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If the base set is finite, then = ℘ since every subset of , and in particular every complement, is then finite.This case is sometimes excluded by definition or else called the improper filter on . [2] Allowing to be finite creates a single exception to the Fréchet filter’s being free and non-principal since a filter on a finite set cannot be free and a non-principal filter cannot contain ...
A is a subset of B (denoted ) and, conversely, B is a superset of A (denoted ). In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B.
In mathematics, a filter on a set is a family of subsets such that: [1]. and ; if and , then ; If and , then ; A filter on a set may be thought of as representing a "collection of large subsets", [2] one intuitive example being the neighborhood filter.
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S {\displaystyle S} of integers and a target-sum T {\displaystyle T} , and the question is to decide whether any subset of the integers sum to precisely T {\displaystyle T} . [ 1 ]
For example, the set of odd numbers is a sum-free subset of the integers, and the set {N + 1, ..., 2N } forms a large sum-free subset of the set {1, ..., 2N }. Fermat's Last Theorem is the statement that, for a given integer n > 2, the set of all nonzero n th powers of the integers is a sum-free set.
Given two sets A and B, A is a subset of B if every element of A is also an element of B. In particular, each set B is a subset of itself; a subset of B that is not equal to B is called a proper subset. If A is a subset of B, then one can also say that B is a superset of A, that A is contained in B, or that B contains A.
A thin set, in general, is a subset of a finite union of thin sets of types I and II . The terminology thin may be justified by the fact that if A is a thin subset of the line over Q then the number of points of A of height at most H is ≪ H : the number of integral points of height at most H is O ( H 1 / 2 ) {\displaystyle O\left({H^{1/2 ...
The behaviour of a prime ideal = of A under extension is one of the central problems of algebraic number theory. The following is sometimes useful: [ 22 ] a prime ideal p {\displaystyle {\mathfrak {p}}} is a contraction of a prime ideal if and only if p = p e c {\displaystyle {\mathfrak {p}}={\mathfrak {p}}^{ec}} .