Editing Universe (mathematics) (section)

==In a specific context==
{{Main|Domain of discourse}}
Perhaps the simplest version is that ''any'' set can be a universe, so long as the object of study is confined to that particular set. If the object of study is formed by the [[real number]]s, then the [[real line]] '''R''', which is the real number set, could be the universe under consideration. Implicitly, this is the universe that [[Georg Cantor]] was using when he first developed modern [[naive set theory]] and [[cardinality]] in the 1870s and 1880s in applications to [[real analysis]]. The only sets that Cantor was originally interested in were [[subset]]s of '''R'''.

This concept of a universe is reflected in the use of [[Venn diagram]]s. In a Venn diagram, the action traditionally takes place inside a large rectangle that represents the universe ''U''. One generally says that sets are represented by circles; but these sets can only be subsets of ''U''. The [[complement (set theory)|complement]] of a set ''A'' is then given by that portion of the rectangle outside of ''A'''s circle. Strictly speaking, this is the [[complement (set theory)|relative complement]] ''U'' \ ''A'' of ''A'' relative to ''U''; but in a context where ''U'' is the universe, it can be regarded as the [[complement (set theory)|absolute complement]] ''A''<sup>C</sup> of ''A''. Similarly, there is a notion of the [[nullary intersection]], that is the [[intersection (set theory)|intersection]] of [[0 (number)|zero]] sets (meaning no sets, not [[null set]]s).

Without a universe, the nullary intersection would be the set of absolutely everything, which is generally regarded as impossible; but with the universe in mind, the nullary intersection can be treated as the set of everything under consideration, which is simply ''U''. These conventions are quite useful in the algebraic approach to basic set theory, based on [[Boolean lattice]]s. Except in some non-standard forms of [[axiomatic set theory]] (such as [[New Foundations]]), the [[class (set theory)|class]] of all sets is not a Boolean lattice (it is only a [[relatively complemented lattice]]).

In contrast, the class of all subsets of ''U'', called the [[power set]] of ''U'', is a Boolean lattice. The absolute complement described above is the complement operation in the Boolean lattice; and ''U'', as the nullary intersection, serves as the [[Greatest element|top element]] (or nullary [[meet (mathematics)|meet]]) in the Boolean lattice. Then [[De Morgan's laws]], which deal with complements of meets and [[join (mathematics)|join]]s (which are [[union (set theory)|union]]s in set theory) apply, and apply even to the nullary meet and the nullary join (which is the [[empty set]]).