Naive Set Theory - Universal Sets and Absolute Complements

Universal Sets and Absolute Complements

In certain contexts we may consider all sets under consideration as being subsets of some given universal set. For instance, if we are investigating properties of the real numbers R (and subsets of R), then we may take R as our universal set. A true universal set is not included in standard set theory (see Paradoxes below), but is included in some non-standard set theories.

Given a universal set U and a subset A of U, we may define the complement of A (in U) as

AC := {x ∈ U : x ∉ A}.

In other words, AC ("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 OC is the set of even integers, while if R is the universal set, then OC is the set of all real numbers that are either even integers or not integers at all.