de Morgan's laws relate the negation of a conjunction or disjunction to the negations
of its components. Let 
represent OR, 
represent AND, and 
represent NOT.
Then, for two logical units 
and 
,
 |
 |
These laws also apply in the more general context of Boolean algebra and, in particular, in the Boolean algebra
of set theory, in which case 
would denote union, 
intersection,
and 
complementation with respect to any superset
of 
and 
.
The Venn diagrams above illustrate these identities for sets 
and 
in a universal set 
, represented by the rectangle.
In each row, the two diagrams have the same shaded region.
In the top row, an element is outside the union 
iff
it is outside both sets, so 
. In the bottom row, an element is outside the intersection 
iff
it is outside at least one of the sets, so 
.
