Given a set, S, a subset A of S is set containing none or more elements of S. So by definition, the subset A is a set.
If there exists some element that is in S but not in A then A is a pro[er subset of S.
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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.
Assume that set A is a subset of set B. If sets A and B are equal (they contain the same elements), then A is NOT a proper subset of B, otherwise, it is.
If set A and set B are two sets then A is a subset of B whose all members are also in set B.
No.
That term, over there!