# Defining uncountably infinite set

Sets can be divided in to 3 types: finite, countably infinite, uncountable. All these are mutually disjoint and the definitions are as follows:

Finite set: Set with finite number of elements,

Countably infinite set: Infinite set which has the existence of bijection with the set of natural numbers,

Unountable: Infinite set which has no existence of bijection with the set of natural numbers,

Whether it is valid if I define uncountable set as follows:

Unountable: Infinite set which has existence of bijection with the set of Real numbers.

If not, what is the counter example for the above definition?

It is not valid. You can look at the set of the parts of the real numbers $$\mathcal{P}(\mathbb{R})=\{A\vert A\subseteq\mathbb{R}\}.$$ It is a fairly standard exercice to show that for any set $$X$$, you never have a bijection $$X\to\mathcal{P}(X)$$.