Proper Subset

As well known, a subset of A, say, set B, will have some or all the elements contained in the set A. But if the latter option is removed, then the subset is a proper set. Therefore the difference between a subset - proper subset is that the latter has at least one element less than the number of elements of the main set.
We can also define proper subset as strict subset, because of the analogy to the inequalities. The symbols ‘<’ and ‘>’ mean strict inequalities whereas the symbols ‘=’ and ‘=’ mean only just inequalities as they allow a situation of equality as well.  
On the other hand if B is the subset of A, then A is called super subset of B and if B is the strict subset of A, then A is called as proper superset of B. The subset proper set differences can be identified when the relations are expressed symbolically.
B ?  A (B is subset of A)  B ?  A (B is strict subset of A)B ?  A (B is subset of A)   B ?  A (B is strict subset of A)
Let us look into some proper subsets examples. The set {1, 2, 3} is a strict subset of the set {1, 2, 3, 4} because the former does not have the number 4 which is present in the latter. This is a simple example just to understand the basic difference. Let us see where we can see the difference more effectively. We know the domain of a logarithmic function y = ln (x) is (0,8),in interval form.
Suppose we introduce a set D as{0,8}. One can immediately recognize that x is a strict subset of D because, ‘x’ cannot assume the value of 0.
Similarly, we know the range of a sine function y = sin (x) is [-1, 1], in interval form. Suppose we introduce a set R as {-1, 1}. One can immediately recognize that x is a not a strict subset of R because, ‘y’ can assume all the value of between -1 and 1, both included.
Another practical example can be given from number system. We know all natural numbers are integers. But the converse is not true. That is, the set of integers contain natural numbers and in additionalso contain 0 and negative of whole numbers.
Therefore, set of natural numbers is a strict subset of set of integers. Same way, set of irrational numbers