Search results
Results from the WOW.Com Content Network
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 ...
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 ...
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 ...
Every set can be the basis of a free abelian group, which is unique up to group isomorphisms. The free abelian group for a given basis set can be constructed in several different but equivalent ways: as a direct sum of copies of the integers, as a family of integer-valued functions, as a signed multiset, or by a presentation of a group.
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 ...
Although the complement of a fuzzy set has a single most common definition, the other main operations, union and intersection, do have some ambiguity. For a given fuzzy set A {\displaystyle A} , its complement ¬ A {\displaystyle \neg {A}} (sometimes denoted as A c {\displaystyle A^{c}} or c A {\displaystyle cA} ) is defined by the following ...
A disjoint union of a family of pairwise disjoint sets is their union. In category theory , the disjoint union is the coproduct of the category of sets , and thus defined up to a bijection . In this context, the notation ∐ i ∈ I A i {\textstyle \coprod _{i\in I}A_{i}} is often used.
North and West Branch Railroad: PRR: 1871 1881 North and West Branch Railway: North and West Branch Railway: PRR: 1881 1900 Schuylkill and Juniata Railroad: Northampton Railroad: LNE: 1901 1903 Lehigh and New England Railroad: Northampton and Bath Railroad: NB 1902 1979 N/A North East Pennsylvania Railroad: RDG: 1870 1945 Reading Company ...