Singleton Set¶

Definition¶

A singleton set is a set that has only one element.

Interpretation¶

A singleton set isolates a single object as a collection: $\{a\}$ has exactly one element, $a$ . It represents the simplest nonempty set, and it lets set operations be applied to one object.

The difference between $a$ and $\{a\}$ matters: $a\in\{a\}$ ${a}$ , and $\{a\}\subseteq B$ if and only if $a\in B$ $B$ . A list with repeated entries can describe a singleton, as $\{1,1\}$ in the Examples shows, and so can a condition with one solution: $\{n\in\mathbb{N}\colon 2n=6\}=\{3\}$ ${3}$ . Removing one element is a difference with a singleton, as in $\mathbb{N}\setminus\{0\}=\mathbb{N}^+$ $N^{+}$ .

A common slip is to think that $\{\varnothing\}$ is empty. It has one element, the empty set, so it is a singleton, whereas $\varnothing$ has no elements at all.

Key Results¶

A singleton contains exactly one element.
Singleton sets are finite.

Examples¶

$\{1\}$ is a singleton set.
$\{1,\,1\}$ is also a singleton set since it contains exactly one element, which is $1\text{.}$
$\{\varnothing\}$ is a singleton set which contains the only element $\varnothing\text{.}$

Prerequisites¶

Set

Empty Set

Sources¶

Math I: Calculus · Lecture 1

Aliases: singleton, unit set, one-point set Symbols: element of, natural numbers, subset

Last updated: 20 October 2026 03:02