Inclusion (Boolean algebra)

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

The inclusion relation 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 Boolean algebra whose carrier set is restricted to the elements in an interval is itself a Boolean algebra.


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