How to write Scientific Research proposal

how to write scientific research paper introduction and how scientific research can be manipulated for different purposes how does scientific research become published
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DrJohnRyder,United Kingdom,Researcher
Published Date:07-07-2017
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Ars longa, vita brevis INTRODUCTION Mastering the discipline “Foundations of Scientific Research” (Foundations of Research Activities) is aimed at training students in methodological foundations and organization of scientific research; organization of reference and information retrieval on the topic of research in system of scientific and technical libraries and by local and global computer information networks; analysis and evaluation of information and research and development processes in civil aviation and in another fields of national economy; guidance, principles and facilities of optimization of scientific research; preparation of facts, which documenting results of research scientific work (scientific report, article, talk, theses, etc.) The main tasks of the discipline are to familiarize students with basic terminology, theoretical and experimental methods of scientific research as well as methods of analysis of observed results, their practical use and documentation facilities. The tasks of mastering the discipline “Foundations of scientific research” are the following:  to learn professional terminology of scientific research;  to be able to perform the reference and information retrieval on the topic of research;  to be able to formulate methodological foundations of scientific research on specialty;  to understand the organization of scientific research;  to make scientific report (talk) on professional and socio-political topics defined by this syllabus. Practical skills in the foundations of scientific research enable students to be aware of world scientific results and new technologies, to understand novel scientific results, papers, computer manuals, software documentation, and additional literature with the aim of professional decisions-making. Prolific knowledge and good practical skills in the foundations of scientific research allow students to study in novel scientific results, 4 make investigations, reports, summaries and comments, develop scientific projects and be engaged in foundations of scientific research. As a result of mastering the discipline a student shall KNOW:  basic professional and technical terminology on the disciplines defined by the academic curriculum;  categorical apparatus of scientific research;  main rules of handling scientific and technical literature;  aim and tasks of scientific research;  methodology and methods of scientific research;  classification of methods by the level of investigation, by the state of the organization of scientific research, by the character of cognitive activity;  types of exposition results of scientific research;  peculiarities of students research activities. LEARNING OUTCOMES:  organize and carry out scientific research by oneself;  carry out information retrieval of scientific literature;  competently work with scientific information sources;  take out optimal research methods by the content and aim of the scientific task. The ideas in this manual have been derived from many sources 1-19,25. Here I will try to acknowledge those that are explicitly attributable to other authors. Most of the other ideas are part of Scientific Research folklore. To try to attribute them to anyone would be impossible. Also in the manual we use texts from Wikipedia and some another papers and books. The author thanks his students A. Babaryka, V. Burenkov, K.Vasyanovich, D. Eremenko, A. Kachinskaya, L. Mel’nikova, O. Samusenko, 5 I. Tatomyr, V. Trush and others for texts of lectures, labs and homeworks on the discipline “Foundations of Scientific Research”. The list of references is indicated in Literature section at the end of the manual. 1. GENERAL NOTIONS ABOUT SCIENTIFIC RESEARCH Science is the process of gathering, comparing, and evaluating proposed models against observables. A model can be a simulation, mathematical or chemical formula, or set of proposed steps. Under science we will understand natural sciences, mathematical sciences and applied sciences with special emphasis on computer sciences. In sone cases we will distinguish mathematics as the language of science. From school and university mathematical cources we know that reseachers (in the case these are schoolgirls, schoolboys, students) can clearly distinguish what is known from what is unknown at each stage of mathematical discovery. Science is like mathematics in that researchers in both disciplines can clearly distinguish what is known from what is unknown at each stage of scientific discovery. Models, in both science and mathematics, need to be internally consistent and also ought to be falsifiable (capable of disproof). In mathematics, a statement need not yet be proven; at such a stage, that statement would be called a conjecture. But when a statement has attained mathematical proof, that statement gains a kind of immortality which is highly prized by mathematicians, and for which some mathematicians devote their lives. The hypothesis that people understand the world also by building mental models raises fundamental issues for all the fields of cognitive science. For instance in the framework of computer science there are a questions: How can a person's model of the word be reflected in a computer system? What languages and tools are needed to describe such models and relate them to outside systems? Can the models support a computer