# Definition:Non-Empty Set

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## 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**.

## Also see

## Sources

- 1964: Walter Rudin:
*Principles of Mathematical Analysis*(2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Introduction: $1.3$. Notation - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems