Definition:Empty Set/Notation

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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.


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.