Definition:Empty Set

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Definition

The empty set is a set which has no elements.

That is, $x \in \O$ is false, whatever $x$ is.


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


Axiomatic Set Theory

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


The Axiom of the Empty Set posits the existence of a set which has no elements.

Depending on whether this axiom is declared in the context of set theory or class theory, it exists in different forms.

Set Theory

$\exists x: \forall y: \paren {\neg \paren {y \in x} }$

Class Theory

In class theory, the existence of the empty class is not axiomatic, as it has been derived from previous axioms.

Hence the Axiom of the Empty Set takes this form:


The empty class $\O$ is a set, that is:

$\O \in V$

where $V$ denotes the basic universe.


Also known as

The empty set is sometimes called the null set, but this name is discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$ 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.


Notation

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 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.


Existence of Empty Set

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

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 $\O$, 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 $\O$ the status of a set.
The best attitude towards the empty set $\O$ is, perhaps, to regard it as an interesting curiosity, a convenient fiction. To say that $x \in \O$ simply means that $x$ does not exist. Note that it is conveniently agreed that $\O$ is a subset of every set, for elements of $\O$ are supposed to possess every property.
Now some students are bothered with the notion of an "empty set". "How", they say, "can you have a set with nothing in it?" ... The empty set is only a convention, and mathematics could very well get along without it. But it is a very convenient convention, for it saves us a good deal of awkwardness in stating theorems and proving them.

Such a philosophical position is considered by many mathematicians to be a timid attitude harking back to the mediaeval distrust of zero.


In any case, its convenience cannot be doubted:

One may regard [the vacuous set] as a zero element that is adjoined to the collection of "real" subsets.
One practical advantage in admitting $\O$ as a set is that we may wish to talk about a set without knowing a priori whether it has any members.
The courtesy of regarding this as a set has several advantages ... In allowing $\O$ the status of a set, we gain the advantage of being able to talk about a set without knowing at the outset whether or not it has any elements.


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.


Examples

Real Roots of $x^2 + 1 = 0$

The set:

$S = \set {x \in \R: x^2 + 1 = 0}$

is an instance of a specification of the empty set.


Also see

  • Results about the empty set can be found here.


Historical Note

The concept of the empty set was stated by Leibniz in his initial conception of symbolic logic.


The use of $\O$ has relevance to the computing industry, when $\emptyset$ is usually used in the rendering of source code 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.


Linguistic Note

The word vacuous literally means empty.

It derives from the Latin word vacuum, meaning empty space.


Technical Note

The $\LaTeX$ code for \(\O\) is \O .

The same symbol is also generated by \varnothing or \empty, but these are more unwieldy, and \O is preferred.


Sources

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