Editing 0 (section)

===Other uses in mathematics===
[[File:Nullset.svg|thumb|upright=0.4|The empty set has zero elements]]The role of 0 as the smallest counting number can be generalized or extended in various ways. In [[set theory]], 0 is the [[cardinality]] of the [[empty set]] (notated as "{ }", "<math display="inline">\emptyset</math>", or "∅"): if one does not have any apples, then one has 0 apples. In fact, in certain axiomatic developments of mathematics from set theory, 0 is ''[[definition|defined]]'' to be the empty set.{{sfn|Cheng|2017|p=60}} When this is done, the empty set is the [[von Neumann cardinal assignment]] for a set with no elements, which is the empty set. The cardinality function, applied to the empty set, returns the empty set as a value, thereby assigning it 0 elements.

Also in set theory, 0 is the lowest [[ordinal number]], corresponding to the empty set viewed as a [[well-order|well-ordered set]]. In [[order theory]] (and especially its subfield [[lattice theory]]), 0 may denote the [[least element]] of a [[Lattice (order)|lattice]] or other [[partially ordered set]].

The role of 0 as additive identity generalizes beyond elementary algebra. In [[abstract algebra]], 0 is commonly used to denote a [[zero element]], which is the [[identity element]] for addition (if defined on the structure under consideration) and an [[absorbing element]] for multiplication (if defined). (Such elements may also be called [[zero element]]s.) Examples include identity elements of [[additive group]]s and [[vector space]]s. Another example is the '''zero function''' (or '''zero map''') on a domain {{mvar|D}}. This is the [[constant function]] with 0 as its only possible output value, that is, it is the function {{mvar|f}} defined by {{math|''f''(''x'') {{=}} 0}} for all {{mvar|x}} in {{mvar|D}}. As a function from the real numbers to the real numbers, the zero function is the only function that is both [[Even function|even]] and [[Odd function|odd]].

The number 0 is also used in several other ways within various branches of mathematics:
* A ''[[zero of a function]]'' ''f'' is a point ''x'' in the domain of the function such that {{math|''f''(''x'') {{=}} 0}}.
* In [[propositional logic]], 0 may be used to denote the [[truth value]] false.
* In [[probability theory]], 0 is the smallest allowed value for the probability of any event.{{sfn|Kardar|2007|p=35}}
* [[Category theory]] introduces the idea of a [[zero object]], often denoted 0, and the related concept of [[zero morphism]]s, which generalize the zero function.<ref>{{cite book|last=Riehl |first=Emily |title=Category Theory in Context |author-link=Emily Riehl |page=103 |url=https://math.jhu.edu/~eriehl/context/ |publisher=Dover |year=2016 |isbn=978-0-486-80903-8}}</ref>