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The Cantor set is an example of an uncountable null set. It is uncountable because it contains all numbers between 0 and 1 whose ternary form contains only 0's and 2's in their decimal expansion, and it is null because it is constructed by beginning with the closed interval of real numbers from 0 to 1 and multiplying the length by 2/3 continuously.
The empty set is the set containing no elements. In mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. [1] Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced.
A null space of a mapping is the part of the domain that is mapped into the null element of the image (the inverse image of the null element). For example, in linear algebra, the null space of a linear mapping, also known as kernel, is the set of vectors which map to the null vector under that mapping. In statistics, a null hypothesis is a ...
The empty set is also occasionally called the null set, [11] though this name is ambiguous and can lead to several interpretations. The power set of a set A, denoted (), is the set whose members are all of the possible subsets of A. For example, the power set of {1, 2} is { {}, {1}, {2}, {1, 2} }.
ℵ 0 (aleph-nought, aleph-zero, or aleph-null) is the cardinality of the set of all natural numbers, and is an infinite cardinal.The set of all finite ordinals, called ω or ω 0 (where ω is the lowercase Greek letter omega), also has cardinality ℵ 0.
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
In axiomatic set theory, the axiom of empty set, [1] [2] also called the axiom of null set [3] and the axiom of existence, [4] [5] is a statement that asserts the existence of a set with no elements. [3]
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...