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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.
If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value). [4] One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple.
In response to a query, the database returns a result set. Often, data from multiple tables are combined into one, by doing a join. Conceptually, this is done by taking all possible combinations of rows (the Cartesian product), and then filtering out everything except the answer. There are a number of relational operations in addition to join.
Actual SQL implementations normally use other approaches, such as hash joins or sort-merge joins, since computing the Cartesian product is slower and would often require a prohibitively large amount of memory to store. SQL specifies two different syntactical ways to express joins: the "explicit join notation" and the "implicit join notation".
The cartesian product is implemented in SQL as the Cross join operator. The remaining operators proposed by Codd involve special operations specific to relational databases: The selection, or restriction, operation (σ) retrieves tuples from a relation, limiting the results to only those that meet a specific criterion, i.e. a subset in terms of ...
The sort-merge join (also known as merge join) is a join algorithm and is used in the implementation of a relational database management system. The basic problem of a join algorithm is to find, for each distinct value of the join attribute, the set of tuples in each relation which display that value. The key idea of the sort-merge algorithm is ...
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Throughout this article, capital letters (such as ,,,,, and ) will denote sets.On the left hand side of an identity, typically, will be the L eft most set,; will be the M iddle set, and