What Is the Empty Set in Set Theory?

An equation for a null or empty set.
A set with no elements is not considered nothing. Rather, it is a set with nothing contained within it. C.K.Taylor

When can nothing be something? It seems like a silly question, and quite paradoxical.  In the mathematical field of set theory, it is routine for nothing to be something other than nothing. How can this be?

When we form a set with no elements, we no longer have nothing. We have a set with nothing in it. There is a special name for the set which contains no elements. This is called the empty or null set.

A Subtle Difference

The definition of the empty set is quite subtle and requires a little bit of thought. It is important to remember that we think of a set as a collection of elements. The set itself is different from the elements that it contains.

For example, we will look at {5}, which is a set containing the element 5. The set {5} is not a number. It is a set with the number 5 as an element, whereas 5 is a number.

In a similar way, the empty set is not nothing. Instead, it is the set with no elements. It helps to think of sets as containers, and the elements are those things that we put in them. An empty container is still a container and is analogous to the empty set.

The Uniqueness of the Empty Set

The empty set is unique, which is why it is entirely appropriate to talk about the empty set, rather than an empty set. This makes the empty set distinct from other sets. There are infinitely many sets with one element in them. The sets {a}, {1}, {b} and {123} each have one element, and so they are equivalent to one another.  Since the elements themselves are different from one another, the sets are not equal.

There is nothing special about the examples above each having one element. With one exception, for any counting number or infinity, there are infinitely many sets of that size. The exception is for the number zero. There is only one set, the empty set, with no elements in it.

The mathematical proof of this fact is not difficult. We first assume that the empty set is not unique, that there are two sets with no elements in them, and then use a few properties from set theory to show that this assumption implies a contradiction.

Notation and Terminology for the Empty Set

The empty set is denoted by the symbol ∅, which comes from a similar symbol in the Danish alphabet. Some books refer to the empty set by its alternate name of null set.

Properties of the Empty Set

Since there is only one empty set, it is worthwhile to see what happens when the set operations of intersection, union, and complement are used with the empty set and a general set that we will denote by X. It is also interesting to consider subset of the empty set and when is the empty set a subset. These facts are collected below:

  • The intersection of any set with the empty set is the empty set. This is because there are no elements in the empty set, and so the two sets have no elements in common. In symbols, we write X ∩ ∅ = ∅.
  • The union of any set with the empty set is the set we started with. This is because there are no elements in the empty set, and so we are not adding any elements to the other set when we form the union. In symbols, we write X U ∅ = X.
  • The complement of the empty set is the universal set for the setting that we are working in. This is because the set of all elements that are not in the empty set is just the set of all elements.
  • The empty set is a subset of any set. This is because we form subsets of a set X by selecting (or not selecting) elements from X. One option for a subset is to use no elements at all from X. This gives us the empty set.
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Taylor, Courtney. "What Is the Empty Set in Set Theory?" ThoughtCo, Apr. 5, 2023, thoughtco.com/empty-set-3126581. Taylor, Courtney. (2023, April 5). What Is the Empty Set in Set Theory? Retrieved from https://www.thoughtco.com/empty-set-3126581 Taylor, Courtney. "What Is the Empty Set in Set Theory?" ThoughtCo. https://www.thoughtco.com/empty-set-3126581 (accessed June 1, 2023).