The relational model for database management is a database model based on predicate logic and set theory. It was first formulated and proposed in 1969 by Edgar Codd with aims that included avoiding, without loss of completeness, the need to write computer programs to express database queries and enforce database integrity constraints. "Relation" is a mathematical term for "table", and thus "relational" roughly means "based on tables". It does not refer to the links or "keys" between tables, contrary to popular belief. This article is about computing. ... A database model is a theory or specification describing how a database is structured and used. ... ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Also: 1969 (Stargate SG-1) episode. ... Edgar Frank Ted Codd (August 23, 1923 – April 18, 2003) was a British computer scientist who made seminal contributions to the theory of relational databases. ...

Codd's original description of the relational model, "Derivability, Redundancy, and Consistency of Relations Stored in Large Data Banks", was published in an IBM Research Report in 1969. A revised version of this paper, the highly acclaimed "A Relational Model of Data for Large Shared Data Banks", was published in Communications of the ACM the following year.^[1] Communications of the ACM (CACM) is the flagship monthly magazine of the Association for Computing Machinery. ...

The model

The fundamental assumption of the relational model is that all data are represented as mathematical n-ary relations, an n-ary relation being a subset of the Cartesian product of n domains. In the mathematical model, reasoning about such data is done in two-valued predicate logic, meaning there are two possible evaluations for each proposition: either true or false (and in particular no third value such as unknown, or not applicable, either of which are often associated with the concept of NULL). Some think that logic (which is inherently two-valued) is an important part of the relational model, where others think that a system that uses a form of three-valued logic can still be considered relational. For other uses, see Data (disambiguation). ... In mathematics, the concept of a relation is a generalization of 2-place relations, such as the relation of equality, denoted by the sign = in a statement like 5 + 7 = 12, or the relation of order, denoted by the sign < in a statement like 5 < 12. Relations that involve two... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics, the Cartesian product is a direct product of sets. ... Reasoning is the mental (cognitive) process of looking for reasons to support beliefs, conclusions, actions or feelings. ... ... Evaluation is the systematic determination of merit, worth, and significance of something or someone. ... This article is about the word proposition as it is used in logic, philosophy, and linguistics. ... The Greek lowercase omega (ω) character is historically used by academics to represent Null in relational databases. ... Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... A ternary, three-valued or trivalent logic is a multi-valued logic in which there are three truth values indicating true, false and some third value. ...

Data are operated upon by means of a relational calculus or algebra, these being equivalent in expressive power. The relational calculus refers to the two calculi, the tuple calculus and the domain calculus, that are part of the relational model for databases and that provide a declarative way to specify database queries. ... Relational algebra, an offshoot of first-order logic, is a set of relations closed under operators. ... Expressive power is a relatively generic term used by Abelson and Sussman in Structure and Interpretation of Computer Programs to describe the conciseness with which a particular logical design may be translated into a computer program in a given programming language. ...

The relational model of data permits the database designer to create a consistent, logical representation of information. Consistency is achieved by including declared constraints in the database design, which is usually referred to as the logical schema. The theory includes a process of database normalization whereby a design with certain desirable properties can be selected from a set of logically equivalent alternatives. The access plans and other implementation and operation details are handled by the DBMS engine, and are not reflected in the logical model. This contrasts with common practice for SQL DBMSs in which performance tuning often requires changes to the logical model. The ASCII codes for the word Wikipedia represented in binary, the numeral system most commonly used for encoding computer information. ... Database normalization is a design technique for structuring relational database tables. ... A query plan (or query execution plan) is an ordered set of steps used to access information in a SQL relational database management system. ... A database management system (DBMS) is a computer program (or more typically, a suite of them) designed to manage a database, a large set of structured data, and run operations on the data requested by numerous users. ... Performance tuning is the improvement of system performance. ...