interface that people would find easy to use ? Here we will consider basic notions about scientific research, research methods, stages of scientific research, motion of scientific research, scientific search. In some 6 cases biside with the term “scientific research” we will use the term “scientific activety”. At first we illustrate the ontology based approach to design the course Foundations of Research Activities. This is a course with the problem domains “Computer sciences”, “Software Engeneering“, “Electromagnetism”, “Relativity Theory (Gravitation)” and “Quantum Mechenics” that enables the student to both apply and expand previous content knowledge toward the endeavour of engaging in an open-ended, student- centered investigation in the pursuit of an answer to a question or problem of interest. Some background in concept analtsis, electromagnetism, special and general relativity and quantum theory are presented. The particular feature of the course is studying and applying computer-assisted methods and technologies to justification of conjectures (hypotheses). In our course, justification of conjectures encompasses those tasks that include gathering and analysis of data, go into testing conjectures, taking account of mathematical and computer-assisted methods of mathematical proof of the conjecture. Justification of conjectures is critical to the success of the solution of a problem. Design involves problem-solving and creativity. Then, following to Wiki and some another sources, recall more traditional information about research and about scientific research. At first recall definitions of two terms (Concept Map, Conception (Theory)) that will use in our course. Concept Map: A schematic device for representing the relationships between concepts and ideas. The boxes represent ideas or relevant features of the phenomenon (i.e. concepts) and the lines represent connections between these ideas or relevant features. The lines are labeled to indicate the type of connection. Conception (Theory): A general term used to describe beliefs, knowledge, preferences, mental images, and other similar aspects of a t lecturer’s mental structure. 7 Research is scientific or critical investigation aimed at discovering and interpreting facts. Research may use the scientific method, but need not do so. Scientific research relies on the application of the scientific method, a harnessing of curiosity. This research provides scientific information and theories for the explanation of the nature and the properties of the world around us. It makes practical applications possible. Scientific research is funded by public authorities, by charitable organisations and by private groups, including many companies. Scientific research can be subdivided into different classifications according to their academic and application disciplines. Recall some classifications: Basic research. Applied research. Exploratory research. Constructive research. Empirical research Primary research. Secondary research. Generally, research is understood to follow a certain structural process. The goal of the research process is to produce new knowledge, which takes three main forms (although, as previously discussed, the boundaries between them may be fuzzy):  Exploratory research, which structures and identifies new problems  Constructive research, which develops solutions to a problem  Empirical research, which tests the feasibility of a solution using empirical evidence Research is often conducted using the hourglass model. The hourglass model starts with a broad spectrum for research, focusing in on the required information through the methodology of the project (like the neck of the hourglass), then expands the research in the form of discussion and results. Though step order may vary depending on the subject matter and researcher, the following steps are usually part of most formal research, both basic and applied: 8  Formation of the topic  Hypothesis  Conceptual definitions  Operational definitions  Gathering of data  Analysis of data  Test, revising of hypothesis  Conclusion, iteration if necessary A common misunderstanding is that by this method a hypothesis can be proven or tested. Generally a hypothesis is used to make predictions that can be tested by observing the outcome of an experiment. If the outcome is inconsistent with the hypothesis, then the hypothesis is rejected. However, if the outcome is consistent with the hypothesis, the experiment is said to support the hypothesis. This careful language is used because researchers recognize that alternative hypotheses may also be consistent with the observations. In this sense, a hypothesis can never be proven, but rather only supported by surviving rounds of scientific testing and, eventually, becoming widely thought of as true (or better, predictive), but this is not the same as it having been proven. A useful hypothesis allows prediction and within the accuracy of observation of the time, the prediction will be verified. As the accuracy of observation improves with time, the hypothesis may no longer provide an accurate prediction. In this case a new hypothesis will arise to challenge the old, and to the extent that the new hypothesis makes more accurate predictions than the old, the new will supplant it. 1.1. Scientific method Scientific method 1-4,6-8 refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering observable, empirical and measurable evidence subject to specific principles of 9 reasoning. A scientific method consists of the collection of data through observation and experimentation, and the formulation and testing of hypotheses. As have indicated in cited references knowledge is more than a static encoding