Empty Set

The empty set is the unique set that contains no elements. Usually denoted by $\varnothing$, $\emptyset$, or $\{\}$, it is a basic object in set theory. Its cardinality, or number of elements, is zero. Although it has no members, it is itself a mathematical object and can belong to other sets. It provides a precise representation of an empty collection, including the collection of solutions to a problem that has no solution. (math.cmu.edu)

Definition, notation, and uniqueness

The defining property of the empty set is

The notation $\{\}$ expresses this directly: there are no objects between the braces. By contrast, $\{\varnothing\}$ contains one element—the empty set—and therefore is not empty. Similarly, $\{0\}$ has one element, even though that element is the number zero. Membership and the number of members must therefore be distinguished. (math.cmu.edu)

Uniqueness follows from the axiom of extensionality, which states that sets with exactly the same elements are equal. If $E$ and $F$ both have no elements, their membership conditions agree for every object, so $E=F$. Thus different descriptions of empty collections identify the same set, not separate empty sets. (people.maths.ox.ac.uk)

In Zermelo–Fraenkel set theory, existence can be stated by an empty-set axiom. Alternatively, once some set $A$ exists, the separation schema produces

which has no elements. Consequently, an explicit empty-set axiom is unnecessary in formulations whose other axioms already establish these conditions. (maths.tcd.ie)

Subsets and vacuous truth

The empty set is a subset of every set:

The definition of inclusion requires every element of $\varnothing$ to belong to $A$. There are no such elements, so none can violate the requirement. In particular, $\varnothing\subseteq\varnothing$, while $\varnothing\in\varnothing$ is false. These statements concern different relations: inclusion compares members of sets, whereas membership identifies an object as an element. (math.cmu.edu)

This illustrates vacuous truth in logic. For any predicate $P$,

is true, while

is false. A universal statement over an empty domain has no counterexample; an existential statement has no witness. Both “every element of the empty set satisfies $P$” and “every element satisfies not-$P$” may therefore be true without contradiction. Neither asserts that an element exists. (math.cmu.edu)

The only subset of $\varnothing$ is $\varnothing$ itself. Its power set, the set of all its subsets, is consequently

Taking a power set thus turns this empty collection into a nonempty one. (people.maths.ox.ac.uk)

Set operations

The empty set obeys the following identities for every set $A$:

It is therefore an identity for union and an absorbing element for intersection. Two sets are disjoint precisely when their intersection is empty. Relative to an ambient set $U$, the complement of $\varnothing$ is $U$, and the complement of $U$ is $\varnothing$. (math.cmu.edu)

The Cartesian product satisfies

because an ordered pair in either product would require a member of the empty set. The union of an empty family of sets is also empty: there is no member set contributing an element. These are direct consequences of the membership definitions of the operations. (people.maths.ox.ac.uk)

Functions with an empty domain

For every set $B$, exactly one function exists from $\varnothing$ to $B$. Called the empty function, it has no inputs, values, or ordered pairs in its graph. The requirement that each input have exactly one output is satisfied because there are no inputs. There is, however, no function from a nonempty set to $\varnothing$: an input could not be assigned an output. (math.cmu.edu)

The empty function is an injective function, since no distinct inputs can share an output. It is a surjective function exactly when $B=\varnothing$. The unique map from $\varnothing$ to itself is therefore a bijective function and its identity map. These conclusions follow directly from the respective definitions. (math.cmu.edu)

Construction of numbers

In the von Neumann construction of the natural numbers, zero is represented by $\varnothing$, and each successor is defined by $n+1=n\cup\{n\}$. Thus

Each number contains exactly its predecessors. The empty set is also the least ordinal number. Identifying zero with the empty set here is a particular foundational construction, rather than a rule that every use of zero denotes an empty collection. (people.maths.ox.ac.uk)

Topology, measure, and probability

In every topological space, the empty set is both an open set and a closed set. It is also compact: any open cover of it has the empty family as a finite subcover. These properties reflect definitions that permit empty collections rather than requiring points to exist. (ocw.mit.edu)

In measure theory, every measure satisfies $\mu(\varnothing)=0$. Nevertheless, a null set, meaning a measurable set of measure zero, need not be empty. Likewise, in probability, the empty event has probability zero, but a nonempty event may also have probability zero. Emptiness is a statement about membership; measure zero is a statement about the value assigned by a measure. (math.mit.edu)