The basic relational building block is the domain or data type, usually abbreviated nowadays to type. A tuple is an unordered set of attribute values. An attribute is an ordered pair of attribute name and type name. An attribute value is a specific valid value for the type of the attribute. This can be either a scalar value or a more complex type. In data management and database analysis, a data domain refers to all the unique values which a data element may contain. ... A data type is a constraint placed upon the interpretation of data in a type system in computer programming. ... In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... An attribute is the following: Generally, an attribute is an abstraction characteristic of an entity In database management, an attribute is a property inherent in an entity or associated with that entity for database purposes. ...

A relation consists of a heading and a body. A heading is a set of attributes. A body (of an n-ary relation) is a set of n-tuples. The heading of the relation is also the heading of each of its tuples.

A relation is defined as a set of n-tuples. In both mathematics and the relational database model, a set is an unordered collection of items, although some DBMSs impose an order to their data. In mathematics, a tuple has an order, and allows for duplication. E.F. Codd originally defined tuples using this mathematical definition.^[2] Later, it was one of E.F. Codd's great insights that using attribute names instead of an ordering would be so much more convenient (in general) in a computer language based on relations ^{[citation needed]}. This insight is still being used today. Though the concept has changed, the name "tuple" has not. An immediate and important consequence of this distinguishing feature is that in the relational model the Cartesian product becomes commutative. In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ... Edgar Frank Ted Codd (August 23, 1923 – April 18, 2003) was a British computer scientist who made seminal contributions to the theory of relational databases. ... Edgar Frank Ted Codd (August 23, 1923 – April 18, 2003) was a British computer scientist who made seminal contributions to the theory of relational databases. ... In mathematics, the Cartesian product is a direct product of sets. ... Mathematical meaning A map or binary operation is said to be commutative when, for any x in A and any y in B . ...

A table is an accepted visual representation of a relation; a tuple is similar to the concept of row, but note that in the database language SQL the columns and the rows of a table are ordered. In relational databases, SQL databases, and flat file databases, a table is a set of data elements (values) that is organized using a model of horizontal rows and vertical columns. ... In the context of a relational database, a row, also called a record or tuple, represents a single, implicitly structured data item in a table. ... SQL (IPA: or IPA: ), commonly expanded as Structured Query Language, is a computer language designed for the retrieval and management of data in relational database management systems, database schema creation and modification, and database object access control management. ...

A relvar is a named variable of some specific relation type, to which at all times some relation of that type is assigned, though the relation may contain zero tuples. An abbreviation for relation variable in relational databases. ...

The basic principle of the relational model is the Information Principle: all information is represented by data values in relations. In accordance with this Principle, a relational database is a set of relvars and the result of every query is presented as a relation. This article does not cite any references or sources. ... The ASCII codes for the word Wikipedia represented in binary, the numeral system most commonly used for encoding computer information. ... A relational database is a database that conforms to the relational model, and refers to a databases data and schema (the databases structure of how that data is arranged). ...

The consistency of a relational database is enforced, not by rules built into the applications that use it, but rather by constraints, declared as part of the logical schema and enforced by the DBMS for all applications. In general, constraints are expressed using relational comparison operators, of which just one, "is subset of" (⊆), is theoretically sufficient. In practice, several useful shorthands are expected to be available, of which the most important are candidate key (really, superkey) and foreign key constraints. A database management system (DBMS) is a computer program (or more typically, a suite of them) designed to manage a database, a large set of structured data, and run operations on the data requested by numerous users. ... In the relational model a candidate key of a relation variable (relvar) is a set of attributes of that relvar such that (1) at all times it holds in the relation assigned to that variable that there are no two distinct tuples with the same values for these attributes and... A superkey is defined in the relational model as a set of attributes of a relation variable (relvar) for which it holds that in all relations assigned to that variable there are no two distinct tuples (rows) that have the same values for the attributes in this set. ... In the context of relational databases, a foreign key is a referential constraint between two tables[1]. The foreign key identifies a column or a set of columns in one (referencing) table that refers to a column or set of columns in another (referenced) table. ...

Interpretation

To fully appreciate the relational model of data it is essential to understand the intended interpretation of a relation.

