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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-related op ... [...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 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 abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nephew And Niece
In the lineal kinship system used in the English-speaking world, a niece or nephew is a child of the subject's sibling or sibling-in-law. The converse relationship, the relationship from the niece or nephew's perspective, is that of an aunt or uncle. A niece is female and a nephew is male. The term nibling has been used in place of the common, gender-specific terms in some specialist literature. As aunt/uncle and niece/nephew are separated by one generation, they are an example of a second-degree relationship. They are 25% related by blood. Lexicology The word nephew is derived from the French word ''neveu'' which is derived from the Latin ''nepos''. The term ''nepotism'', meaning familial loyalty, is derived from this Latin term. ''Niece'' entered Middle English from the Old French word ''nece'', which also derives from Latin ''nepotem''. The word ''nibling'' is a neologism suggested by Samuel Martin in 1951 as a cover term for "nephew or niece"; it is not c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heterogeneous Relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in . It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element is ''related'' to an element , if and only if the pair belongs to the set of ordered pairs that defines the ''binary relation''. A binary relation is the most studied special case of an -ary relation over sets , which is a subset of the Cartesian product X_1 \times \cdots \times X_n. An example of a binary relation is the "divides" relation over the set of prime numbers \mathbb and the set of integers \mathbb, in which each prime is related to each integer that is a multiple of , but not to an integer that is not a multiple of . In this relation, for instance, the prime number 2 is related to numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantale
In mathematics, quantales are certain partially ordered algebraic structures that generalize locales ( point free topologies) as well as various multiplicative lattices of ideals from ring theory and functional analysis (C*-algebras, von Neumann algebras). Quantales are sometimes referred to as ''complete residuated semigroups''. Overview A quantale is a complete lattice ''Q'' with an associative binary operation ∗ : ''Q'' × ''Q'' → ''Q'', called its multiplication, satisfying a distributive property such that :x*\left(\bigvee_\right) = \bigvee_(x*y_i) and :\left(\bigvee_\right)*=\bigvee_(y_i*x) for all ''x'', ''yi'' in ''Q'', ''i'' in ''I'' (here ''I'' is any index set). The quantale is unital if it has an identity element ''e'' for its multiplication: :x*e = x = e*x for all ''x'' in ''Q''. In this case, the quantale is naturally a monoid with respect to its multiplication ∗. A unital quantale may be defined equivalently as a monoid in the catego ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-adjoint
In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if x^*=y then since y^*=x^=x in a star-algebra, the set is a self-adjoint set even though ''x'' and ''y'' need not be self-adjoint elements. In functional analysis, a linear operator A : H \to H on a Hilbert space is called self-adjoint if it is equal to its own adjoint ''A''. See self-adjoint operator for a detailed discussion. If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator ''A'' is self-adjoint if and only if the matrix describing ''A'' with respect to this basis is Hermitian, i.e. if it is equal to its own conjugate transpose. Hermitian matrices are also called self-adjoint. In a dagger ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Relation
A symmetric relation is a type of binary relation. An example is the relation "is equal to", because if ''a'' = ''b'' is true then ''b'' = ''a'' is also true. Formally, a binary relation ''R'' over a set ''X'' is symmetric if: :\forall a, b \in X(a R b \Leftrightarrow b R a) , where the notation aRb means that (a,b)\in R. If ''R''T represents the converse of ''R'', then ''R'' is symmetric if and only if ''R'' = ''R''T. Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation. Examples In mathematics * "is equal to" ( equality) (whereas "is less than" is not symmetric) * "is comparable to", for elements of a partially ordered set * "... and ... are odd": :::::: Outside mathematics * "is married to" (in most legal systems) * "is a fully biological sibling of" * "is a homophone of" * "is co-worker of" * "is teammate of" Relationship to asymmetric and antisymmetric relations By definition, a nonempty relation cannot be b ... [...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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Function
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when is the identity function, the equality is true for all values of to which can be applied. Definition Formally, if is a set, the identity function on is defined to be a function with as its domain and codomain, satisfying In other words, the function value in the codomain is always the same as the input element in the domain . The identity function on is clearly an injective function as well as a surjective function, so it is bijective. The identity function on is often denoted by . In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or ''diagonal'' of . Algebraic properties If is any function, then we have ... [...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 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; S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary operation ''on a set'' is a binary operation whose two domains and the codomain are the same set. Examples include the familiar arithmetic operations of addition, subtraction, and multiplication. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups. An operation of arity two that involves several sets is sometimes also called a ''binary operation''. For example, scalar multiplication of vector spaces takes a scalar and a vector to produce a vector, and scalar product takes two vectors to produce a scalar. Such binary operations may be called simply binary functions. Binary operations are the keystone of most algebraic structures that are studied ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endorelation
In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''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 over people. Common types of endorelations include orders, graphs, and equivalences. Specialized studies order theory and graph theory have developed understanding of endorelations. Terminology particular for graph theory is used for description, with an ordinary 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'' corresponds to an edge between ''x'' and ''y'' in the graph, and to a 1 in the square matrix of ''R''. It is called an adjacency matrix in graph terminology. Parti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics. The functions from a set into itself form a monoid with respect to function composition. More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. In computer science and computer programming, the set of strings built from a given set of characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for process calculi and concurrent computing. In theoretical computer science, the study of monoids is fundamental for autom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
