Editing Algebra of sets (section)

== Some additional laws for complements ==

The following proposition states five more important laws of set algebra, involving complements.

'''PROPOSITION 4''': Let {{tmath|1= A }} and {{tmath|1= B }} be [[subset]]s of a universe {{tmath|1= \boldsymbol{U} }}, then:
: [[De Morgan's laws]]:
:* {{tmath|1= (A \cup B)^\complement = A^\complement \cap B^\complement }}
:* {{tmath|1= (A \cap B)^\complement = A^\complement \cup B^\complement }}
: double complement or [[Involution (mathematics)|involution]] law:
:* {{tmath|1= (A^\complement)^\complement = A }}
: complement laws for the universe set and the empty set:
:* {{tmath|1= \varnothing^\complement = \boldsymbol{U} }}
:* {{tmath|1= \boldsymbol{U}^\complement = \varnothing }}

Notice that the double complement law is self-dual.

The next proposition, which is also self-dual, says that the complement  of a set is the only set that satisfies the complement laws.  In other words, complementation is characterized by the complement laws.

'''PROPOSITION 5''': Let {{tmath|1= A }} and {{tmath|1= B }} be subsets of a universe {{tmath|1= \boldsymbol{U} }}, then:
: uniqueness of complements:
:* If {{tmath|1= A \cup B = \boldsymbol{U} }}, and {{tmath|1= A \cap B = \varnothing }}, then {{tmath|1= B = A^\complement }}