The body of a relation is sometimes called its extension. This is because it is to be interpreted as a representation of the extension of some predicate, this being the set of true propositions that can be formed by replacing each free variable in that predicate by a name (a term that designates something). The extension of a predicate is the set of true propositions that can be formed by substituting a term for each of its free variables. ... In linguistics and logic, a predicate is an expression that can be true of something. ... This article is about the word proposition as it is used in logic, philosophy, and linguistics. ... In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation for a place or places in an expression, into which some definite substitution may take place, or with respect to which some operation (summation or quantification, to give two...

There is a one-to-one correspondence between the free variables of the predicate and the attribute names of the relation heading. Each tuple of the relation body provides attribute values to instantiate the predicate by substituting each of its free variables. The result is a proposition that is deemed, on account of the appearance of the tuple in the relation body, to be true. Contrariwise, every tuple whose heading conforms to that of the relation but which does not appear in the body is deemed to be false. This assumption is known as the closed world assumption. The closed world assumption is the presumption that what is not currently known to be true is false. ...

For a formal exposition of these ideas, see the section Set Theory Formulation, below.

Application to databases

A type as used in a typical relational database might be the set of integers, the set of character strings, the set of dates, or the two boolean values true and false, and so on. The corresponding type names for these types might be the strings "int", "char", "date", "boolean", etc. It is important to understand, though, that relational theory does not dictate what types are to be supported; indeed, nowadays provisions are expected to be available for user-defined types in addition to the built-in ones provided by the system.

Attribute is the term used in the theory for what is commonly referred to as a column. Similarly, table is commonly used in place of the theoretical term relation (though in SQL the term is by no means synonymous with relation). A table data structure is specified as a list of column definitions, each of which specifies a unique column name and the type of the values that are permitted for that column. An attribute value is the entry in a specific column and row, such as "John Doe" or "35".

A tuple is basically the same thing as a row, except in an SQL DBMS, where the column values in a row are ordered. (Tuples are not ordered; instead, each attribute value is identified solely by the attribute name and never by its ordinal position within the tuple.) An attribute name might be "name" or "age".

A relation is a table structure definition (a set of column definitions) along with the data appearing in that structure. The structure definition is the heading and the data appearing in it is the body, a set of rows. A database relvar (relation variable) is commonly known as a base table. The heading of its assigned value at any time is as specified in the table declaration and its body is that most recently assigned to it by invoking some update operator (typically, INSERT, UPDATE, or DELETE). The heading and body of the table resulting from evaluation of some query are determined by the definitions of the operators used in the expression of that query. (Note that in SQL the heading is not always a set of column definitions as described above, because it is possible for a column to have no name and also for two or more columns to have the same name. Also, the body is not always a set of rows because in SQL it is possible for the same row to appear more than once in the same body.)

Competition

Other models are the hierarchical model and network model. Some systems using these older architectures are still in use today in data centers with high data volume needs or where existing systems are so complex and abstract it would be cost prohibitive to migrate to systems employing the relational model; also of note are newer object-oriented databases, even though many of them are DBMS-construction kits, rather than proper DBMSs. An abstract model (or conceptual model) is a theoretical construct that represents something, with a set of variables and a set of logical and quantitative relationships between them. ... In a hierarchical data model, data are organized into a tree-like structure. ... The network model is a database model conceived as a flexible way of representing objects and their relationships. ... For other uses, see System (disambiguation). ... This article is about building architecture. ... A data center is a facility used for housing a large amount of electronic equipment, typically computers and communications equipment. ... In an object oriented database, information is represented in the form of objects as used in Object-Oriented Programming. ... A database management system (DBMS) is a computer program (or more typically, a suite of them) designed to manage a database, a large set of structured data, and run operations on the data requested by numerous users. ...

The relational model was the first formal database model. After it was defined, informal models were made to describe hierarchical databases (the hierarchical model) and network databases (the network model). Hierarchical and network databases existed before relational databases, but were only described as models after the relational model was defined, in order to establish a basis for comparison.

