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2 definitions found
 for Boolean algebra
From WordNet (r) 3.0 (2006) :

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

From The Free On-line Dictionary of Computing (30 December 2018) :

  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)
  

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