Algebra of Sets - Some Additional Laws For Complements

Some Additional Laws For Complements

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

PROPOSITION 4: Let A and B be subsets of a universe U, then:

De Morgan's laws:

double complement or Involution law:

complement laws for the universal set and the empty set:

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 A and B be subsets of a universe U, then:

uniqueness of complements:

• If, and, then