History

The relational model was invented by E.F. (Ted) Codd as a general model of data, and subsequently maintained and developed by Chris Date and Hugh Darwen among others. In The Third Manifesto (first published in 1995) Date and Darwen show how the relational model can accommodate certain desired object-oriented features without compromising its fundamental principles. Edgar Frank Ted Codd (August 23, 1923 – April 18, 2003) was a British computer scientist who made seminal contributions to the theory of relational databases. ... Christopher J. Date is an independent author, lecturer, researcher, and consultant, specializing in relational database technology. ... Hugh Darwen, employee of IBM UK from 1967 to 2004, has been involved in the history of the relational model since the beginning. ... The Third Manifesto (1995) is Christopher J. Dates and Hugh Darwens proposal for future relational database management systems that would avoid Object-Relational Impedance Mismatch between object-oriented programming languages and RDBMSs by fully supporting all the capabilities of the relational model. ... Object-oriented programming (OOP) is a computer programming paradigm in which a software system is modeled as a set of objects that interact with each other. ...

The foundation for the relational model included important works published by Georg Cantor (1874) and D.L. Childs (1968). Cantor was a 19th century German mathematician who published a number of articles and was the principal creator of set theory. Childs is an American mathematician whose "Description of a Set-Theoretic Data Structure" was cited by Codd in his seminal 1970 paper "A Relational Model of Data for Large Shared Data Banks". Childs' STDS uses set theory as the basis for querying data using set operations such as union, intersection, domain, range, restriction, cardinality and Cartesian product. The use of sets and set operations provided independence from physical data structures, a pioneering concept at the time. Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

SQL standard

SQL, initially pushed as the standard language for relational databases, deviates from the relational model in several places. The current ISO SQL standard doesn't mention the relational model or use relational terms or concepts. However, it is possible to create a database conforming to the relational model using SQL if one does not use certain SQL features. SQL (IPA: or IPA: ), commonly expanded as Structured Query Language, is a computer language designed for the retrieval and management of data in relational database management systems, database schema creation and modification, and database object access control management. ... Standardization, in the context related to technologies and industries, is the process of establishing a technical standard among competing entities in a market, where this will bring benefits without hurting competition. ... A relational database is a database that conforms to the relational model, and refers to a databases data and schema (the databases structure of how that data is arranged). ... This article does not cite any references or sources. ...

The following deviations from the relational model have been noted in SQL. Note that few database servers implement the entire SQL standard and in particular do not allow some of these deviations. Whereas NULL is nearly ubiquitous, for example, allowing duplicate column names within a table or anonymous columns is uncommon. SQL (IPA: or IPA: ), commonly expanded as Structured Query Language, is a computer language designed for the retrieval and management of data in relational database management systems, database schema creation and modification, and database object access control management. ...

Duplicate rows

The same row can appear more than once in an SQL table. The same tuple cannot appear more than once in a relation.

Anonymous columns

A column in an SQL table can be unnamed and thus unable to be referenced in expressions. The relational model requires every attribute to be named and referenceable.

Duplicate column names

Two or more columns of the same SQL table can have the same name and therefore cannot be referenced, on account of the obvious ambiguity. The relational model requires every attribute to be referenceable.

Column order significance

The order of columns in an SQL table is defined and significant, one consequence being that SQL's implementations of Cartesian product and union are both noncommutative. The relational model requires there to be no significance to any ordering of the attributes of a relation.

Views without CHECK OPTION

Updates to a view defined without CHECK OPTION can be accepted but the resulting update to the database does not necessarily have the expressed effect on its target. For example, an invocation of INSERT can be accepted but the inserted rows might not all appear in the view, or an invocation of UPDATE can result in rows disappearing from the view. The relational model requires updates to a view to have the same effect as if the view were a base relvar.

Columnless tables unrecognized

SQL requires every table to have at least one column, but there are two relations of degree zero (of cardinality one and zero) and they are needed to represent extensions of predicates that contain no free variables.

