Union (set theory)
Algebraic properties

Binary union is an associative operation; that is, for any sets A, B, and C {\displaystyle A,B,{\text{ and }}C},

Thus, the parentheses may be omitted without ambiguity: either of the above can be written as A ∪ B ∪ C {\displaystyle A\cup B\cup C}. Also, union is commutative, so the sets can be written in any order. The empty set is an identity element for the operation of union. That is, A ∪ ∅ = A {\displaystyle A\cup \varnothing =A}, for any set A {\displaystyle A}. Also, the union operation is idempotent: A ∪ A = A {\displaystyle A\cup A=A}. All these properties follow from analogous facts about logical disjunction.

Intersection distributes over union

and union distributes over intersection

The power set of a set U {\displaystyle U}, together with the operations given by union, intersection, and complementation, is a Boolean algebra. In this Boolean algebra, union can be expressed in terms of intersection and complementation by the formula

where the superscript ∁ {\displaystyle {}^{\complement }} denotes the complement in the universal set U {\displaystyle U}.
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