Empty set

The empty set is the set with no elements. The empty set is written down either as ∅ (which derives from the Norwegian letter Ø) or simply as {}. Its features are:

• For any set A, the empty set is asubset of A.
• For any set A, the union of A with the empty set is A.
• For any set A, the intersection of A with the empty set is the empty set.
• The only subset of the empty set is the empty set itself.
• The cardinality of the empty set is zero.

The empty set is very simple, ironically so simple that many mathematics students (and even professional mathematicians!) have a difficult time applying it correctly. For example, take the first feature listed above, that the empty set is a subset of any set A. If you look up subset, then you'll see that this claim means that for every element x of {}, x belongs to A. Since "for every" is a strong condition, we intuitively expect that it must be necessary to find many elements of {} that also belong to A. Of course, we can't find any elements of {} that also belong to A. So many people think that {} is not a subset of A after all. But this is a mistake. In fact "for every" may not be a very strong requirement at all if it says "for every element of {}". Since there are no elements of {}, "for every element of {}" is no requirement at all. Every statement satisfies that requirement, which is actually the weakest condition possible. We say that something is vacuously true if it is required of all the elements of {} (or otherwise required of nothing whatsoever). So it is vacuously true that {} is a subset of A. The concept of vacuous truth can be a difficult one to wrap one's brain around, and this leads to difficulties in applying the empty set correctly.