of facts, it also includes the ability to use those facts in interacting with the world. There is the operative definition: Knowledge - attach purpose and competence to information potential to generate action Although procedures vary from one field of inquiry to another, identifiable features distinguish scientific inquiry from other methodologies of knowledge. Scientific researchers propose hypotheses as explanations of phenomena, and design experimental studies to test these hypotheses. These steps must be repeatable in order to dependably predict any future results. Theories that encompass wider domains of inquiry may bind many independently-derived hypotheses together in a coherent, supportive structure. This in turn may help form new hypotheses or place groups of hypotheses into context. Among other facets shared by the various fields of inquiry is the conviction that the process be objective to reduce biased interpretations of the results. Another basic expectation is to document, archive and share all data and methodology so they are available for careful scrutiny by other scientists, thereby allowing other researchers the opportunity to verify results by attempting to reproduce them. This practice, called full disclosure, also allows statistical measures of the reliability of these data to be established. 1.2. Basic research Does string theory provide physics with a grand unification theory? The solution of the problem is the main goal of String Theory and basic research in the field 4. 10 Basic research (also called fundamental or pure research) has as its primary objective the advancement of knowledge and the theoretical understanding of the relations among variables (see statistics). It is exploratory and often driven by the researcher’s curiosity, interest, and intuition. Therefore, it is sometimes conducted without any practical end in mind, although it may have unexpected results pointing to practical applications. The terms “basic” or “fundamental” indicate that, through theory generation, basic research provides the foundation for further, sometimes applied research. As there is no guarantee of short-term practical gain, researchers may find it difficult to obtain funding for basic research. Traditionally, basic research was considered as an activity that preceded applied research, which in turn preceded development into practical applications. Recently, these distinctions have become much less clear-cut, and it is sometimes the case that all stages will intermix. This is particularly the case in fields such as biotechnology and electronics, where fundamental discoveries may be made alongside work intended to develop new products, and in areas where public and private sector partners collaborate in order to develop greater insight into key areas of interest. For this reason, some now prefer the term frontier research. 1.2.1. Publishing Academic publishing describes a system that is necessary in order for academic scholars to peer review the work and make it available for a wider audience 21-24,26. The 'system', which is probably disorganised enough not to merit the title, varies widely by field, and is also always changing, if often slowly. Most academic work is published in journal article or book form. In publishing, STM publishing is an abbreviation for academic publications in science, technology, and medicine. 1.3. Information supply of scientific research. Scientist’s bibliographic activity includes: organization, technology, control. 11 Information retrieval systems and Internet. It is very important now to have lot’s of possibilities to have access to different kind of information. There are several ways. Indicate two of them and consider more carefully more modern: 1) go to library or 2) use Internet. As indicate many students: “I think that it is not difficult to understand why Internet is more preferable for me.” So, let as consider how works the best nowadays’s web-search Google and how a student can find article “A mathematical theory of communication” by C.E. Shannon. 1.3.1. How does Google work. Google runs on a distributed network of thousands of low-cost computers and can therefore carry out fast parallel processing. Parallel processing is a method of computation in which many calculations can be performed simultaneously, significantly speeding up data processing. Google has three distinct parts: Googlebot, a web crawler that finds and fetches web pages. The indexer that sorts every word on every page and stores the resulting index of words in a huge database. The query processor, which compares your search query to the index and recommends the documents that it considers most relevant. Let’s take a closer look at each part. 1.3.2. Googlebot, Google’s Web Crawler Googlebot is Google’s web crawling robot, which finds and retrieves pages on the web and hands them off to the Google indexer. It’s easy to imagine Googlebot as a little spider scurrying across the strands of cyberspace, but in reality Googlebot doesn’t traverse the web at all. It functions much like your web browser, by sending a request to a web server for a web page, downloading the entire page, then handing it off to Google’s indexer. 