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Graph models exist for ZF and also set theories different from Zermelo set theory, such as non-well founded theories. Such models have more intricate edge structure. In graph theory, the graph whose vertices correspond to hereditarily finite sets and edges correspond to set membership is the Rado graph or random graph.
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.
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.
Basis (linear algebra) The same vector can be represented in two different bases (purple and red arrows). In mathematics, a set B of vectors in a vector space V is called a basis (pl.: bases) if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are ...
Affine hull. In mathematics, the affine hull or affine span of a set S in Euclidean space Rn is the smallest affine set containing S, [1] or equivalently, the intersection of all affine sets containing S. Here, an affine set may be defined as the translation of a vector subspace. The affine hull aff (S) of S is the set of all affine ...
In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is ...
The empty set is a spanning set of {(0, 0, 0)}, since the empty set is a subset of all possible vector spaces in , and {(0, 0, 0)} is the intersection of all of these vector spaces. The set of monomials x n, where n is a non-negative integer, spans the space of polynomials.
Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallest T 1 topology on any infinite set. [13] Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is ...