Boolean algebra
    n 1: a system of symbolic logic devised by George Boole; used in
         computers [syn: Boolean logic, Boolean algebra]

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
   well.

   (1997-02-27)