Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...

, the inclusion relation

a\le b

is defined as

ab'=0

and is the Boolean analogue to the

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...

relation in

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

. Inclusion is a

partial order In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable ...

. The inclusion relation

a 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

relation; in arithmetic Boolean algebra,

divisibility In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a ''Multiple (mathematics), multiple'' of m. An integer n is divis ...

; in the algebra of propositions, material implication; in the two-element algebra, the set . 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

* , ''Boolean Reasoning: The Logic of Boolean Equations'', 2nd edition, 2003
p. 34, 52
{{isbn, 0486164594 Boolean algebra