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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 other theories, its existence can be deduced.
Partial orders. A reflexive, weak, [1] or non-strict partial order, [2] commonly referred to simply as a partial order, is a homogeneous relation ≤ on a set that is reflexive, antisymmetric, and transitive. That is, for all it must satisfy: Reflexivity: , i.e. every element is related to itself.
In axiomatic set theory, the axiom of empty set, [1][2] also called the axiom of null set[3] and the axiom of existence, [4][5] is a statement that asserts the existence of a set with no elements. [3] It is an axiom of Kripke–Platek set theory and the variant of general set theory that Burgess (2005) calls "ST," and a demonstrable truth in ...
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number is equal to itself (reflexive). If , then (symmetric).
For a cardinal number κ, define the following statement: . MA(κ)For any partial order P satisfying the countable chain condition (hereafter ccc) and any set D = {D i} i∈I of dense subsets of P such that |D| ≤ κ, there is a filter F on P such that F ∩ D i is non-empty for every D i ∈ D.
Hausdorff distance. In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, [1][2] measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right. It is named after Felix Hausdorff and Dimitrie ...
Intersection (set theory) The intersection of two sets and represented by circles. is in red. The intersection of and is the set of elements that lie in both set and set . In set theory, the intersection of two sets and denoted by [1] is the set containing all elements of that also belong to or equivalently, all elements of that also belong to [2]
Reflexive relation. In mathematics, a binary relation on a set is reflexive if it relates every element of to itself. [1][2] An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess ...