[1] : WordNet (r) 2.0

boolean algebra
     n : a system of symbolic logic devised by george boole; used in
         computers syn: boolean logic
see also:
boolean logic 

[2] : The Free On-line Dictionary of Computing (27 SEP 03)

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
     
     
see also:
george boole two-valued logic models theory algebra lattice 
distributive lattice algebras set logical complement partial ordering 
linear ordering latex 

Algebra >> A Words
Algebra, definition of term: Algebra
algebra_pag1.html

Boolean >> B Words
Boolean, definition of term: Boolean
boolean_pag1.html

Boolean Logic >> B Words
Boolean Logic, definition of term: Boolean Logic
boolean+logic_pag1.html

Linear Algebra >> L Words
Linear Algebra, definition of term: Linear Algebra
linear+algebra_pag1.html

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