NULL

This special mark can appear instead of a value wherever a value can appear in SQL, in particular in place of a column value in some row. The deviation from the relational model arises from the fact that the implementation of this ad hoc concept in SQL involves the use of three-valued logic, under which the comparison of NULL with itself does not yield true but instead yields the third truth value, unknown; similarly the comparison NULL with something other than itself does not yield false but instead yields unknown. It is because of this behaviour in comparisons that NULL is described as a mark rather than a value. The relational model depends on the law of excluded middle under which anything that is not true is false and anything that is not false is true; it also requires every tuple in a relation body to have a value for every attribute of that relation. This particular deviation is disputed by some if only because E.F. Codd himself eventually advocated the use of special marks and a 4-valued logic, but this was based on his observation that there are two distinct reasons why one might want to use a special mark in place of a value, which led opponents of the use of such logics to discover more distinct reasons and at least as many as 19 have been noted, which would require a 21-valued logic. SQL itself uses NULL for several purposes other than to represent "value unknown". For example, the sum of the empty set is NULL, meaning zero, the average of the empty set is NULL, meaning undefined, and NULL appearing in the result of a LEFT JOIN can mean "no value because there is no matching row in the right-hand operand".

Concepts

SQL uses concepts "table", "column", "row" instead of "relvar", "attribute", "tuple". These are not merely differences in terminology. For example, a "table" may contain duplicate rows, whereas the same tuple cannot appear more than once in a relation.

“Excluded middle” redirects here. ...

Implementation

There have been several attempts to produce a true implementation of the relational database model as originally defined by Codd and explained by Date, Darwen and others, but none have been popular successes so far. Rel is one of the more recent attempts to do this. Edgar Frank Ted Codd (August 23, 1923 – April 18, 2003) was a British computer scientist who made seminal contributions to the theory of relational databases. ... Christopher J. Date is an independent author, lecturer, researcher, and consultant, specializing in relational database technology. ... Hugh Darwen, employee of IBM UK from 1967 to 2004, has been involved in the history of the relational model since the beginning. ... Rel is an Open Source true relational database management system that implements a significant portion of Chris Date and Hugh Darwens Tutorial D query language. ...

Controversies

Codd himself, some years after publication of his 1970 model, proposed a three-valued logic (True, False, Missing or NULL) version of it in order to deal with missing information, and in his The Relational Model for Database Management Version 2 (1990) he went a step further with a four-valued logic (True, False, Missing but Applicable, Missing but Inapplicable) version. But these have never been implemented, presumably because of attending complexity. SQL's NULL construct was intended to be part of a three-valued logic system, but fell short of that due to logical errors in the standard and in its implementations. See the section "SQL standard", above. Look up null in Wiktionary, the free dictionary. ... The relational model for database management is a database model based on predicate logic and set theory. ...

Design

Database normalization is usually performed when designing a relational database, to improve the logical consistency of the database design and the transactional performance. Database normalization is a design technique for structuring relational database tables. ... A relational database is a database that conforms to the relational model, and refers to a databases data and schema (the databases structure of how that data is arranged). ... Look up Consistency in Wiktionary, the free dictionary. ... Database design is the process of producing a detailed data model of a database. ...

There are two commonly used systems of diagramming to aid in the visual representation of the relational model: the entity-relationship diagram (ERD), and the related IDEF diagram used in the IDEF1X method created by the U.S. Air Force based on ERDs. For other uses, see System (disambiguation). ... Diagramming software consists of computer programs that are may be used to produce graphical diagrams. ... The Entity-Relationship model is a data model for high-level descriptions of conceptual data models, and it provides a graphical notation for representing such data models in the form of entity-relationship diagrams. ... The family of ICAM Definition Languages, short IDEF, were initiated in the 1970s and finished being developed in the 1980s. ... Seal of the Air Force. ...

The tree structure of data may enforce hierarchical model organization, with parent-child relationship table. In a hierarchical data model, data are organized into a tree-like structure. ...

