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The set of all size-k subsets contains: (1) all size-k subsets that do contain the distinguished element, and (2) all size-k subsets that do not contain the distinguished element. If a size- k subset of a size-( n + 1) set does contain the distinguished element, then its other k − 1 elements are chosen from among the other n elements of our ...
Let (M, d) be a metric space, let K be a subset of M, and let r be a positive real number.Let B r (x) denote the ball of radius r centered at x.A subset C of M is an r-external covering of K if:
In mathematics, the packing dimension is one of a number of concepts that can be used to define the dimension of a subset of a metric space.Packing dimension is in some sense dual to Hausdorff dimension, since packing dimension is constructed by "packing" small open balls inside the given subset, whereas Hausdorff dimension is constructed by covering the given subset by such small open balls.
These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition. The set of rational numbers is a proper subset of the set of real ...
A generating set of a module is said to be minimal if no proper subset of the set generates the module. If R is a field, then a minimal generating set is the same thing as a basis. Unless the module is finitely generated, there may exist no minimal generating set. [1]
In algebra, the length of a module over a ring is a generalization of the dimension of a vector space which measures its size. [1] page 153 It is defined to be the length of the longest chain of submodules. For vector spaces (modules over a field), the length equals the dimension.