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In mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. [1] In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. [2]
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.
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} }. Notably, () contains both A and the empty set.
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 family consisting only of the empty set and the set , called the minimal or trivial σ-algebra over . The power set of X , {\displaystyle X,} called the discrete σ-algebra . The collection { ∅ , A , X ∖ A , X } {\displaystyle \{\varnothing ,A,X\setminus A,X\}} is a simple σ-algebra generated by the subset A . {\displaystyle A.}
At stage 0, there are no sets yet. At each following stage, a set is added to the universe if all of its elements have been added at previous stages. Thus the empty set is added at stage 1, and the set containing the empty set is added at stage 2. [11] The collection of all sets that are obtained in this way, over all the stages, is known as V.
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 ...
Given a finite non-empty set , the power set consisting of all subsets of is union-closed. Each element of U {\displaystyle U} is contained in exactly half of the subsets of U {\displaystyle U} . Therefore, in general we cannot ask for an element contained in more than half of the sets of the family: the bound of the conjecture is sharp.