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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
In constructive mathematics, "not empty" and "inhabited" are not equivalent: every inhabited set is not empty but the converse is not always guaranteed; that is, in constructive mathematics, a set that is not empty (where by definition, "is empty" means that the statement () is true) might not have an inhabitant (which is an such that ).
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
In mathematics, a field of sets is a mathematical structure consisting of a pair (,) consisting of a set and a family of subsets of called an algebra over that contains the empty set as an element, and is closed under the operations of taking complements in , finite unions, and finite intersections.
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 ...
The simple theorems in the algebra of sets are some of the elementary properties of the algebra of union (infix operator: ∪), intersection (infix operator: ∩), and set complement (postfix ') of sets. These properties assume the existence of at least two sets: a given universal set, denoted U, and the empty set, denoted {}.
The empty set is the unique initial object in Set, the category of sets. Every one-element set is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category.
In mathematics, the affine hull or affine span of a set S in Euclidean space R n 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 combinations of elements ...