Null set

In set theory, the null set (also known as the empty set) is the unique set containing no elements. It is a fundamental concept in set theory and serves as the additive identity for the union operation on sets.

The null set is often denoted by the symbol ∅ (Unicode character U+2205) or sometimes by {}. It is important to distinguish between the null set ∅ and the set containing the null set {∅}. The former contains no elements, while the latter contains one element, which is itself the null set.

The null set is a subset of every set, including itself. This can be proven using vacuous truth; since there are no elements in the null set, it is impossible to find an element in the null set that is not also in another set. Therefore, the condition for subset inclusion is always met.

The cardinality (number of elements) of the null set is zero. This is a direct consequence of the definition of the null set as containing no elements.

The null set plays a crucial role in various set-theoretic operations, such as intersection and complement. The intersection of any set with the null set is the null set itself. The complement of the universal set (the set containing all elements under consideration) is the null set.

Formally, the existence of the null set can be established axiomatically within set theory, most commonly within Zermelo-Fraenkel set theory (ZFC). One common axiom implying its existence is the Axiom of Infinity combined with the Axiom Schema of Separation.

The concept of the null set extends beyond pure mathematics. It is used in other areas such as logic, computer science, and database theory to represent situations where no elements satisfy a certain condition or query.