Inclusion (Boolean algebra) (original) (raw)

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In Boolean algebra, the inclusion relation a ≤ b {\displaystyle a\leq b} $a\leq b$ is defined as a b ′ = 0 {\displaystyle ab'=0} $ab'=0$ 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} $a<b$ can be expressed in many ways:

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

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

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

Frank Markham Brown [d], Boolean Reasoning: The Logic of Boolean Equations, 2nd edition, 2003, p. 34, 52 ISBN 0486164594