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The preimage of an output value is the set of input values that produce . More generally, evaluating f {\displaystyle f} at each element of a given subset A {\displaystyle A} of its domain X {\displaystyle X} produces a set, called the " image of A {\displaystyle A} under (or through) f {\displaystyle f} ".
This set is not Borel. However, it is analytic (all Borel sets are also analytic), and complete in the class of analytic sets. For more details see descriptive set theory and the book by A. S. Kechris (see References), especially Exercise (27.2) on page 209, Definition (22.9) on page 169, Exercise (3.4)(ii) on page 14, and on page 196.
In words, given two programs, if the first program is in the set of programs satisfying the property and two programs are computing the same thing, then also the second program satisfies the property. This means that if one program with a certain property is in the set, all programs computing the same function must also be in the set).
The indicator or characteristic function of a subset A of some set X maps elements of X to the codomain {,}. This mapping is surjective only when A is a non-empty proper subset of X . If A = X , {\displaystyle A=X,} then 1 A ≡ 1. {\displaystyle \mathbf {1} _{A}\equiv 1.}
This schema is used in 4.-7. below to cut down the set that is stated to exist to the set containing precisely the intended elements, rather than some larger set with extraneous elements. For example, the axiom of pairing applied to the sets A {\displaystyle A} and B {\displaystyle B} only guarantees there is some set X {\displaystyle X} such ...
In naive set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even integers. Clearly, the set of even numbers is infinitely large; there is no requirement that a ...
The set of all dense open subsets of a (non–empty) topological space is a proper π –system and so also a prefilter. If the space is a Baire space, then the set of all countable intersections of dense open subsets is a π –system and a prefilter that is finer than .
A Vitali set is a subset of the interval [,] of real numbers such that, for each real number , there is exactly one number such that is a rational number.Vitali sets exist because the rational numbers form a normal subgroup of the real numbers under addition, and this allows the construction of the additive quotient group / of these two groups which is the group formed by the cosets + of the ...