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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 xByPz \text xUz. 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 composition R; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and mathematical analysis, analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanford Encyclopedia Of Philosophy
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from many academic institutions worldwide. Authors contributing to the encyclopedia give Stanford University the permission to publish the articles, but retain the copyright to those articles. Approach and history As of August 5th, 2022, the ''SEP'' has 1,774 published entries. Apart from its online status, the encyclopedia uses the traditional academic approach of most encyclopedias and academic journals to achieve quality by means of specialist authors selected by an editor or an editorial committee that is competent (although not necessarily considered specialists) in the field covered by the encyclopedia and peer review. The encyclopedia was created ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup With Involution
In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a group because this involution, considered as unary operator, exhibits certain fundamental properties of the operation of taking the inverse in a group: uniqueness, double application "cancelling itself out", and the same interaction law with the binary operation as in the case of the group inverse. It is thus not a surprise that any group is a semigroup with involution. However, there are significant natural examples of semigroups with involution that are not groups. An example from linear algebra is the multiplicative monoid of real square matrices of order ''n'' (called the full linear monoid). The map which sends a matrix to its transpose is an involution because the transpose is well defined for any matrix and obeys the law , which has the same form of interactio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Converse Relation
In mathematics, the converse relation, or transpose, 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^ = \. The notation is analogous with that for an inverse function. Although many functions do not have an inverse, every relation does have a unique converse. The unary operation that maps a relation to the converse relation is an involution, so it induces the structure of a semigroup with involution on the binary relations on a set, or, more generally, induces a dagger category on the category of relations as detailed below. As a unary operation, taking the converse (sometimes called conversion or transposition) commutes with the order-rel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Property
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: \begin (2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\ 2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 . \end Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition of morphisms is the composition of functions. Many other categories (such as the category of groups, with group homomorphisms as arrows) add structure to the objects of the category of sets and/or restrict the arrows to functions of a particular kind. Properties of the category of sets The axioms of a category are satisfied by Set because composition of functions is associative, and because every set ''X'' has an identity function id''X'' : ''X'' → ''X'' which serves as identity element for function composition. The epimorphisms in Set are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps. The empty set serves as the initial object in Set with empty functions as morphisms. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Relations
In mathematics, the category Rel has the class of sets as objects and binary relations as morphisms. A morphism (or arrow) ''R'' : ''A'' → ''B'' in this category is a relation between the sets ''A'' and ''B'', so . The composition of two relations ''R'': ''A'' → ''B'' and ''S'': ''B'' → ''C'' is given by :(''a'', ''c'') ∈ ''S'' o ''R'' ⇔ for some ''b'' ∈ ''B'', (''a'', ''b'') ∈ ''R'' and (''b'', ''c'') ∈ ''S''. Rel has also been called the "category of correspondences of sets". Properties The category Rel has the category of sets Set as a (wide) subcategory, where the arrow in Set corresponds to the relation defined by .This category is called SetRel by Rydeheard and Burstall. A morphism in Rel is a relation, and the corresponding morphism in the opposite category to Rel has arrows reversed, so it is the converse relation. Thus Rel contains its opposite and is self-dual. The involution represented by taking the converse relation provides the dagger to mak ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Z Notation
The Z notation is a formal specification language used for describing and modelling computing systems. It is targeted at the clear specification of computer programs and computer-based systems in general. History In 1974, Jean-Raymond Abrial published "Data Semantics". He used a notation that would later be taught in the University of Grenoble until the end of the 1980s. While at EDF ( Électricité de France), working with Bertrand Meyer, Abrial also worked on developing Z. The Z notation is used in the 1980 book ''Méthodes de programmation''. Z was originally proposed by Abrial in 1977 with the help of Steve Schuman and Bertrand Meyer. It was developed further at the Programming Research Group at Oxford University, where Abrial worked in the early 1980s, having arrived at Oxford in September 1979. Abrial has said that Z is so named "Because it is the ultimate language!" although the name " Zermelo" is also associated with the Z notation through its use of Zermelo–F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Juxtaposition
Juxtaposition is an act or instance of placing two elements close together or side by side. This is often done in order to compare/contrast the two, to show similarities or differences, etc. Speech Juxtaposition in literary terms is the showing contrast by concepts placed side by side. An example of juxtaposition are the quotes "Ask not what your country can do for you; ask what you can do for your country", and "Let us never negotiate out of fear, but let us never fear to negotiate", both by John F. Kennedy, who particularly liked juxtaposition as a rhetorical device. Jean Piaget specifically contrasts juxtaposition in various fields from syncretism, arguing that "juxtaposition and syncretism are in antithesis, syncretism being the predominance of the whole over the details, juxtaposition that of the details over the whole". Piaget writes: In grammar, juxtaposition refers to the absence of linking elements in a group of words that are listed together. Thus, where English us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composition Of Functions
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and are composed to yield a function that maps in domain to in codomain . Intuitively, if is a function of , and is a function of , then is a function of . The resulting ''composite'' function is denoted , defined by for all in . The notation is read as " of ", " after ", " circle ", " round ", " about ", " composed with ", " following ", " then ", or " on ", or "the composition of and ". Intuitively, composing functions is a chaining process in which the output of function feeds the input of function . The composition of functions is a special case of the composition of relations, sometimes also denoted by \circ. As a result, all properties of composition of relations are true of composition of functions, such as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
