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Natural Join
In relational algebra, a join is a binary operation, written as R \bowtie S where R and S represent relations, that combines their data where they have a common attribute. Natural join Natural join (⨝) is a binary operator that is written as (''R'' ⨝ ''S'') where ''R'' and ''S'' are relations. The result of the natural join is the set of all combinations of tuples in ''R'' and ''S'' that are equal on their common attribute names. For an example consider the tables ''Employee'' and ''Dept'' and their natural join: Note that neither the employee named Mary nor the Production department appear in the result. Mary does not appear in the result because Mary's Department, "Human Resources", is not listed in the Dept relation and the Production department does not appear in the result because there are no tuples in the Employee relation that have "Production" as their DeptName attribute. This can also be used to define composition of relations. For example, the compositi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relational Algebra
In database theory, relational algebra is a theory that uses algebraic structures for modeling data and defining queries on it with well founded semantics (computer science), semantics. The theory was introduced by Edgar F. Codd. The main application of relational algebra is to provide a theoretical foundation for relational databases, particularly query languages for such databases, chief among which is SQL. Relational databases store tabular data represented as relation (database), relations. Queries over relational databases often likewise return tabular data represented as relations. The main purpose of relational algebra is to define Operator (mathematics), operators that transform one or more input relations to an output relation. Given that these operators accept relations as input and produce relations as output, they can be combined and used to express complex queries that transform multiple input relations (whose data are stored in the database) into a single output rela ... [...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] |
Relation (database)
In database theory, a relation, as originally defined by E. F. Codd, is a set of tuples (d1,d2,...,dn), where each element dj is a member of Dj, a data domain. Codd's original definition notwithstanding, and contrary to the usual definition in mathematics, there is no ordering to the elements of the tuples of a relation. Instead, each element is termed an attribute value. An attribute is a name paired with a domain (nowadays more commonly referred to as a type or data type). An attribute value is an attribute name paired with an element of that attribute's domain, and a tuple is a ''set'' of attribute values in which no two distinct elements have the same name. Thus, in some accounts, a tuple is described as a function, mapping names to values. A set of attributes in which no two distinct elements have the same name is called a heading. It follows from the above definitions that to every tuple there corresponds a unique heading, being the set of names from the tuple, paired wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Relation
In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs (x, y), where x is an element of X and y is an element of Y. It encodes the common concept of relation: an element x is ''related'' to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation. 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 p is related to each integer z that is a Divisibility, multiple of p, but not to an integer that is not a Multiple (mathematics), multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as -4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13. Binary relations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unicode
Unicode or ''The Unicode Standard'' or TUS is a character encoding standard maintained by the Unicode Consortium designed to support the use of text in all of the world's writing systems that can be digitized. Version 16.0 defines 154,998 Character (computing), characters and 168 script (Unicode), scripts used in various ordinary, literary, academic, and technical contexts. Unicode has largely supplanted the previous environment of a myriad of incompatible character sets used within different locales and on different computer architectures. The entire repertoire of these sets, plus many additional characters, were merged into the single Unicode set. Unicode is used to encode the vast majority of text on the Internet, including most web pages, and relevant Unicode support has become a common consideration in contemporary software development. Unicode is ultimately capable of encoding more than 1.1 million characters. The Unicode character repertoire is synchronized with Univers ... [...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] |
Category Theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber Product
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is written :. Usually the morphisms and are omitted from the notation, and then the pullback is written :. The pullback comes equipped with two natural morphisms and . The pullback of two morphisms and need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, may intuitively be thought of as consisting of pairs of elements with in , in , and . For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative square. The dual concept of the pullback is the '' pushout''. Universal property Explicitly, a pullback of the morphisms f and g consists of an object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idempotence
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency). The term was introduced by American mathematician Benjamin Peirce in 1870 in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from + '' potence'' (same + power). Definition An element x of a set S equipped with a binary operator \cdot is said to be ''idempotent'' under \cdot if : . The ''binary operation'' \cdot is said to be ''idempotent'' if : . Examples * In the monoid (\mathbb, \times) of the natural numbers with multiplication, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foreign Key
A foreign key is a set of attributes in a table that refers to the primary key of another table, linking these two tables. In the context of relational databases, a foreign key is subject to an inclusion dependency constraint that the tuples consisting of the foreign key attributes in one relation, R, must also exist in some other (not necessarily distinct) relation, S; furthermore that those attributes must also be a candidate key in S. In other words, a foreign key is a set of attributes that a candidate key. For example, a table called TEAM may have an attribute, MEMBER_NAME, which is a foreign key referencing a candidate key, PERSON_NAME, in the PERSON table. Since MEMBER_NAME is a foreign key, any value existing as the name of a member in TEAM must also exist as a person's name in the PERSON table; in other words, every member of a TEAM is also a PERSON. Summary The table containing the foreign key is called the child table, and the table containing the candidate ke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predicate (mathematics)
In logic, a predicate is a symbol that represents a property or a relation. For instance, in the first-order formula P(a), the symbol P is a predicate that applies to the individual constant a. Similarly, in the formula R(a,b), the symbol R is a predicate that applies to the individual constants a and b. According to Gottlob Frege, the meaning of a predicate is exactly a function from the domain of objects to the truth values "true" and "false". In the semantics of logic, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula R(a,b) would be true on an interpretation if the entities denoted by a and b stand in the relation denoted by R. Since predicates are non-logical symbols, they can denote different relations depending on the interpretation given to them. While first-order logic only includes predicates that apply to individual objects, other logics may allow predicates that apply to collections of objects defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relation (mathematics)
In mathematics, a relation denotes some kind of ''relationship'' between two mathematical object, objects in a Set (mathematics), set, which may or may not hold. As an example, "''is less than''" is a relation on the set of natural numbers; it holds, for instance, between the values and (denoted as ), and likewise between and (denoted as ), but not between the values and nor between and , that is, and both evaluate to false. As another example, "''is sister of'' is a relation on the set of all people, it holds e.g. between Marie Curie and Bronisława Dłuska, and likewise vice versa. Set members may not be in relation "to a certain degree" – either they are in relation or they are not. Formally, a relation over a set can be seen as a set of ordered pairs of members of . The relation holds between and if is a member of . For example, the relation "''is less than''" on the natural numbers is an infinite set of pairs of natural numbers that contains both and , b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
