mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure...

axiomatic set theory, the axiom of empty set is a statement that asserts the existence of a set with no elements. It is an axiom of Kripke–Platek set theory...

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...

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...

set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set,...

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...

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...

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...

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...

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...

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...

of real numbers and the empty set—the unique set containing no elements. The empty set is also occasionally called the null set, though this name is ambiguous...

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...

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...

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| =...

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...

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....

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...

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...

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...

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...

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...

language is empty if its set of valid sentences is the empty set. The emptiness problem is the question of determining whether a language is empty given some...

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...

semilattice over a finite set is the set of its non-empty subsets, with the join operation being given by set union. In Zermelo–Fraenkel set theory without the...

otherwise it is the empty set. So if f is a function and x is in its domain, then f ′ x is f(x). f ″ X f ″ X is the image of a set X by f. If f is a function...

sets are the empty set and X {\displaystyle X} itself, we have that the closure of the empty set is the empty set, and for every non-empty subset A {\displaystyle...

following conditions is satisfied: A is equal to B, or A or B is the empty set. For example: A = {1,2}; B = {3,4} A × B = {1,2} × {3,4} = {(1,3), (1...

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...