Properties
- There exists an enumeration for a set (in this sense) if and only if the set is countable.
- If a set is enumerable it will have an uncountable infinity of different enumerations, except in the degenerate cases of the empty set or (depending on the precise definition) sets with one element. However, if one requires enumerations to be injective and allows only a limited form of partiality such that if ƒ(n) is defined then ƒ(m) must be defined for all m < n, then a finite set of N elements has exactly N! enumerations.
- An enumeration e of a set S with domain induces a well-order ≤ on that set defined by s ≤ t if and only if min e−1(s) ≤ min e−1(t). Although the order may have little to do with the underlying set, it is useful when some order of the set is necessary.
Read more about this topic: Enumeration, Enumeration in Set Theory, Enumeration in Countable Vs. Uncountable Context
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)