12 Googlebot consists of many computers requesting and fetching pages much more quickly than you can with your web browser. In fact, Googlebot can request thousands of different pages simultaneously. To avoid overwhelming web servers, or crowding out requests from human users, Googlebot deliberately makes requests of each individual web server more slowly than it’s capable of doing. Googlebot finds pages in two ways: through an add URL form, www.google.com/addurl.html, and through finding links by crawling the web. Unfortunately, spammers figured out how to create automated bots that bombarded the add URL form with millions of URLs pointing to commercial propaganda. Google rejects those URLs submitted through its Add URL form that it suspects are trying to deceive users by employing tactics such as including hidden text or links on a page, stuffing a page with irrelevant words, cloaking (aka bait and switch), using sneaky redirects, creating doorways, domains, or sub-domains with substantially similar content, sending automated queries to Google, and linking to bad neighbors. So now the Add URL form also has a test: it displays some squiggly letters designed to fool automated “letter-guessers”; it asks you to enter the letters you see — something like an eye-chart test to stop spambots. When Googlebot fetches a page, it culls all the links appearing on the page and adds them to a queue for subsequent crawling. Googlebot tends to encounter little spam because most web authors link only to what they believe are high-quality pages. By harvesting links from every page it encounters, Googlebot can quickly build a list of links that can cover broad reaches of the web. This technique, known as deep crawling, also allows Googlebot to probe deep within individual sites. Because of their massive scale, deep crawls can reach almost every page in the web. Because the web is vast, this can take some time, so some pages may be crawled only once a month. Although its function is simple, Googlebot must be programmed to handle several challenges. First, since Googlebot sends out simultaneous requests for thousands of pages, the queue of “visit soon” URLs must be constantly examined and compared with URLs already in Google’s index. Duplicates in the queue must be eliminated to prevent Googlebot from fetching the same page again. Googlebot must determine how 13 often to revisit a page. On the one hand, it’s a waste of resources to re-index an unchanged page. On the other hand, Google wants to re-index changed pages to deliver up-to-date results. To keep the index current, Google continuously recrawls popular frequently changing web pages at a rate roughly proportional to how often the pages change. Such crawls keep an index current and are known as fresh crawls. Newspaper pages are downloaded daily, pages with stock quotes are downloaded much more frequently. Of course, fresh crawls return fewer pages than the deep crawl. The combination of the two types of crawls allows Google to both make efficient use of its resources and keep its index reasonably current. 1.3.3. Google’s Indexer Googlebot gives the indexer the full text of the pages it finds. These pages are stored in Google’s index database. This index is sorted alphabetically by search term, with each index entry storing a list of documents in which the term appears and the location within the text where it occurs. This data structure allows rapid access to documents that contain user query terms. To improve search performance, Google ignores (doesn’t index) common words called stop words (such as the, is, on, or, of, how, why, as well as certain single digits and single letters). Stop words are so common that they do little to narrow a search, and therefore they can safely be discarded. The indexer also ignores some punctuation and multiple spaces, as well as converting all letters to lowercase, to improve Google’s performance. 1.3.4. Google’s Query Processor The query processor has several parts, including the user interface (search box), the “engine” that evaluates queries and matches them to relevant documents, and the results formatter. 14 PageRank is Google’s system for ranking web pages. A page with a higher PageRank is deemed more important and is more likely to be listed above a page with a lower PageRank. Google considers over a hundred factors in computing a PageRank and determining which documents are most relevant to a query, including the popularity of the page, the position and size of the search terms within the page, and the proximity of the search terms to one another on the page. A patent application discusses other factors that Google considers when ranking a page. Visit SEOmoz.org’s report for an interpretation of the concepts and the practical applications contained in Google’s patent application. Google also applies machine-learning techniques to improve its performance automatically by learning relationships and associations within the stored data. For example, the spelling-correcting system uses such techniques to figure out likely alternative spellings. Google closely guards the formulas it uses to calculate relevance; they’re tweaked to improve quality and performance, and to outwit the latest devious techniques used by spammers. Indexing the full text of the web allows Google to go beyond simply matching single search terms. Google gives more priority to pages that have search terms near each other and in the same order as the query. Google can also match multi-word phrases and sentences. Since Google indexes HTML code in addition to the text on the page, users can restrict searches on the basis of where query words appear, e.g., in the title, in the URL, in the body, and in links to the page, options offered by Google’s Advanced Search Form and Using Search Operators (Advanced Operators). 