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The relational calculus consists of two calculi, the tuple relational calculus and the domain relational calculus, that are part of the relational model for databases and provide a declarative way to specify database queries. The raison d'être of the relational calculus is the formalization of query optimization, which is finding more efficient manners to execute the same query.
The relational calculus is similar to the relational algebra, which is also part of the relational model: While the relational calculus is meant as a declarative language which prescribes no execution order on the subexpressions of a relational calculus expression, the relational algebra is meant as an imperative language: the sub-expressions of a relational algebraic expressions are meant to be executed from left-to-right and inside-out following their nesting.
Per Codd's theorem, the relational algebra and the domain-independent relational calculus are logically equivalent.