The Empty Set
The empty set is the set that contains no elements. The empty set can be denoted using one of the following notations.
\[\emptyset \;\;\; \text{or} \;\;\; \{\}\]The first notation is more modern and is more commonly used. Before typesetting software, the second notation was more common. This is why older textbooks may use the second notation to denote the empty set.
Sets Are Containers
It is important to realize that a set is a container that can hold elements. The empty set is a container that holds no elements. It is incorrect to refer to the empty set as “nothing” or “null”. The empty set is a container that holds no elements, but it is still a container. The empty set is still a set, even though it contains no elements. It is like having an empty box. The box exists, but it contains nothing. The box is still a box, even though it is empty. The empty set is still a set, even though it is empty.
Uniqueness of the Empty Set
Mathematicians often refer to the empty set as “the empty set”. When two sets contain the same elements, we say that the two sets are equal. Since the empty set contains no elements, there can only be one empty set. Every empty set is effectively equivalent to all other empty sets.
Empty Set as a Subset
The empty set is always a subset of every set. This is a vacuously true statement.
A vacuously true statement is a statement that is true simply because there are no counterexamples. Restated, a vacuously true statement is a statement that is true because the alternative leads to a contradiction.
A false statement is a statement that leads to a contradiction. A true statement is any statement that is not false. Restated, a true statement is any statement that does not lead to a contradiction. This can result in some very awkward logical statements
The empty set provides an excellent example of a vacuously true statement. Not that the statement below must be true or false.
\[\emptyset \subset A\]Let’s assume to the contrary that the statement above is false. If the statement above is false, then there must be an element of $\emptyset$ that is not an element of $A$. However, there are no elements of $\emptyset$. The existence of an element in a set with no elements is a contradiction. The only assumption was that the statement above is false and it resulted in a contradiction. Since we only had two options (true or false) and the false option led to a contradiction, we can conclude that the statement above (empty set is always a subset) is true. It feels perhaps a bit flimsy, but this is the nature of vacuously true statements.
Notice the following must also be true
\[\emptyset \subset \emptyset\]Empty Set as an Element
Sets can be elements of other sets. Consider the following set.
\[A = \{\emptyset\}\]The set $A$ contains one element, and that element is the empty set. The empty set is an element of $A$, but the empty set is not a subset of $A$. The empty set is a subset of every set, but it is not an element of every set. The empty set is an element of $A$, but it is not an element of every set.