15 2. ONTOLOGIES AND UPPER ONTOLOGIES There are several definitions of the notion of ontology 10-13. By T. R. Gruber (Gruber, 1992) “An ontology is a specification of a conceptualization”. By B. Smith and his colleagues, (Smith, 2004) “an ontology is a representational artefact whose representational units are intended to designate universals in reality and the relations between them”. By our opinion the definitions reflect critical goals of ontologies in computer science. For our purposes we will use more specific definition of ontology: concepts with relations and rules define ontology (Gruber, 1992; Ontology, 2008; Wikipedia, 2009 ). Ontology Development aims at building reusable semantic structures that can be informal vocabularies, catalogs, glossaries as well as more complex finite formal structures representing the entities within a domain and the relationships between those entities. Ontologies, have been gaining interest and acceptance in computational audiences: formal ontologies are a form of software, thus software development methodologies can be adapted to serve ontology development. A wide range of applications is emerging, especially given the current web emphasis, including library science, ontology-enhanced search, e-commerce and configuration. Knowledge Engineering (KE) and Ontology Development (OD) aims at becoming a major meeting point for researchers and practitioners interested in the study and development of methodologies and technologies for Knowledge Engineering and Ontology Development. There are next relations among concepts: associative partial order higher subordinate 16 subsumption relation (is a, is subtype of, is subclass of) part-of relation. More generally, we may use Description Logic (DL) 5 for constructing consepts and knowledge base (Franz Baader, Werner Nutt. Basic Description Logics). See the section: Scientific research in Artificial Intelligence Different spaces are used in aforementioned courses. In our framework we treat ontology of spaces and ontology of symmetries as upper ontologies. 2.1. Concepts of Foundations of Research Activities Foundations of Research Activities Concepts: (a) Scientific Method. (b) Ethics of Research Activity. (c) Embedded Technology and Engineering. (d) Communication of Results (Dublin Core). In the section we consider briefly (a). Investigative processes, which are assumed to operate iteratively, involved in the research method are the follows: (i) Hypothesis, Low, Assumption, Generalization; (ii) Deduction; (iii) Observation, Confirmation; (iv) Induction. Indicate some related concepts: Problem. Class of Scientific Data. Scientific Theory. Formalization. Interpretation. Analyzing and Studying of Classic Scientific Problems. Investigation. Fundamental (pure) Research. Formulation of a Working Hypothesis to Guide Research. Developing Procedures to Testing a Hypothesis. Analysis of Data. Evaluation of Data. 17 2.2. Ontology components Contemporary ontologies share many structural similarities, regardless of the language in which they are expressed. As mentioned above, most ontologies describe individuals (instances), classes (concepts), attributes, and relations. In this subsection each of these components is discussed in turn. Common components of ontologies include:  Individuals: instances or objects (the basic or "ground level" objects) 10  Classes: sets, collections, concepts, types of objects, or kinds of things.  Attributes: aspects, properties, features, characteristics, or parameters that objects (and classes) can have  Relations: ways in which classes and individuals can be related to one another  Function terms: complex structures formed from certain relations that can be used in place of an individual term in a statement  Restrictions: formally stated descriptions of what must be true in order for some assertion to be accepted as input  Rules: statements in the form of an if-then (antecedent-consequent) sentence that describe the logical inferences that can be drawn from an assertion in a particular form  Axioms: assertions (including rules) in a logical form that together comprise the overall theory that the ontology describes in its domain of application. This definition differs from that of "axioms" in generative grammar and formal logic. In those disciplines, axioms include only statements asserted as a priori knowledge. As used here, "axioms" also include the theory derived from axiomatic statements.  Events: the changing of attributes or relations. 18 2.3. Ontology for the visualization of a lecture Upper ontology: visualization. Visualization of the text (white text against the dark background) is subclass of visualization. Visible page, data visualization, flow visualization, image visualization, spatial visualization, surface rendering, two-dimensional field visualization, three- dimensional field visualization, video content. 3. ONTOLOGIES OF OBJECT DOMAINS 3.1 Elements of the ontology of spaces and symmetries There is the well known from mathematics space - ring_of_functions_on_the_space duality. In the subsection we only mention some concepts, relations and rules of the ontology of spaces and symmetries. Two main concepts are space and symmetry. 3 3 3-dimensional real space R ; Linear group GL(3, R) of automorphisms of R ; Classical physical world has three spatial dimensions, so electric and magnetic fields are 3-component vectors defined at every point of space. 1,3 Minkowski space-time M is a 4-dimensional real manifold with a 2 2 2 2 1,3 4 pseudoriemannian metric t – x – y – z . From M it is possible to pass to R by means of the substitution t  iu and an overall sign-change in the metric. A 4 4 compactification of R by means of a stereographic projection gives S . 