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In [[Boolean algebra (structure)]], the '''inclusion relation''' <math>a\le b</math> is defined as <math>ab'=0</math> and is the Boolean analogue to the [[subset]] relation in [[set theory]]. Inclusion is a [[partial order]].

The inclusion relation <math>a<b</math> can be expressed in many ways:
* <math>a<b</math>
* <math>ab'=0</math>
* <math>a'+b=1</math>
* <math>b'<a'</math>
* <math>a+b=b</math>
* <math>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, [[divisor|divisibility]]; in the [[Propositional formula#An algebra of propositions.2C the propositional calculus|algebra of propositions]], [[Material conditional|material implication]]; in the two-element algebra, the set { (0,0), (0,1), (1,1) }.

Some useful properties of the inclusion relation are:
* <math>a\le a+b</math>
* <math>ab\le a</math>

The inclusion relation may be used to define '''Boolean intervals''' such that <math>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, ''Boolean Reasoning: The Logic of Boolean Equations'', 2nd edition, 2003, p. 52

[[Category:Boolean algebra]]

Commutator subspace - Revision history

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