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A set such as {{,,}} is a singleton as it contains a single element (which itself is a set, but not a singleton). A set is a singleton if and only if its cardinality is 1. In von Neumann's set-theoretic construction of the natural numbers, the number 1 is defined as the singleton {}.
For example, is a set with no elements, while {} is a singleton with as its unique element. A set is finite if there exists a natural number n {\displaystyle n} such that the n {\displaystyle n} first natural numbers can be put in one to one correspondence with the elements of the set.
The set {A,A} is abbreviated {A}, called the singleton containing A. Note that a singleton is a special case of a pair. 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 x = { x } {\displaystyle x=\{x\}} from the Axiom ...
The empty set has exactly one partition, namely . (Note: this is the partition, not a member of the partition.) For any non-empty set X, P = { X} is a partition of X, called the trivial partition. Particularly, every singleton set {x} has exactly one partition, namely { {x} }.
As a less trivial example, consider the space of all rational numbers with their ordinary topology, and the set of all positive rational numbers whose square is bigger than 2. Using the fact that 2 {\displaystyle {\sqrt {2}}} is not in Q , {\displaystyle \mathbb {Q} ,} one can show quite easily that A {\displaystyle A} is a clopen subset of Q ...
Other notations include () and (). [4] The inverse image of a singleton set, denoted by [{}] or by (), is also called the fiber or fiber over or the level set of . The set of all the fibers over the elements of Y {\displaystyle Y} is a family of sets indexed by Y . {\displaystyle Y.}
A set is called bornivorous if it absorbs every bounded subset. First examples. Every set absorbs the empty set but the empty set does not absorb any non-empty set. The singleton set {} containing the origin is the one and only singleton subset that absorbs itself.
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.