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Relation Of Degree Zero
A relation of degree zero, 0-ary relation, or nullary relation is a relation with zero attributes. There are exactly two relations of degree zero. One has cardinality zero; that is, contains no tuples at all. The other has cardinality 1 and contains only the unique 0-tuple.:56 The zero-degree relations represent true and false in relational algebra.:57 Under the closed-world assumption, an ''n''-ary relation is interpreted as the extension of some ''n''-adic predicate: all and only those ''n''-tuples whose values, substituted for corresponding free variables in the predicate, yield propositions that hold true, appear in the relation. A zero-degree relation is therefore interpreted as the extension of the 0-adic predicate . The zero-degree relation with cardinality zero therefore represents false because it contains no tuples that yield a true proposition, and the zero-degree relation with cardinality 1 represents true because it contains the unique 0-tuple that yields a tr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nullary
In logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ..., mathematics, and computer science, arity () is the number of argument of a function, arguments or operands taken by a function (mathematics), function, operation (mathematics), operation or relation (mathematics), relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and philosophy, arity may also be called adicity and degree. In linguistics, it is usually named valency (linguistics), valency. Examples In general, functions or operators with a given arity follow the naming conventions of ''n''-based numeral systems, such as Binary numeral system, binary and hexadecimal. A Latin prefix is combined with the -ary suffix. For example: * A nullary function takes no argum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Operator
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 particular in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as group (mathematics), groups and ring (mathematics), rings. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. Definitions Let be a set equipped with a binary operation ∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an . An identity with respect to addition is called an Additive identity, (often denoted as 0) and an identity with respect to m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty Set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set. Any set other than the empty set is called ''non-empty''. In some textbooks and popularizations, the empty set is referred to as the "null set". However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). Notation Common notations for the empty set include "", "\emptyset", and "∅". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø () in the Danish orthography, Danish and Norwegian orthography, Norwegian a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tweedledum And Tweedledee
Tweedledum and Tweedledee are characters in an English nursery rhyme and in Lewis Carroll's 1871 book '' Through the Looking-Glass, and What Alice Found There''. Their names may have originally come from an epigram written by poet John Byrom. The nursery rhyme has a Roud Folk Song Index number of 19800. The names have since become synonymous in western popular culture slang for any two people whose appearances and actions are identical. Lyrics Common versions of the nursery rhyme include: :Tweedledum and Tweedledee : Agreed to have a battle; :For Tweedledum said Tweedledee : Had spoiled his nice new rattle. :Just then flew down a monstrous crow, : As black as a tar-barrel; :Which frightened both the heroes so, : They quite forgot their quarrel.I. Opie and P. Opie, ''The Oxford Dictionary of Nursery Rhymes'' (Oxford University Press, 1951, 2nd edn., 1997), p. 418. Origins The words "Tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hugh Darwen
Hugh Darwen is a computer scientist who was an employee of IBM United Kingdom from 1967. to 2004, and has been involved in the development of the relational model. Work From 1978 to 1982 he was a chief architect on Business System 12, a database management system that faithfully embraced the principles of the relational model.. He worked closely with Christopher J. Date and represented IBM at the ISO SQL committees (JTC1 SC32 WG3 Database languages,. WG4 SQL/MM.) until his retirement from IBM. Darwen is the author of The Askew Wall and co-author of The Third Manifesto, a proposal for serving object-oriented programs with purely relational databases without compromising either side and getting the best of both worlds, arguably even better than with so-called object-oriented databases. From 2004 to 2013 he lectured on relational databases at the Department of Computer Science, University of Warwick (UK), and from 1989 to 2014 was a tutor and consultant for the Open Universit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projection (relational Algebra)
In relational algebra, a projection is a unary operation written as \Pi_( R ), where R is a relation and a_1,...,a_n are attribute names. Its result is defined as the set obtained when the components of the tuples in R are restricted to the set \ – it ''discards'' (or ''excludes'') the other attributes. In practical terms, if a relation is thought of as a table, then projection can be thought of as picking a subset of its columns. For example, if the attributes are (name, age), then projection of the relation onto attribute list (age) yields – we have discarded the names, and only know what ages are present. Projections may also modify attribute values. For example, if R has attributes a, b, c, where the values of b are numbers, then \Pi_( R ) is like R, but with all b-values halved.http://www.csee.umbc.edu/~pmundur/courses/CMSC661-02/rel-alg.pdf ''See Problem 3.8.B on page 3'' Related concepts The closely related concept in set theory (see: projection (set theory)) di ... [...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] |
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] |
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] |
Identity Element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as group (mathematics), groups and ring (mathematics), rings. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. Definitions Let be a set equipped with a binary operation ∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an . An identity with respect to addition is called an Additive identity, (often denoted as 0) and an identity with respect to m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predicate (mathematical Logic)
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 defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
