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. Some axiomatic set theories...

set theory, the axiom of empty set, also called the axiom of null set and the axiom of existence, is a statement that asserts the existence of a set with...

other sets. A set may have a finite number of elements or be an infinite set. There is a unique set with no elements, called the empty set; a set with...

A\cap B=B\cap A.} The intersection of any set with the empty set results in the empty set; that is, that for any set A {\displaystyle A} , A ∩ ∅ = ∅ {\displaystyle...

null set should not be confused with the empty set as defined in set theory. Although the empty set has Lebesgue measure zero, there are also non-empty sets...

members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological...

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. In axiomatic set theory (as developed...

every line in a line segment, single point, or the empty set. For example, a solid cube is a convex set, but anything that is hollow or has an indent, for...

sets of central importance are the set of natural numbers, the set of real numbers and the empty set—the unique set containing no elements. The empty...

of the empty product we must take the decategorification of the empty product in the category of finite sets. Dually, the coproduct of an empty family...

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...

codomain R {\displaystyle \mathbb {R} } (the set of all real numbers), this implies that the diameter of the empty set (the case S = ∅ {\displaystyle S=\varnothing...

is a set with no members at all. Because a set is determined completely by its elements, there can be only one empty set. (See axiom of empty set.) Although...

Equivalently, two disjoint sets are sets whose intersection is the empty set. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4...

empty set. For explanation of the symbols used in this article, refer to the table of mathematical symbols. The union of two sets A and B is the set of...

mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero. The natural way to extend non-empty sums is to let the empty sum be...

spaces" their name. In any topological space X , {\displaystyle X,} the empty set and the whole space X {\displaystyle X} are both clopen. Now consider...

every set is open, every set is equal to its interior. In any indiscrete space X , {\displaystyle X,} since the only open sets are the empty set and X...

set is defined inductively as the smallest ordinal number greater than the ranks of all members of the set. In particular, the rank of the empty set is...

closed sets is closed (this includes intersections of infinitely many closed sets) The union of finitely many closed sets is closed. The empty set is closed...

finite sets to be countable.) The free semilattice over a finite set is the set of its non-empty subsets, with the join operation being given by set union...

the axiom of the empty set. On the other hand, the axiom schema of specification can be used to prove the existence of the empty set, denoted ∅ {\displaystyle...

which is a formal language (i.e. a set of strings) that contains no strings, not even the empty string. The empty string has several properties: |ε| =...

None may refer to: Zero, the mathematical concept of the quantity "none" Empty set, the mathematical concept of the collection of things represented by "none"...

is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined...

again dense and open. The empty set is a dense subset of itself. But every dense subset of a non-empty space must also be non-empty. By the Weierstrass approximation...

set X, there is a unique function, called the empty function, or empty map, from the empty set to X. The graph of an empty function is the empty set....

i\geq 3} . From the properties of the empty set ( ∅ {\displaystyle \varnothing } ), it is easy to see that the sets E i {\displaystyle E_{i}} are pairwise...

addition modulo 2. The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element...

special cases: The empty set is computable. The entire set of natural numbers is computable. Each natural number (as defined in standard set theory) is computable;...