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Triadic Relation
In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place. Just as a binary relation is formally defined as a set of ''pairs'', i.e. a subset of the Cartesian product of some sets ''A'' and ''B'', so a ternary relation is a set of triples, forming a subset of the Cartesian product of three sets ''A'', ''B'' and ''C''. An example of a ternary relation in elementary geometry can be given on triples of points, where a triple is in the relation if the three points are collinear. Another geometric example can be obtained by considering triples consisting of two points and a line, where a triple is in the ternary relation if the two points determine (are incident with) the line. Examples Binary functions A function in two variables, mapping two values from sets ''A'' and ''B'', respectively, to a value in ''C'' ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composition Of Relations
In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplication, and its result is called a relative product. Function composition is the special case of composition of relations where all relations involved are functions. The word uncle indicates a compound relation: for a person to be an uncle, he must be the brother of a parent. In algebraic logic it is said that the relation of Uncle (x U z) is the composition of relations "is a brother of" (x B y) and "is a parent of" (y P z). U = BP \quad \text \quad xUz \text \exists y\ xByPz. Beginning with Augustus De Morgan, the traditional form of reasoning by syllogism has been subsumed by relational logical expressions and their composition. Definition If R \subseteq X \times Y and S \subseteq Y \times Z are two binary relations, then their compo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Books
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Schröder (mathematician)
Friedrich Wilhelm Karl Ernst Schröder (; 25 November 1841 – 16 June 1902) was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic, by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce. He is best known for his monumental ''Vorlesungen über die Algebra der Logik'' (''Lectures on the Algebra of Logic'', 1890–1905), in three volumes, which prepared the way for the emergence of mathematical logic as a separate discipline in the twentieth century by systematizing the various systems of formal logic of the day. Life Schröder learned mathematics at Heidelberg, Königsberg, and Zürich, under Otto Hesse, Gustav Kirchhoff, and Franz Neumann. After teaching school for a few years, he moved to the Technische Hochschule Darmstadt in 1874. Two years later, he took up a chair in mathematics at the Karlsruhe Polytechnische Schule, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Augustus De Morgan
Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He is best known for De Morgan's laws, relating logical conjunction, disjunction, and negation, and for coining the term "mathematical induction", the underlying principles of which he formalized. De Morgan's contributions to logic are heavily used in many branches of mathematics, including set theory and probability theory, as well as other related fields such as computer science. Biography Childhood Augustus De Morgan was born in Madurai, in the Carnatic Sultanate, Carnatic region of India, in 1806. His father was Lieutenant-Colonel John De Morgan (1772–1816), who held various appointments in the service of the East India Company, and his mother, Elizabeth (née Dodson, 1776–1856), was the granddaughter of James Dodson (mathematician), James Dodson, who computed a table of anti-logarithms (inverse logarithms). Augustus De Morgan became blind in one eye within a few months of his bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, : for all in the domain of . Equivalently, applying twice produces the original value. General properties Any involution is a bijection. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (), reciprocation (), and complex conjugation () in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The composition of two involutions and is an involution if and only if they commute: . Involutions on finite sets The number of involutions, including the identity involution, on a set with elements is given by a recurrence relation found by Heinrich August Rothe in 1800: : a_0 = a_1 = 1 and a_n = a_ + (n - 1)a_ for n > 1. The first few terms of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Converse Relation
In mathematics, the converse of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if X and Y are sets and L \subseteq X \times Y is a relation from X to Y, then L^ is the relation defined so that yL^x if and only if xLy. In set-builder notation, :L^ = \. Since a relation may be represented by a logical matrix, and the logical matrix of the converse relation is the transpose of the original, the converse relation is also called the transpose relation. It has also been called the opposite or dual of the original relation, the inverse of the original relation,Gerard O'Regan (2016): ''Guide to Discrete Mathematics: An Accessible Introduction to the History, Theory, Logic and Applications'' or the reciprocal L^ of the relation L. Other notations for the converse relation include L^, L^, \breve, L^, or L^. The notati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 for some set ''X'', while a heterogeneous relation is found in the power set of , where . Whether a given relation holds for two individual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inclusion (set Theory)
In mathematics, a set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. When quantified, A \subseteq B is represented as \forall x \left(x \in A \Rightarrow x \in B\right). One can prove the statement A \subseteq B by applying a proof technique known as the element argument:Let sets ''A'' and ''B'' be given. To prove that A \subseteq B, # suppose that ''a'' is a particular but arbitrarily chosen element of A # show that ''a'' is an element of ''B''. The validity of this technique can be seen as a consequence of univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Relation
In mathematics, a homogeneous relation (also called endorelation) on a set ''X'' is a binary relation between ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation over ''X''". An example of a homogeneous relation is the relation of kinship, where the relation is between people. Common types of endorelations include order (mathematics), orders, graph (discrete mathematics), graphs, and equivalence relation, equivalences. Specialized studies of order theory and graph theory have developed understanding of endorelations. Terminology particular for graph theory is used for description, with an ordinary (undirected) graph presumed to correspond to a symmetric relation, and a general endorelation corresponding to a directed graph. An endorelation ''R'' corresponds to a logical matrix of 0s and 1s, where the expression ''xRy'' (''x'' is ''R''-related to ''y'') corresponds to an edge between ''x'' and ''y' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitary Relation
In mathematics, a finitary relation over a sequence of sets is a subset of the Cartesian product ; that is, it is a set of ''n''-tuples , each being a sequence of elements ''x''''i'' in the corresponding ''X''''i''. Typically, the relation describes a possible connection between the elements of an ''n''-tuple. For example, the relation "''x'' is divisible by ''y'' and ''z''" consists of the set of 3-tuples such that when substituted to ''x'', ''y'' and ''z'', respectively, make the sentence true. The non-negative integer ''n'' that gives the number of "places" in the relation is called the ''arity'', ''adicity'' or ''degree'' of the relation. A relation with ''n'' "places" is variously called an ''n''-ary relation, an ''n''-adic relation or a relation of degree ''n''. Relations with a finite number of places are called ''finitary relations'' (or simply ''relations'' if the context is clear). It is also possible to generalize the concept to ''infinitary relations'' with Sequence, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number a is equal to itself (reflexive). If a = b, then b = a (symmetric). If a = b and b = c, then a = c (transitive). Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definitions A binary relation \,\si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
