null set

In mathematical sets, the null set, also called the empty set, is the set that does not contain anything. It is symbolized or { }. There is only one null set. This is because there is logically only one way that a set can contain nothing.

The null set makes it possible to explicitly define the results of operations on certain sets that would otherwise not be explicitly definable. The intersection of two disjoint sets (two sets that contain no elements in common) is the null set. For example:

{1, 3, 5, 7, 9, ...} {2, 4, 6, 8, 10, ...} =

The null set provides a foundation for building a formal theory of numbers. In axiomatic mathematics, zero is defined as the cardinality of (that is, the number of elements in) the null set. From this starting point, mathematicians can build the set of natural numbers, and from there, the sets of integers and rational numbers.