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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] |
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] |
Properties
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Philosophy and science * Property (philosophy), in philosophy and logic, an abstraction characterizing an object * Material properties, properties by which the benefits of one material versus another can be assessed * Chemical property, a material's properties that becomes evident during a chemical reaction *Physical property, any property that is measurable whose value describes a state of a physical system * Thermodynamic properties, in thermodynamics and materials science, intensive and extensive physical properties of substances * Mathematical property, a property is any characteristic that applies to a given set * Semantic property * Mental property, a property of the mind studied by many sciences and parasciences Computer science * Property (programming), a type of class member in object-oriented programming * .properties, a Java Pro ... [...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 n ... [...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 (its codomain is also its range), 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 propert ... [...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] |
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, a binary operation on a set is a binary function that maps every pair of elements of the set to an element of the set. Examples include the familiar arithmetic operations like addition, subtraction, multiplication, set operations like union, complement, intersection. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups. A binary function 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. Binary operations are the keystone of most structures that are studied in algebra, in parti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endorelation
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 orders, graphs, and 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'' in the graph, and to a 1 in the square matrix of ''R''. It is calle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoid
In abstract algebra, 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 . 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 automata theory (Krohn–Rhodes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sibling
A sibling is a relative that shares at least one parent with the other person. A male sibling is a brother, and a female sibling is a sister. A person with no siblings is an only child. While some circumstances can cause siblings to be raised separately (such as foster care or adoption), most societies have siblings grow up together. This causes the development of strong emotional bonds, with siblinghood considered a unique type of relationship. The emotional bond between siblings is often complicated and is influenced by factors such as parental treatment, birth order, personality, and personal experiences outside the family. Medically, a full-sibling is a first-degree relative and a half-sibling is a second-degree relative as they are related by 50% and 25%, respectively. Definitions The word ''sibling'' was reintroduced in 1903 in an article in '' Biometrika'', as a translation for the German ''Geschwister'', having not been used since Middle English, specifically 142 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aunt
An aunt is a woman who is a sibling of a parent or married to a sibling of a parent. Aunts who are related by birth are second-degree relatives. Alternate terms include auntie or aunty. Aunt, auntie, and aunty also may be titles bestowed by parents and children to close friends of one or both parents who assume a sustained caring or nurturing role for the children. Children in some cultures and families may refer to the cousins of their parents as aunt or uncle due to the age and generation gap. The word comes from via Old French ''ante'' and is a family">-4; we might wonder whether there's a point at which it's appropriate to talk of the beginnings of French, that is, when it wa ... ''ante'' and is a family relationship within an extended or immediate family. The male counterpart of an aunt is an uncle, and the reciprocal relationship is that of a niece and nephew, nephew or niece. The gender-neutral term pibling, a shortened form of ''parent's sibling'', may refer to eit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uncle
An uncle is usually defined as a male relative who is a sibling of a parent or married to a sibling of a parent, as well as the parent of the cousins. Uncles who are related by birth are second-degree relatives. The female counterpart of an uncle is an aunt, and the reciprocal relationship is that of a nephew or niece. The word comes from , the diminutive of ''avus'' (grandfather), and is a family relationship within an extended or immediate family. A popular colloquial term in English is Unc. In some cultures and families, children may refer to the cousins of their parents as uncle (or aunt). It is also used as a title of respect for older relatives, neighbours, acquaintances, family friends, and even total strangers in some cultures, for example Aboriginal Australian elders. Using the term in this way is a form of fictive kinship. Any social institution where a special relationship exists between a man and his sisters' children is known as an avunculate (or avunculism ... [...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 an individual's sibling or sibling-in-law. A niece is female and a nephew is male, and they would call their parents' siblings aunt or uncle. The gender-neutral term nibling has been used in place of the common terms, especially in specialist literature. As aunt/uncle and niece/nephew are separated by one generation, they are an example of a second-degree relationship. Unless related by marriage, they are 25% or more related by blood if the aunt/uncle is a full sibling of one of the parents, or 12.5% if they are a half-sibling. Lexicology The word nephew is derived from the French word which is derived from the Latin . The term ''nepotism'', meaning familial loyalty, is derived from this Latin term. ''Niece'' entered Middle English from the Old French word , which also derives from Latin . The word ''nibling'', derived from ''sibling'', is a neologism suggested by Samue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
