Editing Naive set theory (section)

== Universal sets and absolute complements ==
In certain contexts, one may consider all sets under consideration as being subsets of some given [[universe (mathematics)|universal set]].
For instance, when investigating properties of the [[real number]]s '''R''' (and subsets of '''R'''), '''R''' may be taken as the universal set. A true universal set is not included in standard set theory (see '''[[#Paradoxes|Paradoxes]]''' below), but is included in some non-standard set theories.

Given a universal set '''U''' and a subset ''A'' of '''U''', the '''[[complement (set theory)|complement]]''' of ''A'' (in '''U''') is defined as
:{{math|1=''A''<sup>C</sup> := {{mset|''x'' ∈ '''U''' | ''x'' ∉ ''A''}}}}.
In other words, ''A''<sup>C</sup> ("''A-complement''"; sometimes simply ''A''', "''A-prime''" ) is the set of all members of '''U''' which are not members of ''A''.
Thus with '''R''', '''Z''' and ''O'' defined as in the section on subsets, if '''Z''' is the universal set, then ''O<sup>C</sup>'' is the set of even integers, while if '''R''' is the universal set, then ''O<sup>C</sup>'' is the set of all real numbers that are either even integers or not integers at all.