Inclusion (Boolean algebra)

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

The inclusion relation ${\displaystyle a<b}$ can be expressed in many ways:

• ${\displaystyle a<b}$
• ${\displaystyle ab'=0}$
• ${\displaystyle a'+b=1}$
• ${\displaystyle b'<a'}$
• ${\displaystyle a+b=b}$
• ${\displaystyle ab=a}$

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:

• ${\displaystyle a\leq a+b}$
• ${\displaystyle ab\leq a}$

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