Empty Set

The empty set is a set that contains no elements. It is denoted by a circle with a diagonal slash through it ø or by two curly braces with nothing inside { }. The most common symbol for the empty set in mathematical notation is:

ø

In set theory, it is also known as the null set. The empty set should not be confused with nothingness. Even though it has no elements, it is still a set and thus something to consider.

Properties of the Empty Set

The key properties of the empty set are as follows:

• Uniqueness of the Empty Set. There is only one empty set. This means that the empty set of odd natural numbers divisible by two and the empty set of consonants in the word "aiuola" are the same empty set. Therefore, we always refer to the empty set in the singular form and never use the plural (empty sets).
• Universality of the Empty Set. The empty set is an improper subset of all sets. Given any set A and an empty set, there are no elements in the empty set that do not belong to set A. Thus, the empty set is considered an improper subset of set A.

The Empty Set in Set Operations

In set theory, the empty set is the absorbing element of intersection and the neutral element of union.

• The union of any set X and the empty set Ø is the set X itself: $$ X \cup \emptyset =\emptyset \cup X = X $$
• The intersection of any set X and the empty set Ø is still the empty set: $$ X \cap \emptyset =\emptyset \cap X = \emptyset $$