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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:
Szemerédi's theorem asserts that a subset of the natural numbers with positive upper density contains an arithmetic progression of length k for all positive integers k. An often-used equivalent finitary version of the theorem states that for every positive integer k and real number δ ∈ ( 0 , 1 ] {\displaystyle \delta \in (0,1]} , there ...
The two-element subset {3, 5} is a generating set, since (−5) + 3 + 3 = 1 (in fact, any pair of coprime numbers is, as a consequence of Bézout's identity). The dihedral group of an n-gon (which has order 2n ) is generated by the set { r , s } , where r represents rotation by 2 π / n and s is any reflection across a line of symmetry.
The set D = {1, 2, 3} is a subset (but not a proper subset) of E = {1, 2, 3}, thus is true, and is not true (false). The set { x : x is a prime number greater than 10} is a proper subset of { x : x is an odd number greater than 10}
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.
The interior of a closed subset of is a regular open subset of and likewise, the closure of an open subset of is a regular closed subset of . [2] The intersection (but not necessarily the union) of two regular open sets is a regular open set. Similarly, the union (but not necessarily the intersection) of two regular closed sets is a regular ...
A subset of Baire space has a corresponding subset of Cantor space under the map that takes each function from to to the characteristic function of its graph. A subset of Baire space is given the classification Σ n 0 {\displaystyle \Sigma _{n}^{0}} , Π n 0 {\displaystyle \Pi _{n}^{0}} , or Δ n 0 {\displaystyle \Delta _{n}^{0}} if and only if ...
Rather than generating and storing all subsets of n/2 elements in advance, they partition the elements into 4 sets of n/4 elements each, and generate subsets of n/2 element pairs dynamically using a min heap, which yields the above time and space complexities since this can be done in ( ()) and space () given 4 lists of length k.