Inclusion (Boolean algebra)
In Boolean algebra, the inclusion relation [math]\displaystyle{ a\le b }[/math] is defined as [math]\displaystyle{ ab'=0 }[/math] and is the Boolean analogue to the subset relation in set theory. Inclusion is a partial order. The inclusion relation [math]\displaystyle{ a\lt b }[/math] can be expressed in many ways:
- [math]\displaystyle{ a \lt b }[/math]
- [math]\displaystyle{ ab' = 0 }[/math]
- [math]\displaystyle{ a' + b = 1 }[/math]
- [math]\displaystyle{ b' \lt a' }[/math]
- [math]\displaystyle{ a+b = b }[/math]
- [math]\displaystyle{ ab = a }[/math]
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:
- [math]\displaystyle{ a \le a+b }[/math]
- [math]\displaystyle{ ab \le a }[/math]
The inclusion relation may be used to define Boolean intervals such that [math]\displaystyle{ a\le x\le b }[/math]. A Boolean algebra whose carrier set is restricted to the elements in an interval is itself a Boolean algebra.
References
- Frank Markham Brown (d), Boolean Reasoning: The Logic of Boolean Equations, 2nd edition, 2003, p. 34, 52 ISBN 0486164594
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