Editing Algebra of sets (section)

== Principle of duality ==
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{{See also|Duality (order theory)}}

Each of the identities stated above is one of a pair of identities such that each can be transformed into the other by interchanging {{tmath|1= \cup }} and {{tmath|1= \cap }}, while also interchanging {{tmath|1= \varnothing }} and {{tmath|1= \boldsymbol{U} }}.

These are examples of an extremely important and powerful property of set algebra, namely, the '''principle of duality''' for sets, which asserts that for any true statement about sets, the '''dual''' statement obtained by interchanging unions and intersections, interchanging {{tmath|1= \boldsymbol{U} }} and {{tmath|1= \varnothing }} and reversing inclusions is also true.  A statement is said to be '''self-dual''' if it is equal to its own dual.