Automated Teaching System “Sets” (Research for Organizing the 1st Part of the Project)

The issues of building an automated learning system “Sets” which will allow students to master one of the important topics of the discipline “Discrete Mathematics” and to develop logical and mathematical thinking in this direction are studied. The corresponding topic of the 1st part of the project includes materials related to the concept of a set, operations on sets, algebra of sets, proofs of statements for sets, and the derivation of formulas for the number of set elements. The system is based on a construction of the statements proof editor for a set and of the formulas derivation editor for the number of set elements, both editors are to be used for teaching. The first of these allows students to split the original statement into a number of simpler statements, taken together equivalent to the original statement, to choose a method of proving each simple statement and to conduct their step-by-step proof. The second editor allows (using the inclusion-exclusion principle and the formula of the number of complement elements) to derive a step-by-step formula for the number of set elements through the specified numbers of elements for sets from which the resulting set is constructed. An important part of the system is to monitor the correctness of all actions of students, and on this basis the entire learning system is developed. The logical supervision over the correctness of the selected action in the first editor is performed by a Boolean function created by the system and corresponding to this action and by checking it for identical truth. In the second editor, invariants such as characteristic strings of the set and of its number of elements are used for verification. The rest of the system is related to learning of set algebra and to preparation to editors usage. The main focus here is on the learning strategy in which testing the understanding of the learned material is rather rigorous and eliminating the random choice of answers. The division of the material into sections with verification of the success of teaching not only by tests, but also by exercises and tasks, allows students to master the complex logical and mathematical techniques of proving statements for sets and derivation of formulas for the number of set elements.

Maximal linked systems on families of measurable rectangles

Linked and maximal linked systems (MLS) on π -systems of measurable (in the wide sense) rectangles are considered (π-system is a family of sets closed with respect to finite intersections). Structures in the form of measurable rectangles are used in measure theory and probability theory and usually lead to semi-algebra of subsets of cartesian product. In the present article, sets-factors are supposed to be equipped with π-systems with “zero” and “unit”. This, in particular, can correspond to a standard measurable structure in the form of semialgebra, algebra, or σ-algebra of sets. In the general case, the family of measurable rectangles itself forms a π -system of set-product (the measurability is identified with belonging to a π - system) which allows to consider MLS on a given π -system (of measurable rectangles). The following principal property is established: for all considered variants of π -system of measurable rectangles, MLS on a product are exhausted by products of MLS on sets-factors. In addition, in the case of infinity product, along with traditional, the “box” variant allowing a natural analogy with the base of box topology is considered. For the case of product of two widely understood measurable spaces, one homeomorphism property concerning equipments by the Stone type topologies is established.

LINEAR SEARCH ALGORITHM TO BE SOLVED BY PARAMETER LOOP

There are common occasions for finding out required optional data or element from the given algebra of sets and any kind of systems when we are solving practical and informatics tasks. In order to solve these types of tasks or issues, students or researchers have to know about the methodologies for basic understanding which are called tasks for searching. This process is very important as for finding out basic understanding and methodologies, having detailed knowledge of algorithms and searching methods, although there are modern specific technologies and automatic programming systems. When searching methods are programmedfor informatics tasks, necessary abilities such as seeking essential or import of program language operators and making analysis indifference among characterful solving should be owned by students or specialists. Therefore,I set my purpose of thesis of organizing linear searching algorithms by the parameter loop of the programming language C during the discovering process of identifying differences, patterns, and internal core of linear searching methodologies.

On certain analogues of linkedness and supercompactness

Natural generalizations of properties of the family linkedness and the topological space supercompactness are considered. We keep in mind reinforced linkedness when nonemptyness of intersection of preassigned number of sets from a family is postulated. Analogously, supercompactness is modified: it is postulated the existence of an open subbasis for which, from every covering (by sets of the subbasis), it is possible to extract a subcovering with a given number of sets (more precisely, not more than a given number). It is clear that among all families having the reinforced linkedness, one can distinguish families that are maximal in ordering by inclusion. Under natural and (essentially) “minimal”' conditions imposed on the original measurable structure, among the mentioned maximal families with reinforced linkedness, ultrafilters are certainly contained. These ultrafilters form subspaces in the sense of natural topologies corresponding conceptually to schemes of Wallman and Stone. In addition, maximal families with reinforced linkedness, when applying topology of the Wallman type, have the above-mentioned property generalizing supercompactness. Thus, an analogue of the superextension of the $T_1$-space is realized. The comparability of “Wallman”' and “Stone”' topologies is established. As a result, bitopological spaces (BTS) are realized; for these BTS, under equipping with analogous topologies, ultrafilter sets are subspaces. It is indicated that some cases exist when the above-mentioned BTS is nondegenerate in the sense of the distinction for forming topologies. At that time, in the case of “usual” linkedness (this is a particular case of reinforced linkedness), very general classes of spaces are known for which the mentioned BTS are degenerate (the cases when origial set, i.e., “unit”' is equipped with an algebra of sets or a topology).

On Grothendieck Sets

We call a subset M of an algebra of sets A a Grothendieck set for the Banach space b a ( A ) of bounded finitely additive scalar-valued measures on A equipped with the variation norm if each sequence μ n n = 1 ∞ in b a ( A ) which is pointwise convergent on M is weakly convergent in b a ( A ) , i.e., if there is μ ∈ b a A such that μ n A → μ A for every A ∈ M then μ n → μ weakly in b a ( A ) . A subset M of an algebra of sets A is called a Nikodým set for b a ( A ) if each sequence μ n n = 1 ∞ in b a ( A ) which is pointwise bounded on M is bounded in b a ( A ) . We prove that if Σ is a σ -algebra of subsets of a set Ω which is covered by an increasing sequence Σ n : n ∈ N of subsets of Σ there exists p ∈ N such that Σ p is a Grothendieck set for b a ( A ) . This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a σ -algebra Σ is covered by an increasing sequence Σ n : n ∈ N of subsets, there is p ∈ N such that Σ p is a Nikodým set for b a Σ . This also refines the Grothendieck result stating that for each σ -algebra Σ the Banach space ℓ ∞ Σ is a Grothendieck space. Some applications to classic Banach space theory are given.

Methods of mathematical modeling using polynomials of algebra of sets

The article deals with the construction of discrete mathematical models for solving applied problems arising from the operation of building structures. Security issues in modern high-rise buildings are extremely serious and relevant, and there is no doubt that interest in them will only increase. The territory of the building is divided into zones for which it is necessary to observe. Zones can overlap and have different priorities. Such situations can be described using formulas algebra of sets. Formulas can be programmed, which makes it possible to work with them using computer models.

Analysis of Defeasible Reasoning

This chapter describes implementation of abductive and modified conclusions by means of NTA. The algorithm and rules to form hypotheses for abductive conclusions are proposed. They can be applied not only to NTA objects expressing formulas of propositional calculus, but also to a more general case when attribute domains contain more than two values. Within a specific knowledge system, choosing variables and their values depends on criteria determined by the content of the system. The techniques that we developed simplify generating abductive conclusions for given limitations, for instance, in composition and number of variables. A distinctive feature of the proposed methods is that they are based on the classical foundations of logic, that is, they do not use non-monotonic logic, the logic of defaults, etc., which allowed some violations of laws of Boolean algebra and algebra of sets.