What is countable set and uncountable set?

Answer 1

A countable set is defined as one whose elements can be put into one-to-one correspondence with elements of the set of counting numbers or some subset of it. A countable set can be infinite: for example all even numbers.

This raises the strange concept where a subset (positive even numbers) has the same cardinality as all counting numbers - which should be a set that is twice as large! Even more confusingly (perhaps) is the fact that the set of all rational numbers also has the same cardinality as the set of counting numbers. You need to go to the set of irrationals or bigger before you get to uncountable sets.

So you have the weird situation in which there are more irrationals between 0 and 1 than there are rationals between from 0 and infinity (if infinity can be treated as a value)!

There is a minority definition of countable which means containing a finite number of elements as opposed to uncountable meaning infinitely many elements. However, these definitions are essentially the same as the finite sets and infinite sets and so there is little point in using them.