Empty Set - Properties

Properties

In standard axiomatic set theory, by the principle of extensionality, two sets are equal if they have the same elements; therefore there can be only one set with no elements. Hence there is but one empty set, and we speak of "the empty set" rather than "an empty set".

The mathematical symbols employed below are explained here.

For any set A:

• The empty set is a subset of A:

• The union of A with the empty set is A:

• The intersection of A with the empty set is the empty set:

• The Cartesian product of A and the empty set is empty:

The empty set has the following properties:

• Its only subset is the empty set itself:

• The power set of the empty set is a set containing only the empty set:

• Its number of elements (that is, its cardinality) is zero:

The connection between the empty set and zero goes further, however: in the standard set-theoretic definition of natural numbers, we use sets to model the natural numbers. In this context, zero is modelled by the empty set.

For any property:

• For every element of the property holds (vacuous truth);
• There is no element of for which the property holds.

Conversely, if for some property and some set V, the following two statements hold:

• For every element of V the property holds;
• There is no element of V for which the property holds,

then .

By the definition of subset, the empty set is a subset of any set A, as every element x of belongs to A. If it is not true that every element of is in A, there must be at least one element of that is not present in A. Since there are no elements of at all, there is no element of that is not in A. Hence every element of is in A, and is a subset of A. Any statement that begins "for every element of " is not making any substantive claim; it is a vacuous truth. This is often paraphrased as "everything is true of the elements of the empty set."