2D space, 2D object, 3D space, 3D object. Additiona material for advanced students: 19 1,3 Let SO(1,3) be the pseudoortogonal group. The moving frame in M is a section 1,3 1,3 2 of the trivial bundle M × SO(1,3) . A complex vector bundle M ×C is associated with the frame bundle by the representation SL(2,C) of the Lorentz group SO(1,3). The space-time M in which strings are propagating must have many dimensions 1,3 (10, 26. …) . The ten-dimensional space-time is locally a product M = M ×K of macroscopic four-dimensional space-time and a compact six-dimensional Calabi-Yau manifold K whose size is on the order of the Planck length. Principle bundle over space-time, structure group, associated vector bundle, connection, connection one-form, curvature, curvature form, norm of the curvature. Foregoing concepts with relations and rules define elements of the domain ontology of spaces. 3.1.1 Concepts of Electrodynamics and Classical Gauge Theory Preliminarities: electricity and magnetism. This subsection contains additiona material for advanced students. Short history: Schwarzchild action, Hermann Weyl, F. London, Yang-Mills equations. Quantum Electrodynamics is regarded as physical gauge theory. The set of possible gauge transformations of the entire configuration of a given gauge theory also forms a group, the gauge group of the theory. An element of the gauge group can be parameterized by a smoothly varying function from the points of space-time to the (finite-dimensional) Lie group, whose value at each point represents the action of the gauge transformation on the fiber over that point. Concepts: Gauge group as a (possibly trivial) principle bundle over space-time, gauge, classical field, gauge potential. 20 4. EXAMPLES OF RESEARCH ACTIVITY 4.1. Scientific activity in arithmetics, informatics and discrete mathematics Discrete mathematics becomes now not only a part of mathematics, but also a common language for various fields of cybernetics, computer science, informatics and their applications. Discrete mathematics studies discrete structures, operations with these structures and functions and mappings on the structures. Examples of discrete structures are: finite sets (FSets); sets: N – natural numbers; Z – integer numbers; Q – rational numbers; algebras of matrices over finite, rational and complex fields. Operations with discrete structures:  - union;  - intersection; A\B – set difference and others. Operations with elements of discrete structures: + - addition, - multiplication, scalar product and others. Recall some facts about integer and natural numbers. Sum, difference and product of integer numbers are integers, but the quotient under the division of an integer number a by the integer number b (if b is not equal to zero) maybe as an integer as well as not the integer. In the case when b divides a we will denote it as b a. From school program we know that any integer a is represented uniquely by the positive integer b in the form a = bq + r; 0  r b. The number r is called the residues of a under the division by b. We will study in section 3, that residues under the division of all natural numbers on a natural n form the ring Z/nZ. Below we will consider positive divisors only. Any integer that divides simultaneously integers a, b, c,…m is, called their common divisor. The largest from common divisors is called the greatest common divisor and is denoted by (a, b, c,…m . ) 21 If (a, b, c,…m) = , th1 en a, b, c,…m are called coprime. The number 1 has only one positive divisor. Any integer, greater than 1, has not less than two divisors, namely 1 and itself the integer. If an integer has exactly two positive divisors then the integer is called a prime. Recall now two functions of integer and natural arguments: the Mobius function (a) and Euler’s function (n) . The Mobius function (a) is defined for all positive k integers a : (a) = 0 if a is divided by a square that is not the unit; (a) =(-1) where a is not divided by a square that is not the unit, k is the number of prime divisors of a; =1. (1) Examples of values of the Mobius function: (1)1,(2)1,(3)1,(4)0,(5)1,(6)1. Euler’s function (n) is defined for any natural number n. (n) is the quantity of numbers 0 1 2 . . . n – 1 that are coprime with n. Examples of values of Euler’s function: (1)1,(2)1,(3) 2,(4)2,(5)4,(6)2. 4.2. Algebra of logic and functions of the algebra of logic The area of Algebra of logic and functions of the algebra of logic connects with mathematical logic and computer science. Boolean algebra is a part of the Algebra of logic. Boolean algebra, an abstract mathematical system primarily used in computer science and in expressing the relationships between sets groups of objects or concepts). The notational system was developed by the English mathematician George Boole to permit an algebraic manipulation of logical statements. Such manipulation can demonstrate whether or not a statement is true and show how a complicated statement can be rephrased in a simpler, more convenient form without changing its meaning. 22 Let p , p …p be propositional variables where each variable can take value 0 or 1 2, n 1. The logical operations (connectives) are ,,,, . Their names are: conjunction (AND - function), disjunction (OR - function), negation, equivalence, implication. Definition. A propositional formula is defined inductively as: 1. Each variable p is a formula. i 2. If A and B are formulas, then (AB),(AB),(A),(AB),(AB)are formulas. 3. A is a formula iff it follows from 1 and 2. Remark. The operation of negation has several equivalent notations: , or ‘ (please see below). 4.3. Function of the algebra of logic Let n be the number of Boolean variables. Let P (0,1) be the set of Boolean n n functions in n variables. With respect to a fixed order of all 2 possible arguments, every n such function f:0,10,1 is uniquely represented by its truth table, which is a vector n of length 2 listing the values of f in that order. Boolean functions of one variable (n = 1): x 0 1 x x 0 0 1 0 1 1 0 1 1 0 23

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