Inclusion (Boolean algebra)

In Boolean algebra, the inclusion relation $a\backslash le\; b$ is defined as $ab\text{'}=0$ and is the Boolean analogue to the subset relation in set theory. Inclusion is a partial order.

The inclusion relation

• $ab\text{'}\; =\; 0$
• $a\text{'}\; +\; b\; =\; 1$
• $a+b\; =\; b$
• $ab\; =\; a$

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:

• $a\; \backslash le\; a+b$
• $ab\; \backslash le\; a$

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