Algebra of Sets - The Algebra of Inclusion

The Algebra of Inclusion

The following proposition says that inclusion is a partial order.

PROPOSITION 6: If A, B and C are sets then the following hold:

reflexivity:

antisymmetry:

• and if and only if

transitivity:

• If and, then

The following proposition says that for any set S, the power set of S, ordered by inclusion, is a bounded lattice, and hence together with the distributive and complement laws above, show that it is a Boolean algebra.

PROPOSITION 7: If A, B and C are subsets of a set S then the following hold:

existence of a least element and a greatest element:

existence of joins:

• If and, then

existence of meets:

• If and, then

The following proposition says that the statement is equivalent to various other statements involving unions, intersections and complements.

PROPOSITION 8: For any two sets A and B, the following are equivalent:

The above proposition shows that the relation of set inclusion can be characterized by either of the operations of set union or set intersection, which means that the notion of set inclusion is axiomatically superfluous.