Definition:Empty Set

Definition
The empty set is a set which has no elements.

It is usually denoted by some variant of a zero with a line through it, for example $\varnothing$ or $\empty$, and can always be represented as $\left\{{}\right\}$.

Axiomatic Set Theory
The concept of the empty set is axiomatised in the Axiom of Existence in Zermelo-Fraenkel set theory:

Also known as
The empty set is sometimes called the null set, but this name is discouraged because there is another concept for null set which ought not to be confused with this.

Some sources call the empty set the vacuous set.

Others call it the void set.

Notes on Symbology
The symbols $\varnothing$ and $\empty$ are properly considered as stylings of $0$ (zero), and not variants of the Greek "Phi": $\Phi, \phi, \varphi$.

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

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

The preferred symbol on is $\varnothing$ for its completely unambiguous interpretation and aesthetically pleasing, clean presentation.

Comment
Some authors have problems with the existence (or not) of the empty set:


 * : Introduction $\S 1$: Operations on Sets:
 * One may regard [the vacuous set] as a zero element that is adjoined to the collection of "real" subsets.


 * : $\S 1.3$:
 * If $A, B$ are disjoint, then $A \cap B$ is not really defined, because it has no elements. For this reason we introduce a conventional empty set, denoted $\varnothing$, to be thought of as a 'set with no elements'. Of course this is a set only by courtesy, but it is convenient to allow $\varnothing$ the status of a set.


 * : Appendix:
 * The best attitude towards the empty set $\varnothing$ is, perhaps, to regard it as an interesting curiosity, a convenient fiction. To say that $x \in \varnothing$ simply means that $x$ does not exist. Note that it is conveniently agreed that $\varnothing$ is a subset of every set, for elements of $\varnothing$ are supposed to possess every property.

Such a philosophical position is, to the school of anarchist Bourbakist existentialists such as Matt Westwood, a contemptibly timid attitude harking back to the mediaeval distrust of zero.

Other sources allow the definition of the empty set, but because of the way natural numbers are defined, determine that it is neither finite nor infinite.

Historical Note
The use of $\varnothing$ has relevance to the early days of the computing industry, when $\empty$ was frequently used to mean "zero", in order to distinguish it from $\mathrm O$ (the letter "O").

In the same context, the letter "O" was sometimes seen rendered as $\odot$, so as to ensure its being differentiated from "zero".

The latter has fallen out of use, but it is still common for mathematicians, when writing their mathematics by hand, to strike through their zeroes out of habit.

Also see

 * Empty Set is Unique for a proof that it is justifiable to refer to $\varnothing$ as the empty set.


 * Definition:Non-Empty Set, a common phrasing used to denote any set but the empty set.