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Calculus Of Relations
In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics for these deductive systems) and connected problems like representation and duality. Well known results like the representation theorem for Boolean algebras and Stone duality fall under the umbrella of classical algebraic logic . Works in the more recent abstract algebraic logic (AAL) focus on the process of algebraization itself, like classifying various forms of algebraizability using the Leibniz operator . Calculus of relations A homogeneous binary relation is found in the power set of ''X'' × ''X'' for some set ''X'', while a heterogeneous relation is found in the power set of ''X'' × ''Y'', where ''X'' ≠ ''Y''. Whether a gi ...
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Mathematical Logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory sho ...
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Logical Matrix
A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1) matrix is a matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets. Matrix representation of a relation If ''R'' is a binary relation between the finite indexed sets ''X'' and ''Y'' (so ), then ''R'' can be represented by the logical matrix ''M'' whose row and column indices index the elements of ''X'' and ''Y'', respectively, such that the entries of ''M'' are defined by :M_ = \begin 1 & (x_i, y_j) \in R, \\ 0 & (x_i, y_j) \not\in R. \end In order to designate the row and column numbers of the matrix, the sets ''X'' and ''Y'' are indexed with positive integers: ''i'' ranges from 1 to the cardinality (size) of ''X'', and ''j'' ranges from 1 to the cardinality of ''Y''. See the entry on indexed sets for more detail. Example The binary relation ''R'' on the set is defined so that ''aRb'' holds if and only if ''a' ...
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Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. ...
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Gunther Schmidt
Gunther Schmidt (born 1939, Rüdersdorf) is a German mathematician who works also in informatics. Life Schmidt began studying Mathematics in 1957 at Göttingen University. His academic teachers were in particular Kurt Reidemeister, Wilhelm Klingenberg and Karl Stein. In 1960 he transferred to Ludwig-Maximilians-Universität München where he studied functions of several complex variables with Karl Stein. Schmidt wrote a thesis on analytic continuation of such functions. In 1962 Schmidt began work at TU München with students of Robert Sauer, in the beginning in labs and tutorials, later in mentoring and administration. Schmidt's interests turned toward programming when he collaborated with Hans Langmaack on rewriting and the braid group in 1969. Friedrich L. Bauer and Klaus Samelson were establishing software engineering at the university and Schmidt joined their group in 1974. In 1977 he submitted his Habilitation "Programs as partial graphs". He became a professor in 1980 ...
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Charles Loewner
Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German. Karl Loewner was born into a Jewish family in Lany, about 30 km from Prague, where his father Sigmund Löwner was a store owner. Loewner received his Ph.D. from the University of Prague in 1917 under supervision of Georg Pick. One of his central mathematical contributions is the proof of the Bieberbach conjecture in the first highly nontrivial case of the third coefficient. The technique he introduced, the Loewner differential equation, has had far-reaching implications in geometric function theory; it was used in the final solution of the Bieberbach conjecture by Louis de Branges in 1985. Loewner worked at the University of Berlin, University of Prague, University of Louisville, Brown University, Syracuse University and eventually at Stanford University. His students include Lipman Bers, Roger Horn, Adriano Garsia, and P. M ...
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Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of ...
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Total Relation
In mathematics, a binary relation ''R'' ⊆ ''X''×''Y'' between two sets ''X'' and ''Y'' is total (or left total) if the source set ''X'' equals the domain . Conversely, ''R'' is called right total if ''Y'' equals the range . When ''f'': ''X'' → ''Y'' is a function, the domain of ''f'' is all of ''X'', hence ''f'' is a total relation. On the other hand, if ''f'' is a partial function, then the domain may be a proper subset of ''X'', in which case ''f'' is not a total relation. "A binary relation is said to be total with respect to a universe of discourse just in case everything in that universe of discourse stands in that relation to something else."Functions
from Carnegie Mellon University
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Partial Function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is defined on every element in , then is said to be total. More technically, a partial function is a binary relation over two sets that associates every element of the first set to ''at most'' one element of the second set; it is thus a functional binary relation. It generalizes the concept of a (total) function by not requiring every element of the first set to be associated to ''exactly'' one element of the second set. A partial function is often used when its exact domain of definition is not known or difficult to specify. This is the case in calculus, where, for example, the quotient of two functions is a partial function whose domain of definition cannot contain the zeros of the denominator. For this reason, in calculus, and more ge ...
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Socratic Method
The Socratic method (also known as method of Elenchus, elenctic method, or Socratic debate) is a form of cooperative argumentative dialogue between individuals, based on asking and answering questions to stimulate critical thinking and to draw out ideas and underlying presuppositions. It is named after the Classical Greek philosopher Socrates and is introduced by him in Plato's Theaetetus as midwifery ( maieutics) because it is employed to bring out definitions implicit in the interlocutors' beliefs, or to help them further their understanding. The Socratic method is a method of hypothesis elimination, in that better hypotheses are found by steadily identifying and eliminating those that lead to contradictions. The Socratic method searches for general commonly held truths that shape beliefs and scrutinizes them to determine their consistency with other beliefs. The basic form is a series of questions formulated as tests of logic and fact intended to help a person or group discove ...
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Presupposition
In the branch of linguistics known as pragmatics, a presupposition (or PSP) is an implicit assumption about the world or background belief relating to an utterance whose truth is taken for granted in discourse. Examples of presuppositions include: * ''Jane no longer writes fiction.'' ** Presupposition: Jane once wrote fiction. * ''Have you stopped eating meat?'' ** Presupposition: you had once eaten meat. * ''Have you talked to Hans?'' ** Presupposition: Hans exists. A presupposition must be mutually known or assumed by the speaker and addressee for the utterance to be considered appropriate in context. It will generally remain a necessary assumption whether the utterance is placed in the form of an assertion, denial, or question, and can be associated with a specific lexical item or grammatical feature (presupposition trigger) in the utterance. Crucially, negation of an expression does not change its presuppositions: ''I want to do it again'' and ''I don't want to do it again'' ...
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Question
A question is an utterance which serves as a request for information. Questions are sometimes distinguished from interrogatives, which are the grammatical forms typically used to express them. Rhetorical questions, for instance, are interrogative in form but may not be considered bona fide questions, as they are not expected to be answered. Questions come in a number of varieties. '' Polar questions'' are those such as the English example "Is this a polar question?", which can be answered with "yes" or "no". ''Alternative questions'' such as "Is this a polar question, or an alternative question?" present a list of possibilities to choose from. '' Open questions'' such as "What kind of question is this?" allow many possible resolutions. Questions are widely studied in linguistics and philosophy of language. In the subfield of pragmatics, questions are regarded as illocutionary acts which raise an issue to be resolved in discourse. In approaches to formal semantics such as ...
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Statement (logic)
In logic, the term statement is variously understood to mean either: #a meaningful declarative sentence that is true or false, or #a proposition. Which is the ''assertion'' that is made by (i.e., the meaning of) a true or false declarative sentence. In the latter case, a statement is distinct from a sentence in that a sentence is only one formulation of a statement, whereas there may be many other formulations expressing the same statement. Overview Philosopher of language, Peter Strawson advocated the use of the term "statement" in sense (b) in preference to proposition. Strawson used the term "Statement" to make the point that two declarative sentences can make the same statement if they say the same thing in different ways. Thus in the usage advocated by Strawson, "All men are mortal." and "Every man is mortal." are two different sentences that make the same statement. In either case a statement is viewed as a truth bearer. Examples of sentences that are (or make) ...
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