Inclusion (Boolean algebra)

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

The inclusion relation a<b can be expressed in many ways:

  • a<b
  • ab'=0
  • a'+b=1
  • b'<a'
  • a+b=b
  • 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\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.

References[edit]

  • Frank Markham Brown, Boolean Reasoning: The Logic of Boolean Equations, 2nd edition, 2003, p. 52