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...

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...

sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets...

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...

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...

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...

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...

the empty set is assigned rank 0, while the set containing only the empty set is assigned rank 1. For each ordinal α {\displaystyle \alpha } , the set V...

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...

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...

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...

The empty set and the whole space are convex. The intersection of any collection of convex sets is convex. The union of a collection of convex sets is...

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...

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....

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...

for all sets x. Axiom of empty set: There exists a set with no members, called the empty set and denoted {}. Axiom of pairing: If x, y are sets, then so...

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...

vacuously true that the empty set is a hereditary set, and thus the set { ∅ } {\displaystyle \{\varnothing \}} containing only the empty set ∅ {\displaystyle...

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...

(it's possible in the edge case of an empty domain that the only choice for A {\displaystyle A} is the empty set itself, but that still works). If μ (...

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...

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| =...

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...

topological space X {\displaystyle X} that contains both the empty set both the entire set X {\displaystyle X} and is closed under countable union and...

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...

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...

distinguish sets and urelements. As non-empty sets contain members while urelements do not, the unary relation is only needed to distinguish the empty set from...

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...