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MSDN documentation on UNION in Transact-SQL for SQL Server; Naming of select list items in set operations; UNION in MySQL with Examples; UNION in MySQL; UNION Clause in PostgreSQL; SQL UNION and UNION ALL; Sort order within UNION statement; Designing a data flow that loads a warehouse table; Oracle 11g documentation for UNION (ALL), INTERSECT ...
A multiset may be formally defined as an ordered pair (A, m) where A is the underlying set of the multiset, formed from its distinct elements, and : + is a function from A to the set of positive integers, giving the multiplicity – that is, the number of occurrences – of the element a in the multiset as the number m(a).
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.
SQL allows the selection of rows from a relational table: this operation will in general yield a multiset, unless the keyword DISTINCT is used to force the rows to be all different, or the selection includes the primary (or a candidate) key. In ANSI SQL the MULTISET keyword can be used to transform a subquery into a collection expression:
After some operations of Union, some sets are grouped together. The operation Union(x, y) replaces the set containing x and the set containing y with their union. Union first uses Find to determine the roots of the trees containing x and y. If the roots are the same, there is nothing more to do. Otherwise, the two trees must be merged.
The union is the join/supremum of and with respect to because: L ⊆ L ∪ R {\displaystyle L\subseteq L\cup R} and R ⊆ L ∪ R , {\displaystyle R\subseteq L\cup R,} and if Z {\displaystyle Z} is a set such that L ⊆ Z {\displaystyle L\subseteq Z} and R ⊆ Z {\displaystyle R\subseteq Z} then L ∪ R ⊆ Z . {\displaystyle L\cup R\subseteq Z.}
For example, AVERAGE=SUM/COUNT and RANGE=MAX−MIN. In the MapReduce framework, these steps are known as InitialReduce (value on individual record/singleton set), Combine (binary merge on two aggregations), and FinalReduce (final function on auxiliary values), [ 5 ] and moving decomposable aggregation before the Shuffle phase is known as an ...
HyperLogLog is an algorithm for the count-distinct problem, approximating the number of distinct elements in a multiset. [1] Calculating the exact cardinality of the distinct elements of a multiset requires an amount of memory proportional to the cardinality, which is impractical for very large data sets. Probabilistic cardinality estimators ...