Properties
Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains, thus 1 and {1} are not the same thing, and the empty set is distinct from the set containing only the empty set. A set such as {{1, 2, 3}} is a singleton as it contains a single element (which itself is a set, however not a singleton).
A set is a singleton if and only if its cardinality is 1. In the standard set-theoretic construction of the natural numbers, the number 1 is defined as the singleton {0}.
In axiomatic set theory, the existence of singletons is a consequence of the axiom of pairing: for any set A that axiom applied to A and A asserts the existence of {A,A}, which is the same as the singleton {A} (since it contains A, and no other set, as element).
If A is any set and S is any singleton, then there exists precisely one function from A to S, the function sending every element of A to the one element of S. Thus every singleton is a terminal object in the category of sets.
Read more about this topic: Singleton (mathematics)
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)