In Boolean algebra, the inclusion relation a ≤ b {\displaystyle a\leq b} is defined as a b ′ = 0 {\displaystyle ab'=0} and is the Boolean analogue to the subset relation in set theory. Inclusion is a partial order. The inclusion relation a < b {\displaystyle a<b} can be expressed in many ways:
a < b {\displaystyle a<b}
a b ′ = 0 {\displaystyle ab'=0}
a ′ + b = 1 {\displaystyle a'+b=1}
b ′ < a ′ {\displaystyle b'<a'}
a + b = b {\displaystyle a+b=b}
a b = a {\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:
a ≤ a + b {\displaystyle a\leq a+b}
a b ≤ a {\displaystyle ab\leq a}
The inclusion relation may be used to define Boolean intervals such that a ≤ x ≤ b {\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.
References Frank Markham Brown, Boolean Reasoning: The Logic of Boolean Equations, 2nd edition, 2003, p. 34, 52 ISBN 0486164594
