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Any two objects have a pair. The set {A,A} is abbreviated {A}, called the singleton containing A. Note that a singleton is a special case of a pair. Being able to construct a singleton is necessary, for example, to show the non-existence of the infinitely descending chains = {} from the Axiom of regularity.
A set with precisely two elements is also called a 2-set or (rarely) a binary set. An unordered pair is a finite set; its cardinality (number of elements) is 2 or (if the two elements are not distinct) 1. In axiomatic set theory, the existence of unordered pairs is required by an axiom, the axiom of pairing.
In some cases, the data sets are paired, meaning there is an obvious and meaningful one-to-one correspondence between the data in the first set and the data in the second set, compare Blocking (statistics). For example, paired data can arise from measuring a single set of individuals at different points in time. [1]
In 1990, Regan proposed the first known pairing function that is computable in linear time and with constant space (as the previously known examples can only be computed in linear time if multiplication can be too, which is doubtful). In fact, both this pairing function and its inverse can be computed with finite-state transducers that run in ...
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 ...
A totally ordered set is a partially ordered set in which any two elements are comparable. The Szpilrajn extension theorem states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes ...
So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), [4] but in standard set theory, the universal set does not exist. However, when restricted to the context of subsets of a given fixed set X {\displaystyle X} , the notion of the intersection of an empty collection of ...
A nested set collection or nested set family is a collection of sets that consists of chains of subsets forming a hierarchical structure, like Russian dolls. It is used as reference concept in scientific hierarchy definitions, and many technical approaches, like the tree in computational data structures or nested set model of relational databases .