資料來源 : WordNet®

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

資料來源 : Free On-Line Dictionary of Computing

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 {set}s, 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)