Is the Empty Set a Proper or Improper Subset?

Whether the empty set (\(\emptyset\)) is considered a proper or improper subset of another set depends on the definition of "proper subset" being used.

There are two main interpretations:

• Empty Set as a Proper Subset
  A subset \(B\) of \(A\) is considered proper if it contains some but not all elements of \(A\). Under this definition, the empty set is seen as a proper subset of any non-empty set because it is contained within every other set (\(B \subseteq A\)) and is different from any set that contains at least one element (\(A \neq B\)). However, it is not a proper subset of itself because the empty set is not different from itself (\(\emptyset = \emptyset\)).
• Empty Set as an Improper Subset
  Some sources argue that the empty set is an improper subset of every set, including itself, based on the idea that a "proper subset" must include at least one element of the reference set. Since the empty set has no elements, it does not meet this criterion.

Generally, most mathematicians view the empty set as a proper subset of any non-empty set and an improper subset of itself.

The confusion in texts stems from the different definitions of a proper subset.

Note: The distinction between proper and improper subsets may seem abstract, but it is crucial for more advanced discussions in set theory and other areas of mathematics.

And so on.