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# Inclusion (Boolean algebra)  ( Boolean Algebra )

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In Boolean algebra (structure), the inclusion relation $a\le b$ is defined as $ab\text{'}=0$ and is the Boolean analogue to the relation in . Inclusion is a .

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 relation; in arithmetic Boolean algebra, ; 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:

• $a\le a+b$
• $ab\le a$

The inclusion relation may be used to define Boolean intervals such that $a\le x\le b$ 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

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