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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.
Von Neumann universe. In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC.
Linear span. The cross-hatched plane is the linear span of u and v in both R2 and R3. In mathematics, the linear span (also called the linear hull[ 1 ] or just span) of a set of elements of a vector space is the smallest linear subspace of that contains It is the set of all finite linear combinations of the elements of S, [ 2 ] and the ...
Let X be the set of all linearly independent subsets of V. The set X is nonempty since the empty set is an independent subset of V, and it is partially ordered by inclusion, which is denoted, as usual, by ⊆. Let Y be a subset of X that is totally ordered by ⊆, and let L Y be the union of all the elements of Y (which are themselves certain ...
For example, the empty set is assigned rank 0, while the set {{}} containing only the empty set is assigned rank 1. For each ordinal α {\displaystyle \alpha } , the set V α {\displaystyle V_{\alpha }} is defined to consist of all pure sets with rank less than α {\displaystyle \alpha } .
Let X be a non-empty arbitrary set and V an arbitrary vector space over F. The space of all functions from X to V is a vector space over F under pointwise addition and multiplication. That is, let f : X → V and g : X → V denote two functions, and let α in F. We define (+) = + ()
Interior (topology) The point x is an interior point of S. The point y is on the boundary of S. In mathematics, specifically in topology, the interior of a subset S of a topological space X is the union of all subsets of S that are open in X. A point that is in the interior of S is an interior point of S. The interior of S is the complement of ...
A linear subspace or vector subspace W of a vector space V is a non-empty subset of V that is closed under vector addition and scalar multiplication; that is, the sum of two elements of W and the product of an element of W by a scalar belong to W. [10] This implies that every linear combination of elements of W belongs to W. A linear subspace ...