The family of Rational Numbers has infinite numbers. There can be any number of rational numbers between any two given Rational numbers.
A rational number has a long series of Equivalent Rational Numbers, which is uncountable. For example: 3/5 can be written as equivalent to 6 / 10, 9 / 15 , 12 / 20 , 15 / 25 .... ......so on. So all these numbers represents the same number. Any two rational numbers are equivalent if they are same after simplifying and converting them to standard form.
Now let us talk about the Subsets of rational numbers. The subset means which type of the group of numbers is included in the family of Set of rational numbers. There exists a long list of Types of Numbers which belong to the family of rational numbers.
All the counting numbers ( called Natural Numbers ) , all measuring numbers ( called whole numbers ) , all the integers ( positive and negative ), all fraction numbers and the group of numbers in form of p/q where p is a positive or a negative Integer and q is not equal to zero are all the members of rational number set.
Thus we conclude that the set of rational numbers have the following subsets of rational numbers i.e. a set of natural numbers, a set of whole numbers , a set of integers, a set of fractional numbers. Some of the elements of one set also belong the subset of another set, but still these all subsets comprise of a big set of rational numbers.