# Definition:Non-Empty Set

Jump to navigation Jump to search

## Definition

Let $S$ be a set.

Then $S$ is said to be non-empty if and only if $S$ has at least one element.

By the Axiom of Extension, this may also be phrased as:

$S \ne \O$

where $\O$ denotes the empty set.

Many mathematical theorems and definitions require sets to be non-empty in order to avoid erratic results and inconsistencies.

### Non-Empty Class

In the context of class theory, the definition follows the same lines:

Let $A$ be a class.

Then $A$ is non-empty if and only if $A$ has at least one element.

## Also known as

Some sources prefer to use nonempty, but $\mathsf{Pr} \infty \mathsf{fWiki}$, in striving for consistency, standardises on non-empty.