Editing Complement (set theory) (section)

=== Properties ===
Let {{mvar|A}} and {{mvar|B}} be two sets in a universe {{mvar|U}}. The following identities capture important properties of absolute complements:

[[De Morgan's laws]]:<ref name="Halmos-1960" />
* <math>\left(A \cup B \right)^c= A^c \cap B^c.</math>
* <math>\left(A \cap B \right)^c = A^c \cup B^c.</math>

Complement laws:<ref name="Halmos-1960" />
* <math>A \cup A^c = U.</math>
* <math>A \cap A^c = \empty .</math>
* <math>\empty^c = U.</math>
* <math> U^c = \empty.</math>
* <math>\text{If }A\subseteq B\text{, then }B^c \subseteq A^c.</math>
*: (this follows from the equivalence of a conditional with its [[contrapositive]]).

[[Involution (mathematics)|Involution]] or double complement law:
* <math>\left(A^c\right)^c = A.</math>

Relationships between relative and absolute complements:
* <math>A \setminus B = A \cap B^c.</math>
* <math>(A \setminus B)^c = A^c \cup B = A^c \cup (B \cap A).</math>

Relationship with a set difference:
* <math> A^c \setminus B^c = B \setminus A. </math>

The first two complement laws above show that if {{math|''A''}} is a non-empty, [[proper subset]] of {{math|''U''}}, then {{math|{''A'', ''A''<sup>∁</sup>}{{null}}}} is a [[Partition of a set|partition]] of {{math|''U''}}.