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