Countable Set

An Countable Set is a set (with countable set members) that has a One-to-One Relation with the The N0 Natural Number Sequence.

• AKA: Discrete Set.
• Context:
  • If Y is a Countable Set, and X is a Subset of Y, then X is a Countable Set.
  • It can range from being a Countably Finite Set (if it is also a finite set) to being a Countably Infinite Set.
  • It can range from being an Ordered Countable Set (such as the integer number sequence) to being an Unordered Countable Set (such as a categorical set).
• Example(s):
  • an Empty Set.
  • a Single Member Set.
  • {1,2,3,4,5}.
  • The Integer Number Sequence.
  • a Nominal Set, such as a Nominal Sample Space.
  • a Categorical Set, such as a Discrete Sample Space.
  • …
• Counter-Example(s):
  • an Uncountable Set, such as the real number sequence.
• See: Recursively Enumerable Set.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/countable_set Retrieved:2015-6-1.
  • In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a natural number.

    Some authors use countable set to mean infinitely countable alone. To avoid this ambiguity, the term at most countable may be used when finite sets are included and countably infinite, enumerable, or denumerable otherwise.

    The term countable set was originated by Georg Cantor who contrasted sets which are countable with those which are uncountable (a.k.a. nonenumerable and nondenumerable ). Today, countable sets are researched by a branch of mathematics called discrete mathematics.

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