# 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.

## 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.^{[1]}To avoid this ambiguity, the term at most countable may be used when finite sets are included and**countably infinite**,**enumerable**, or denumerable^{[2]}otherwise.**The term**nonenumerable and*countable set*was originated by Georg Cantor who contrasted sets which are countable with those which are*uncountable*(a.k.a.**nondenumerable**^{[3]}). Today, countable sets are researched by a branch of mathematics called discrete mathematics.

- In mathematics, a