Example database

An idealized, very simple example of a description of some relvars and their attributes:

• Customer(Customer ID, Tax ID, Name, Address, City, State, Zip, Phone)
• Order(Order No, Customer ID, Invoice No, Date Placed, Date Promised, Terms, Status)
• Order Line(Order No, Order Line No, Product Code, Qty)
• Invoice(Invoice No, Customer ID, Order No, Date, Status)
• Invoice Line(Invoice No, Line No, Product Code, Qty Shipped)
• Product(Product Code, Product Description)

In this design we have six relvars: Customer, Order, Order Line, Invoice, Invoice Line and Product. The bold, underlined attributes are candidate keys. The non-bold, underlined attributes are foreign keys. All Saints Chapel in the Cathedral Basilica of St. ... An abbreviation for relation variable in relational databases. ... In the relational model a candidate key of a relation variable (relvar) is a set of attributes of that relvar such that (1) at all times it holds in the relation assigned to that variable that there are no two distinct tuples with the same values for these attributes and... In the context of relational databases, a foreign key is a referential constraint between two tables[1]. The foreign key identifies a column or a set of columns in one (referencing) table that refers to a column or set of columns in another (referenced) table. ...

Usually one candidate key is arbitrarily chosen to be called the primary key and used in preference over the other candidate keys, which are then called alternate keys. In the relational model a candidate key of a relation variable (relvar) is a set of attributes of that relvar such that (1) at all times it holds in the relation assigned to that variable that there are no two distinct tuples with the same values for these attributes and... In database design, a primary key is a value that can be used to identify a unique row in a table. ... Preference (or taste) is a concept, used in the social sciences, particularly economics. ... An alternate key is any candidate key which is not selected to be the primary key. ...

A candidate key is a unique identifier enforcing that no tuple will be duplicated; this would make the relation into something else, namely a bag, by violating the basic definition of a set. Both foreign keys and superkeys (which includes candidate keys) can be composite, that is, can be composed of several attributes. Below is a tabular depiction of a relation of our example Customer relvar; a relation can be thought of as a value that can be attributed to a relvar. Identifiers (IDs) are lexical tokens that name entities. ... In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ... Look up Relation in Wiktionary, the free dictionary In mathematics, a relation is a generalization of arithmetic relations, such as = and <, which occur in statements, such as 5 < 6 or 2 + 2 = 4. See relation (mathematics), binary relation (of set theory and logic) and relational algebra. ... In mathematics, a multiset (sometimes also called a bag) differs from a set in that each member has a multiplicity, which is a cardinal number indicating (loosely speaking) how many times it is a member, or perhaps how many memberships it has in the multiset. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

Example: customer relation

 Customer ID Tax ID Name Address [More fields....] ================================================================================================== 1234567890 555-5512222 Munmun 323 Broadway ... 2223344556 555-5523232 Vijeta 1200 Main Street ... 3334445563 555-5533322 Ekta 871 1st Street ... 4232342432 555-5325523 E. F. Codd 123 It Way ... 

If we attempted to insert a new customer with the ID 1234567890, this would violate the design of the relvar since Customer ID is a primary key and we already have a customer 1234567890. The DBMS must reject a transaction such as this that would render the database inconsistent by a violation of an integrity constraint. A database management system (DBMS) is a computer program (or more typically, a suite of them) designed to manage a database, a large set of structured data, and run operations on the data requested by numerous users. ... A transaction is an agreement, communication, or movement carried out between separate entities or objects. ... This article is about computing. ... The word violation, when used alone, has several possible meanings in the English language. ...

Foreign keys are integrity constraints enforcing that the value of the attribute set is drawn from a candidate key in another relation, for example in the Order relation the attribute Customer ID is a foreign key. A join is the operation that draws on information from several relations at once. By joining relvars from the example above we could query the database for all of the Customers, Orders, and Invoices. If we only wanted the tuples for a specific customer, we would specify this using a restriction condition. In the context of relational databases, a foreign key is a referential constraint between two tables[1]. The foreign key identifies a column or a set of columns in one (referencing) table that refers to a column or set of columns in another (referenced) table. ... In computer science, a value is a sequence of bits that is interpreted according to some data type. ... In the relational model a candidate key of a relation variable (relvar) is a set of attributes of that relvar such that (1) at all times it holds in the relation assigned to that variable that there are no two distinct tuples with the same values for these attributes and... See: JOIN, join command in SQL, a relational database keyword. ... In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... The ASCII codes for the word Wikipedia represented in binary, the numeral system most commonly used for encoding computer information. ...

