Inclusion (Boolean algebra)

In Boolean algebra, the inclusion relation $$a\le b$$ is defined as $$ab'=0$$ 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:
 * $$a < b$$
 * $$ab' = 0$$
 * $$a' + b = 1$$
 * $$b' < a'$$
 * $$a+b = b$$
 * $$ab = a$$

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:
 * $$a \le a+b$$
 * $$ab \le a$$

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