---
tag: 0ECVK59Y03
kind: object
name: Empty set
aliases: [null-set]
msc: ["03E20"]
instance_of: [0BSJ7RQ9BW]
uses: [0SJ3RX9TXA]
sources:
  - url: https://en.wikipedia.org/wiki/Empty_set
    locator: lead section
same_as:
  wikidata: Q226183
  wikipedia: Empty set
---

## Description

The empty set is the [[set|0SJ3RX9TXA]] that has no [[elements|0EZ758XE73]].
It is written $\varnothing$ or $\{\}$.
For each object $x$, $x \notin \varnothing$.
By the [[axiom of extensionality|0TB16T2C97]], there is only one empty set.

## Remarks

The empty set is a [[subset|0FE8Q5S42Y]] of every set. This is a case of [[vacuous truth|0V4QXPGJQK]].
Its [[cardinality|0JTGRV4WT4]] is $0$.
The set $\{\varnothing\}$ is not empty: it has one element.

## Explanations

### L1 human

The empty set is an empty bag.
For example, the set of real numbers $x$ with $x^2 = -1$ is empty.
It is useful in the same way that the number $0$ is useful.