If we wanted to retrieve all of the Orders for Customer 1234567890, we could query the database to return every row in the Order table with Customer ID 1234567890 and join the Order table to the Order Line table based on Order No. In general, a query is a form of questioning, in a line of inquiry. ...

There is a flaw in our database design above. The Invoice relvar contains an Order No attribute. So, each tuple in the Invoice relvar will have one Order No, which implies that there is precisely one Order for each Invoice. But in reality an invoice can be created against many orders, or indeed for no particular order. Additionally the Order relvar contains an Invoice No attribute, implying that each Order has a corresponding Invoice. But again this is not always true in the real world. An order is sometimes paid through several invoices, and sometimes paid without an invoice. In other words there can be many Invoices per Order and many Orders per Invoice. This is a many-to-many relationship between Order and Invoice (also called a non-specific relationship). To represent this relationship in the database a new relvar should be introduced whose role is to specify the correspondence between Orders and Invoices: Database design is the process of producing a detailed data model of a database. ... An attribute is the following: Generally, an attribute is an abstraction characteristic of an entity In database management, an attribute is a property inherent in an entity or associated with that entity for database purposes. ... For other uses, see Reality (disambiguation). ... Many-to-many is a term that describes the third major Internet computing paradigm. ... A function is part of an answer to a question about why some object or process occurred in a system that evolved or was designed with some goal. ... Correspondence may refer to: In the theology of Emanuel Swedenborg, correspondence is the relationship between spiritual and physical realities. ...

OrderInvoice(Order No,Invoice No)

Now, the Order relvar has a one-to-many relationship to the OrderInvoice table, as does the Invoice relvar. If we want to retrieve every Invoice for a particular Order, we can query for all orders where Order No in the Order relation equals the Order No in OrderInvoice, and where Invoice No in OrderInvoice equals the Invoice No in Invoice. ...

Set-theoretic formulation

Basic notions in the relational model are relation names and attribute names. We will represent these as strings such as "Person" and "name" and we will usually use the variables and a,b,c to range over them. Another basic notion is the set of atomic values that contains values such as numbers and strings.

Our first definition concerns the notion of tuple, which formalizes the notion of row or record in a table:

Tuple

A tuple is a partial function from attribute names to atomic values.

Header

A header is a finite set of attribute names.

Projection

The projection of a tuple t on a finite set of attributes A is .

The next definition defines relation which formalizes the contents of a table as it is defined in the relational model. In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ... In mathematics, a partial function is a relation that associates each element of a set (sometimes called its domain) with at most one element of another (possibly the same) set, called the codomain. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

Relation

A relation is a tuple (H,B) with H, the header, and B, the body, a set of tuples that all have the domain H.

Such a relation closely corresponds to what is usually called the extension of a predicate in first-order logic except that here we identify the places in the predicate with attribute names. Usually in the relational model a database schema is said to consist of a set of relation names, the headers that are associated with these names and the constraints that should hold for every instance of the database schema. Look up Relation in Wiktionary, the free dictionary In mathematics, a relation is a generalization of arithmetic relations, such as = and <, which occur in statements, such as 5 < 6 or 2 + 2 = 4. See relation (mathematics), binary relation (of set theory and logic) and relational algebra. ... First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ... A Logical schema is a data model of a specific problem domain that has more detail than a conceptual schema, but does not include the design considerations and physical storage parameters found in a physical schema. ...

Relation universe

A relation universe U over a header H is a non-empty set of relations with header H.

Relation schema

A relation schema (H,C) consists of a header H and a predicate C(R) that is defined for all relations R with header H. A relation satisfies a relation schema (H,C) if it has header H and satisfies C.

Key constraints and functional dependencies

One of the simplest and most important types of relation constraints is the key constraint. It tells us that in every instance of a certain relational schema the tuples can be identified by their values for certain attributes.

Superkey

A superkey is written as a finite set of attribute names.

A superkey K holds in a relation (H,B) if:

• and
• there exist no two distinct tuples such that t₁[K] = t₂[K].

A superkey holds in a relation universe U if it holds in all relations in U.

