Empty Set¶

Definition¶

A set is called the empty set, denoted $\varnothing\text{,}$ if it contains no elements.

Interpretation¶

The empty set represents the absence of elements. Among sets it plays a role like that of $0$ among numbers: it is the answer whenever a condition has no solutions, such as the set of natural numbers that are both even and odd.

Both Key Results follow from the definitions of a subset and of set equality. The statement "every element of $\varnothing$ lies in $A$" has no element to check, so it cannot fail: $\varnothing\subseteq A$ for every set $A$. Two empty sets have the same elements, namely none, so they are equal. Under the set operations, $A\cup\varnothing=A$ and $A\cap\varnothing=\varnothing$$\varnothing$, much like $a+0=a$ and $a\cdot0=0$$0$.

A common slip is to confuse $\varnothing$ with $\{\varnothing\}$ or with $\{0\}$. Both of the latter are singleton sets: each has exactly one element, so neither is empty.

Key Results¶

• The empty set is a subset of every set.
• There is exactly one empty set.

Examples¶

• The solution set of $x^2+1=0$ in $\mathbb{R}$ is $\varnothing\text{.}$
• $\{x\in\mathbb{N}\colon x<0\}=\varnothing\text{.}$

Prerequisites¶

• Set

Related Concepts¶

• Singleton Set
• Subset
• Union of Sets
• Intersection of Sets

Sources¶

• Math I: Calculus · Lecture 1

Aliases: void set Keywords: null set Symbols: element of, intersection, natural numbers, real numbers, subset, union

Last updated: 20 October 2026 03:02