Inclusion (Boolean algebra)

In Boolean algebra (structure), the inclusion relation a ≤ b is defined as a b ′ = 0 and is the Boolean analogue to the subset relation in set theory. Inclusion is a partial order.

The inclusion relation a < b can be expressed in many ways:

The inclusion relation has a natural interpretation in various Boolean algebras: in the subset algebra, the subset relation; in arithmetic Boolean algebra, divisibility; in the algebra of propositions, material implication; in the two-element algebra, the set { (0,0), (0,1), (1,1) }.

Some useful properties of the inclusion relation are:

The inclusion relation may be used to define Boolean intervals such that a ≤ x ≤ b A Boolean algebra whose carrier set is restricted to the elements in an interval is itself a Boolean algebra.