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The Erdős–Woods numbers can be characterized in terms of certain partitions of the prime numbers.A number k is an Erdős–Woods number if and only if the prime numbers less than k can be partitioned into two subsets X and Y with the following property: for every pair of positive integers x and y with x + y = k, either x is divisible by a prime in X, or y is divisible by a prime in Y.
The red subset = {1,2,3,4} has two maximal elements, viz. 3 and 4, and one minimal element, viz. 1, which is also its least element. In mathematics , especially in order theory , a maximal element of a subset S {\displaystyle S} of some preordered set is an element of S {\displaystyle S} that is not smaller than any other element in S ...
In the mathematical field of set theory, an ideal is a partially ordered collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal (this codifies the idea that an ideal is a notion of smallness), and the union of any two elements of the ideal must also be in the ideal.
The meagre subsets of a fixed space form a σ-ideal of subsets; that is, any subset of a meagre set is meagre, and the union of countably many meagre sets is meagre. Meagre sets play an important role in the formulation of the notion of Baire space and of the Baire category theorem , which is used in the proof of several fundamental results of ...
Consider the real line with its usual Borel topology. Let denote the Dirac measure, a unit mass at the point in .The collection := {|} is not tight, since the compact subsets of are precisely the closed and bounded subsets, and any such set, since it is bounded, has -measure zero for large enough .
In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.
Wood is an example of an orthotropic material. Material properties in three perpendicular directions (axial, radial, and circumferential) are different. In material science and solid mechanics, orthotropic materials have material properties at a particular point which differ along three orthogonal axes, where each axis has twofold rotational ...
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 ...