Editing Naive set theory (section)

== Unions, intersections, and relative complements ==
Given two sets ''A'' and ''B'', their '''[[union (set theory)|union]]''' is the set consisting of all objects which are elements of ''A'' or of ''B'' or of both (see [[axiom of union]]). It is denoted by {{math|''A'' ∪ ''B''}}.

The '''[[intersection (set theory)|intersection]]''' of ''A'' and ''B'' is the set of all objects which are both in ''A'' and in ''B''. It is denoted by {{math|''A'' ∩ ''B''}}.

Finally, the '''[[complement (set theory)|relative complement]]''' of ''B'' relative to ''A'', also known as the '''set theoretic difference''' of ''A'' and ''B'', is the set of all objects that belong to ''A'' but ''not'' to ''B''. It is written as {{math|''A'' &setminus; ''B''}} or {{math|''A'' − ''B''}}.

Symbolically, these are respectively
:{{math|1=''A'' ∪ B := {{mset|''x'' | (''x'' ∈ ''A'') [[logical disjunction|&or;]] (''x'' ∈ ''B'')}}}};
:{{math|1=''A'' ∩ ''B'' := {{mset|''x'' | (''x'' ∈ ''A'') [[logical conjunction|&and;]] (''x'' ∈ ''B'')}} = {{mset|''x'' ∈ ''A'' | ''x'' ∈ ''B''}} = {{mset|''x'' ∈ ''B'' | ''x'' ∈ ''A''}}}};
:{{math|1=''A'' &setminus; ''B'' := {{mset|''x'' | (''x'' ∈ ''A'') &and; [[negation|&not;]] (''x'' ∈ ''B'') }} = {{mset|''x'' ∈ ''A'' | &not; (''x'' ∈ ''B'')}}}}.

The set ''B'' doesn't have to be a subset of ''A'' for {{math|''A'' &setminus; ''B''}} to make sense; this is the difference between the relative complement and the absolute complement ({{math|1=''A''<sup>C</sup> = ''U'' &setminus; ''A''}}) from the previous section.

To illustrate these ideas, let ''A'' be the set of left-handed people, and let ''B'' be the set of people with blond hair. Then {{math|''A'' ∩ ''B''}} is the set of all left-handed blond-haired people, while {{math|''A'' ∪ ''B''}} is the set of all people who are left-handed or blond-haired or both. {{math|''A'' &setminus; ''B''}}, on the other hand, is the set of all people that are left-handed but not blond-haired, while {{math|''B'' &setminus; ''A''}} is the set of all people who have blond hair but aren't left-handed.

Now let ''E'' be the set of all human beings, and let ''F'' be the set of all living things over 1000 years old. What is {{math|''E'' ∩ ''F''}} in this case? No living human being is [[Oldest people|over 1000 years old]], so {{math|''E'' ∩ ''F''}} must be the [[empty set]] {}.

For any set ''A'', the power set <math>P(A)</math> is a [[Boolean algebra (structure)|Boolean algebra]] under the operations of union and intersection.