yes, if the set being described is empty, we can talk about proper and improper subsets. there are no proper subsets of the empty set. the only subset of the empty set is the empty set itself. to be a proper subset, the subset must be strictly contained. so the empty set is an improper subset of itself, but it is a proper subset of every other set.
No. The null set cannot have a proper subset. For any other set, the null set will be a proper subset. There will also be other proper subsets.
The null set is a proper subset of any non-empty set.
The null set. Every set is a subset of itself and so the null set is a subset of the null set.
A set with only one element in it. The only proper subset of such a set is the null set.
The only proper subset of a set comprising one element, is the null set.
It's an axiom.
Yes. A null set is always a subset of any set. Also, any set is a subset of the [relevant] universal set.
Yes the null set is a subset of every set.
NO- by definition a set is not a proper subset of itself . ( It is a subset, but not a proper one. )
The null set. It is a subset of every set.