(mathematics, logic) | **Boolean algebra** - (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
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