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There is an inclusion–exclusion principle for finite multisets (similar to the one for sets), stating that a finite union of finite multisets is the difference of two sums of multisets: in the first sum we consider all possible intersections of an odd number of the given multisets, while in the second sum we consider all possible ...
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, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if and only if x is in at least one of A, B, and C. A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set ...
Venn diagram showing the union of sets A and B as everything not in white. In combinatorics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as
The coproduct in the category of sets is simply the disjoint union with the maps i j being the inclusion maps.Unlike direct products, coproducts in other categories are not all obviously based on the notion for sets, because unions don't behave well with respect to preserving operations (e.g. the union of two groups need not be a group), and so coproducts in different categories can be ...
In C++, the Standard Template Library (STL) provides the set template class, which is typically implemented using a binary search tree (e.g. red–black tree); SGI's STL also provides the hash_set template class, which implements a set using a hash table. C++11 has support for the unordered_set template class, which is implemented using a hash ...
In mathematics, the symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} and { 3 , 4 } {\displaystyle \{3,4\}} is { 1 , 2 , 4 ...
In mathematics, the Dershowitz–Manna ordering is a well-founded ordering on multisets named after Nachum Dershowitz and Zohar Manna.It is often used in context of termination of programs or term rewriting systems.