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In mathematics, the algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions ...
The relational algebra uses set union, set difference, and Cartesian product from set theory, and adds additional constraints to these operators to create new ones.. For set union and set difference, the two relations involved must be union-compatible—that is, the two relations must have the same set of attributes.
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, leading to the algebra of sets. Furthermore, the calculus of relations includes the operations of taking the converse and composing relations. [7] [8] [9]
Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. [9] The following is a partial list of them: Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both. [10] For example, the union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4}.
In general, the join and meet of a subset of a partially ordered set need not exist. Join and meet are dual to one another with respect to order inversion. A partially ordered set in which all pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice.
Set operation may have one of the following meanings. Any operation with sets; Set operation (Boolean), Boolean set operations in the algebra of sets;
The standard relational algebra and relational calculus, and the SQL operations based on them, are unable to express directly all desirable operations on hierarchies. The nested set model is a solution to that problem. An alternative solution is the expression of the hierarchy as a parent-child relation. Joe Celko called this the adjacency list ...