Editing Union (set theory) (section)

== Algebraic properties ==
{{See also|List of set identities and relations|Algebra of sets}}

Binary union is an [[associative]] operation; that is, for any sets {{tmath|1= A, B, \text{ and } C }},
<math display="block">A \cup (B \cup C) = (A \cup B) \cup C.</math>
Thus, the parentheses may be omitted without ambiguity: either of the above can be written as {{tmath|1= A \cup B \cup C }}. Also, union is [[commutative]], so the sets can be written in any order.<ref>{{Cite book |last=Halmos |first=P. R. |url=https://books.google.com/books?id=jV_aBwAAQBAJ |title=Naive Set Theory |date=2013-11-27 |publisher=Springer Science & Business Media |isbn=9781475716450 |language=en}}</ref>
The [[empty set]] is an [[identity element]] for the operation of union. That is, {{tmath|1= A \cup \varnothing = A }}, for any set {{tmath|1= A }}. Also, the union operation is idempotent: {{tmath|1= A \cup A = A }}.  All these properties follow from analogous facts about [[logical disjunction]].

Intersection distributes over union
<math display="block">A \cap (B \cup C) = (A \cap B)\cup(A \cap C)</math> 
and union distributes over intersection<ref name=":3" />
<math display="block">A \cup (B \cap C) = (A \cup B) \cap (A \cup C).</math>
The [[power set]] of a set {{tmath|1= U }}, together with the operations given by union, [[Intersection (set theory)|intersection]], and [[Complement (set theory)|complementation]], is a [[Boolean algebra (structure)|Boolean algebra]].  In this Boolean algebra, union can be expressed in terms of intersection and complementation by the formula
<math display="block">A \cup B = ( A^\complement \cap B^\complement )^\complement,</math>
where the superscript <math>{}^\complement</math> denotes the complement in the [[Universe (mathematics)|universal set]] {{tmath|1= U }}. Alternatively, intersection can be expressed in terms of union and complementation in a similar way: <math>A \cap B = ( A^\complement \cup B^\complement )^\complement</math>. These two expressions together are called [[De Morgan's laws]].<ref>{{Cite web |title=MathCS.org - Real Analysis: Theorem 1.1.4: De Morgan's Laws |url=https://mathcs.org/analysis/reals/logic/proofs/demorgan.html |access-date=2024-10-22 |website=mathcs.org}}</ref><ref>{{Cite book |last=Doerr |first=Al |url=https://faculty.uml.edu/klevasseur/ads/s-laws-of-set-theory.html |title=ADS Laws of Set Theory |last2=Levasseur |first2=Ken |language=en-US}}</ref><ref>{{Cite web |title=The algebra of sets - Wikipedia, the free encyclopedia |url=https://www.umsl.edu/~siegelj/SetTheoryandTopology/The_algebra_of_sets.html |access-date=2024-10-22 |website=www.umsl.edu}}</ref>