The principle of duality for sets states that for any true statement about sets, the dual statement obtained by interchanging unions and intersections, interchanging
and
and reversing inclusions is also true. A statement is said to be self-dual if it is equal to its own dual.
For example
has dual
has dual
has dual statement
has dual
Notice here that the complement of
does not become
but stays
Set-theoretic union and intersection are dual under the set complement operator C. That is,
Proof:
More generally,