# Definition:Empty Set/Notation

## Notation for the Empty Set

The symbols $\O$ and $\emptyset$ used for **the empty set** are properly considered as stylings of $0$ (zero), and not variants of the Greek **Phi**: $\Phi, \phi, \varphi$.

Despite this, some sources call the symbol **phi** (pronounced **fie**).

Some sources maintain that it is a variant on the Norwegian / Danish / Faeroese letter Ø.

The symbol $\O$ as presented here is a relatively new invention. Books prior to approximately $1960$ or $1970$ tend to use something less distinctive:

- Some such sources use $\Box$ as the symbol for the
**empty set**, but this is rare.

- Other sources use $0$ (that is, the zero digit).

- Yet others use $O$ (the capital letter).

None of these are recommended.

The preferred symbol on $\mathsf{Pr} \infty \mathsf{fWiki}$ is $\O$ for its completely unambiguous interpretation and aesthetically pleasing, clean presentation.

Also note that $\set {}$ can always be used.

## Warning

Be careful not to mix the notations $\O$ and $\set {}$.

$\set \O$ does *not* mean the empty set.

It means: the set which contains $1$ element, that element being the empty set.

Neither is $\O$ to be written $\set 0$ as this is equally incorrect.

## Sources

- 1971: Patrick J. Murphy and Albert F. Kempf:
*The New Mathematics Made Simple*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets: The Empty Set - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.2$: Operations on Sets - 1993: Richard J. Trudeau:
*Introduction to Graph Theory*... (previous) ... (next): $2$. Graphs: Sets - 1996: Winfried Just and Martin Weese:
*Discovering Modern Set Theory. I: The Basics*... (previous) ... (next): Basic Notations