Boolean algebra

<mathematics, logic> (After the logician George Boole)

1. Commonly, and especially in computer science and digital electronics, this term is used to mean two-valued logic.

2. This is in stark contrast with the definition used by pure mathematicians who in the 1960s introduced “Boolean-valued models” into logic precisely because a “Boolean-valued model” is an interpretation of a theory that allows more than two possible truth values!

Strangely, a Boolean algebra (in the mathematical sense) is not strictly an algebra, but is in fact a lattice.

A Boolean algebra is sometimes defined as a “complemented distributive lattice“.

Boole’s work which inspired the mathematical definition concerned algebras of sets, involving the operations of intersection, union and complement on sets.

Such algebras obey the following identities where the operators ^, V, – and constants 1 and 0 can be thought of either as set intersection, union, complement, universal, empty; or as two-valued logic AND, OR, NOT, TRUE, FALSE; or any other conforming system.

a ^ b = b ^ a

a V b

=

b V a

(commutative laws) (a ^ b) ^ c

=

a ^ (b ^ c) (a V b) V c

=

a V (b V c)

(associative laws) a ^ (b V c)

=

(a ^ b) V (a ^ c) a V (b ^ c)

=

(a V b) ^ (a V c)

(distributive laws) a ^ a

=

a

a V a

=

a

(idempotence laws) –a

=

a -(a ^ b)

=

(-a) V (-b) -(a V b)

=

(-a) ^ (-b)

(de Morgan’s laws) a ^ -a

=

0

a V -a

=

1 a ^ 1

=

a

a V 0

=

a a ^ 0

=

0

a V 1

=

1 -1

=

0

-0

=

1

There are several common alternative notations for the “-” or logical complement operator.

If a and b are elements of a Boolean algebra, we define a <= b to mean that a ^ b = a, or equivalently a V b = b.

Thus, for example, if ^, V and – denote set intersection, union and complement then <= is the inclusive subset relation.

The relation <= is a partial ordering, though it is not necessarily a linear ordering since some Boolean algebras contain incomparable values.

Note that these laws only refer explicitly to the two distinguished constants 1 and 0 (sometimes written as LaTeX \top and \bot), and in two-valued logic there are no others, but according to the more general mathematical definition, in some systems variables a, b and c may take on other values as well.