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:
“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)
“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)