Countable Sets

A set with the equal number of elements as present in some subset of the set of natural numbers is referred to as countable set. On the other hand, the set that cannot be counted is referred to as uncountable. Georg Cantor is the one who introduced this term. According to the rule, the elements that are presented in the countable set can be counted one at a time. Though it would be impossible to finish the counting, each and every element in the set will sooner or later be united with a natural number.

Same Cardinality:

The countable set was used by some authors in order to mean a set which has got the same cardinality as in a set of natural numbers. According to the definition, a set S can be called countable if an injective function f exists from the S to the natural numbers.
Where  N = {0,1,2,3,........}

Here S is referred to be countably infinite and the f is bijective since it is also surjective. The terminology mentioned here is not universal. In order to mean what is mentioned as countably infinite some authors do use countable and exclude the finite sets.

If there is an injective function, then S will be countable.

f : S → N

from S to the natural numbers N = {0,1,2,3,........}

Bijection and numbers:

To divide the sets in to various classes, the countable sets that contain one element should be put together, and the set that contains two elements should be put together and finally, the sets which contain all the infinite sets should be put together by considering that they have got the same size. Though looking from the point of view of natural definition for size, this view won’t be tenable. The concept of bijection is needed in order to elaborate this view. Though there are notions which depicts a bijection to be more advanced than a number the developments that took place in the mathematics as a result of the set theory defines the functions before numbers since they are more based on some simpler sets. The concept of the bijection is highlighted in here and the correspondence is defined as a ↔ 1, b ↔ 2, c ↔ 3 since the elements {1,2,3} is paired with the any one element {a,b,c} and vice versa, the bijection is defined. If there is a bijection between them, then we define the two sets to be of the same size and we do generalize this situation.  So, the usual definition of the same size is given to all finite sets.

Questions:

• What are countable sets?
• Who introduced this term?