First-order logic—also called predicate logic, predicate calculus, quantificational logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all men are mortal", in first-order logic one can have expressions in the form "for all x, if x is a man, then x is mortal"; where "for all x" is a quantifier, x is a variable, and "... is a man" and "... is mortal" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.
In mathematics, a finitary relation over a sequence of sets X1, ..., Xn is a subset of the Cartesian product X1 × ... × Xn; that is, it is a set of n-tuples (x1, ..., xn), each being a sequence of elements xi in the corresponding Xi. 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.
A relational database (RDB) is a database based on the relational model of data, as proposed by E. F. Codd in 1970. A database management system used to maintain relational databases is a relational database management system (RDBMS). Many relational database systems are equipped with the option of using SQL for querying and updating the database.
The relational model (RM) is an approach to managing data using a structure and language consistent with first-order predicate logic, first described in 1969 by English computer scientist Edgar F. Codd, where all data is represented in terms of tuples, grouped into relations. A database organized in terms of the relational model is a relational database.
Relation or relations may refer to:
In logic, mathematics, and computer science, arity is the number of arguments or operands taken by a function, operation or 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.
In database theory, relational algebra is a theory that uses algebraic structures for modeling data and defining queries on it with well founded semantics. The theory was introduced by Edgar F. Codd.
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity, just as the empty sum—the result of adding no numbers—is by convention zero, or the additive identity. When numbers are implied, the empty product becomes one.
A join clause in the Structured Query Language (SQL) combines columns from one or more tables into a new table. The operation corresponds to a join operation in relational algebra. Informally, a join stitches two tables and puts on the same row records with matching fields : INNER, LEFT OUTER, RIGHT OUTER, FULL OUTER and CROSS.
In SQL, null or NULL is a special marker used to indicate that a data value does not exist in the database. Introduced by the creator of the relational database model, E. F. Codd, SQL null serves to fulfill the requirement that all true relational database management systems (RDBMS) support a representation of "missing information and inapplicable information". Codd also introduced the use of the lowercase Greek omega (ω) symbol to represent null in database theory. In SQL, NULL is a reserved word used to identify this marker.
In relational algebra, a projection is a unary operation written as , where is a relation and are attribute names. Its result is defined as the set obtained when the components of the tuples in are restricted to the set – it discards the other attributes.
Sixth normal form (6NF) is a normal form used in relational database normalization which extends the relational algebra and generalizes relational operators to support interval data, which can be useful in temporal databases.
In mathematics, an operation is a function from a set to itself. For example, an operation on real numbers will take in real numbers and return a real number. An operation can take zero or more input values to a well-defined output value. The number of operands is the arity of the operation.
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation algebra is the algebra 2 X 2 of all binary relations on a set X, that is, subsets of the cartesian square X2, with R•S interpreted as the usual composition of binary relations R and S, and with the converse of R as the converse relation.
Boolean algebra is a mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation.' Just as group theory deals with groups, and linear algebra with vector spaces, Boolean algebras are models of the equational theory of the two values 0 and 1. Common to Boolean algebras, groups, and vector spaces is the notion of an algebraic structure, a set closed under some operations satisfying certain equations.
In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. They are rarely made explicit in more philosophical treatments of logic.
In category theory, a branch of mathematics, given a morphism f: X → Y and a morphism g: Z → Y, a lift or lifting of f to Z is a morphism h: X → Z such that f = g∘h. We say that f factors through h.
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. 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.
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B. In terms of set-builder notation, that is
