Empty set

[1] However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty).

Common notations for the empty set include "{ }", "

The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø (U+00D8 Ø LATIN CAPITAL LETTER O WITH STROKE) in the Danish and Norwegian alphabets.

[3] The symbol ∅ is available at Unicode point U+2205 ∅ EMPTY SET.

When writing in languages such as Danish and Norwegian, where the empty set character may be confused with the alphabetic letter Ø (as when using the symbol in linguistics), the Unicode character U+29B0 REVERSED EMPTY SET ⦰ may be used instead.

The number of elements of the empty set (i.e., its cardinality) is zero.

" is not making any substantive claim; it is a vacuous truth.

This is often paraphrased as "everything is true of the elements of the empty set."

In the usual set-theoretic definition of natural numbers, zero is modelled by the empty set.

Similarly, the product of the elements of the empty set (the empty product) should be considered to be one, since one is the identity element for multiplication.

[6] A derangement is a permutation of a set without fixed points.

The empty set can be considered a derangement of itself, because it has only one permutation (

), and it is vacuously true that no element (of the empty set) can be found that retains its original position.

For example, when considered as a subset of the real numbers, with its usual ordering, represented by the real number line, every real number is both an upper and lower bound for the empty set.

[7] When considered as a subset of the extended reals formed by adding two "numbers" or "points" to the real numbers (namely negative infinity, denoted

which is defined to be less than every other extended real number, and positive infinity, denoted

which is defined to be greater than every other extended real number), we have that:

That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound (inf or infimum) is positive infinity.

By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for the minimum and infimum operators.

In any topological space X, the empty set is open by definition, as is X.

This empty topological space is the unique initial object in the category of topological spaces with continuous maps.

The von Neumann construction, along with the axiom of infinity, which guarantees the existence of at least one infinite set, can be used to construct the set of natural numbers,

In the context of sets of real numbers, Cantor used

notation was utilized in definitions; for example, Cantor defined two sets as being disjoint if their intersection has an absence of points; however, it is debatable whether Cantor viewed

However, the axiom of empty set can be shown redundant in at least two ways: While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity, whose meaning and usefulness are debated by philosophers and logicians.

This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists.

"[9] The popular syllogism is often used to demonstrate the philosophical relation between the concept of nothing and the empty set.

Darling writes that the contrast can be seen by rewriting the statements "Nothing is better than eternal happiness" and "[A] ham sandwich is better than nothing" in a mathematical tone.

According to Darling, the former is equivalent to "The set of all things that are better than eternal happiness is

[9] Jonathan Lowe argues that while the empty set it is also the case that: George Boolos argued that much of what has been heretofore obtained by set theory can just as easily be obtained by plural quantification over individuals, without reifying sets as singular entities having other entities as members.