Tuple relational calculus

Tuple calculus is a that was introduced by Edgar F. Codd as part of the relational model, in order to provide a declarative database-query language for this data model. It formed the inspiration for the database-query languages QUEL and SQL, of which the latter, although far less faithful to the original relational model and calculus, is now the de facto standard database-query language; a dialect of SQL is used by nearly every relational-database-management system. Lacroix and Pirotte proposed domain calculus, which is closer to first-order logic and together with Codd showed that both of these calculi (as well as relational algebra) are equivalent in expressive power. Subsequently, query languages for the relational model were called relationally complete if they could express at least all of these queries.

Since the calculus is a query language for relational databases we first have to define a relational database. The basic relational building block is the domain, or data type. A tuple is a finite sequence of attributes, which are ordered pairs of domains and values. A relvar (relation variable) is a set of ordered pairs of domains and names; a relvar serves as the header for a relation. A relation is a set of (compatible) tuples. Although these relational concepts are mathematically defined, those definitions map loosely to traditional database concepts. A table is an accepted visual representation of a relation; a tuple is similar to the concept of a row.

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