Inclusion (Boolean algebra) explained

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 .

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.

References

• , Boolean Reasoning: The Logic of Boolean Equations, 2nd edition, 2003, p. 34, 52