory.Van Nostrand Co., N.Y.

....that the appropriate y may actually be feasibly computed from a given x. The following is a proposition expressing the requirement for a feasible integer square root: # poly x : N. #r : N. r 2 # x (r 1 2 . Bibliographic notes Section 1 Types and Equality The Halmos book [57] does not cite the literature since his account is of the most basic concepts. I will not give extensive references either, but I will cite sources that provide addtional references. One of the best books about basic set theory, in my opinion, is still Foundations of Set Theory, by Fraenkel, ....

Paul R. Halmos. Naive Set Theory. Springer-Verlag, New York, 1974.

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P.R. Halmos. Naive Set Theory. Springer Verlag, 1974.

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P. R. Halmos. Naive Set Theory. Springer-Verlag, 1970.

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Paul R. Halmos. Naive Set Theory. Springer-Verlag, 1974.

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P. Halmos. Naive Set Theory. Van Nostrand, Princeton, NJ, 1960.

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P. R. Halmos. Naive Set Theory. Van Nostrand, 1960.

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P. R. Halmos. Naive Set Theory. Van Nostrand, Princeton, New Jersey, 1960.

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Paul R. Halmos. Naive Set Theory. Springer-Verlag, NY, 1994.

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P. R. Halmos, Naive Set Theory, Van Nostrand, New York, 1960.

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Paul R. Halmos. Naive Set Theory. Van Nostrand, 1960.

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