The complement of a subset of a set is the set
(Besides , there are many other notations, such as , , , and so forth.)
Notice that , while by the principle of excluded middle.
The complement of an element of a lattice is (if it exists) the unique element such that and . If all complements exists one speaks of a complemented lattice. Examples are Boolean algebras.
More generally, the pseudocomplement of an element of a Heyting algebra is given by . This satisfies but not in general. This case includes the complement of a subset even in constructive mathematics.
In another direction, the complement of a complemented subobject of an object in a coherent category is the unique subobject such that is the initial object and .
The complement of a truth value (seen as a subset of the point) is called its negation.
Revised on December 24, 2013 06:00:26
by Urs Schreiber