In my book, it says "We agree to regard the empty set as a subset of every set. Thus any non-empty set S has just two improper subsets, the empty set and the set S itself; all other subsets of S are proper." Does this sound right? I thought the improper subset is the set which is the same as the original set. Why is the empty set also an improper subset? Later in the book it said the empty set is a subset of every set. So the empty set is a subset of the universal set. Than it said the complement of the universal set is the empty set. From this information it implies the complement of the universal set is a subset of the universal set. Which is a contradiction?