Theorem: A superkey K holds in a relation universe U over H if and only if and holds in U.

Candidate key

A superkey K holds as a candidate key for a relation universe U if it holds as a superkey for U and there is no proper subset of K that also holds as a superkey for U.

Functional dependency

A functional dependency (FD for short) is written as for X,Y finite sets of attribute names.

A functional dependency holds in a relation (H,B) if:

• and
• tuples ,

A functional dependency holds in a relation universe U if it holds in all relations in U.

Trivial functional dependency

A functional dependency is trivial under a header H if it holds in all relation universes over H.

Theorem: An FD is trivial under a header H if and only if .

Closure

Armstrong's axioms: The closure of a set of FDs S under a header H, written as S ⁺ , is the smallest superset of S such that:

• (reflexivity)
• (transitivity) and
• (augmentation)

Theorem: Armstrong's axioms are sound and complete; given a header H and a set S of FDs that only contain subsets of H, if and only if holds in all relation universes over H in which all FDs in S hold.

Completion

The completion of a finite set of attributes X under a finite set of FDs S, written as X ⁺ , is the smallest superset of X such that:

• 

The completion of an attribute set can be used to compute if a certain dependency is in the closure of a set of FDs.

Theorem: Given a set S of FDs, if and only if .

Irreducible cover

An irreducible cover of a set S of FDs is a set T of FDs such that:

• S ⁺ = T ⁺
• there exists no such that S ⁺ = U ⁺
• is a singleton set and
• .

In the relational model a candidate key of a relation variable (relvar) is a set of attributes of that relvar such that (1) at all times it holds in the relation assigned to that variable that there are no two distinct tuples with the same values for these attributes and... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X &#8838; Y; Y is a superset of (or includes) X; Y... A functional dependency is a constraint between two sets of attributes in a relation from a database. ... Armstrongs axioms are a set of axioms used to infer all the functional dependencies on a relational database. ...

Algorithm to derive candidate keys from functional dependencies

 INPUT: a set S of FDs that contain only subsets of a header H OUTPUT: the set C of superkeys that hold as candidate keys in all relation universes over H in which all FDs in S hold begin C := ∅; // found candidate keys Q := { H }; // superkeys that contain candidate keys while Q <> ∅ do let K be some element from Q; Q := Q - { K }; minimal := true; for each X->Y in S do K' := (K - Y) ∪ X; // derive new superkey if K' ⊂ K then minimal := false; Q := Q ∪ { K' }; end if end for if minimal and there is not a subset of K in C then remove all supersets of K from C; C := C ∪ { K }; end if end while end 

See also

References

1. ^ White, Colin. In the Beginning: An RDBMS History. Teradata Magazine Online. September 2004 edition. URL: [http://www.teradata.com/t/page/127057]
2. ^ Codd, E.F. (1970). "A Relational Model of Data for Large Shared Data Banks". Communications of the ACM 13 (6): 377–387.

• Codd, E. F. (1970). "A relational model of data for large shared data banks". Communications of the ACM, 13(6):377–387. (Retrieved from ACM, September 4, 2004.)
• Date, C. J., Darwen, H. (2000). Foundation for Future Database Systems: The Third Manifesto, 2nd edition, Addison-Wesley Professional. ISBN 0-201-70928-7.
• Date, C. J. (2003). Introduction to Database Systems. 8th edition, Addison-Wesley. ISBN 0-321-19784-4.

Communications of the ACM (CACM) is the flagship monthly magazine of the Association for Computing Machinery. ... Year 1970 (MCMLXX) was a common year starting on Thursday (link shows full calendar) of the Gregorian calendar. ... Communications of the ACM (CACM) is the flagship monthly magazine of the Association for Computing Machinery. ... Year 2000 (MM) was a leap year starting on Saturday (link will display full 2000 Gregorian calendar). ... Pearson can mean Pearson PLC the media conglomerate. ... Year 2003 (MMIII) was a common year starting on Wednesday of the Gregorian calendar. ...

External links

• The Third Manifesto (TTM)
• DMoz category
• Relational Model