If set A is a subset of set B, that means that all elements in set A are also in set B. In the case of a proper subset, there is the additional specification that the two sets are not equal, i.e.,...
A proper subset B of a set A is a set all of whose elements are elements of A nad there are elements of A that are not elements of B. It follows, then, that an improper subset must be the whole set,...
Recall that Improper subset of A is the set that contains all and only elements of A. Namely A. So does the empty set have all of A provided A is not empty? Of course not! The empty set can be only...
There is no difference between improper subset and equal sets. If A is an improper subset of B then A = B. For this reason, the term "improper subset" is rarely used.