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  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)
       
       

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

  boolean algebra
     Αγγλικά n.
     (ετ μαθ en) άλγεβρα Μπουλ

  Boolean algebra
     n.
     1 (lb en algebra) An algebraic structure <math>(Sigma, vee,
  wedge, sim, 0, 1)</math> where <math>vee</math> and
  <math>wedge</math> are idempotent binary operators,
  <math>sim</math> is a unary involutory operator (called
  "complement"), and 0 and 1 are nullary operators (i.e.,
  constants), such that <math>(Sigma, vee, 0)</math> is a
  commutative monoid, <math>(Sigma, wedge, 1)</math> is a
  commutative monoid, <math>wedge</math> and
  <math>vee</math> distribute with respect to each
  other,<!-- the identity of each monoid acts as the zero of the other
  monoid,--> and such that combining two complementary elements through
  one binary operator yields the identity of the other binary operator.
  (See (w: Boolean algebra (structure)#Axiomatics).)
     2 (lb en algebra logic computing) Specifically, an algebra in which
  all elements can take only one of two values (typically 0 and 1, or
  "true" and "false") and are subject to operations
  based on AND, OR and NOT<!-- may need to be split into: "an
  algebra of logical values" and "an algebra of ''two'' logical
  values" -->
     3 (lb en mathematics) The study of such algebras; Boolean logic,
  classical logic.

  Boolean algebra
     n.
     1 (lb en algebra) An algebraic structure <math>(Sigma, vee,
  wedge, sim, 0, 1)</math> where <math>vee</math> and
  <math>wedge</math> are idempotent binary operators,
  <math>sim</math> is a unary involutory operator (called
  "complement"), and 0 and 1 are nullary operators (i.e.,
  constants), such that <math>(Sigma, vee, 0)</math> is a
  commutative monoid, <math>(Sigma, wedge, 1)</math> is a
  commutative monoid, <math>wedge</math> and
  <math>vee</math> distribute with respect to each
  other,<!-- the identity of each monoid acts as the zero of the other
  monoid,--> and such that combining two complementary elements through
  one binary operator yields the identity of the other binary operator.
  (See (w: Boolean algebra (structure)#Axiomatics).)
     2 (lb en algebra logic computing) Specifically, an algebra in which
  all elements can take only one of two values (typically 0 and 1, or
  "true" and "false") and are subject to operations
  based on AND, OR and NOT<!-- may need to be split into: "an
  algebra of logical values" and "an algebra of ''two'' logical
  values" -->
     3 (lb en mathematics) The study of such algebras; Boolean logic,
  classical logic.

  Boolean algebra
     n.
     1 (lb en algebra) An algebraic structure <math>(Sigma, vee,
  wedge, sim, 0, 1)</math> where <math>vee</math> and
  <math>wedge</math> are idempotent binary operators,
  <math>sim</math> is a unary involutory operator (called
  "complement"), and 0 and 1 are nullary operators (i.e.,
  constants), such that <math>(Sigma, vee, 0)</math> is a
  commutative monoid, <math>(Sigma, wedge, 1)</math> is a
  commutative monoid, <math>wedge</math> and
  <math>vee</math> distribute with respect to each
  other,<!-- the identity of each monoid acts as the zero of the other
  monoid,--> and such that combining two complementary elements through
  one binary operator yields the identity of the other binary operator.
  (See (w: Boolean algebra (structure)#Axiomatics).)
     2 (lb en algebra logic computing) Specifically, an algebra in which
  all elements can take only one of two values (typically 0 and 1, or
  "true" and "false") and are subject to operations
  based on AND, OR and NOT<!-- may need to be split into: "an
  algebra of logical values" and "an algebra of ''two'' logical
  values" -->
     3 (lb en mathematics) The study of such algebras; Boolean logic,
  classical logic.

  Boolean algebra
     n.
     1 (lb en algebra) An algebraic structure <math>(Sigma, vee,
  wedge, sim, 0, 1)</math> where <math>vee</math> and
  <math>wedge</math> are idempotent binary operators,
  <math>sim</math> is a unary involutory operator (called
  "complement"), and 0 and 1 are nullary operators (i.e.,
  constants), such that <math>(Sigma, vee, 0)</math> is a
  commutative monoid, <math>(Sigma, wedge, 1)</math> is a
  commutative monoid, <math>wedge</math> and
  <math>vee</math> distribute with respect to each
  other,<!-- the identity of each monoid acts as the zero of the other
  monoid,--> and such that combining two complementary elements through
  one binary operator yields the identity of the other binary operator.
  (See (w: Boolean algebra (structure)#Axiomatics).)
     2 (lb en algebra logic computing) Specifically, an algebra in which
  all elements can take only one of two values (typically 0 and 1, or
  "true" and "false") and are subject to operations
  based on AND, OR and NOT<!-- may need to be split into: "an
  algebra of logical values" and "an algebra of ''two'' logical
  values" -->
     3 (lb en mathematics) The study of such algebras; Boolean logic,
  classical logic.

  Boolean algebra /bˈuːliən ˈaldʒɪbɹə/
  Boole'sche Algebra
   see: algebra, abstract algebra, homological algebra, Lie algebra
  

  Boolean algebra /bˈuːliən ˈaldʒɪbɹə/
  Schaltalgebra 
     Synonym: logic algebra
  

  boolean algebra /bˈuːliən ˈaldʒɪbɹə/
  Booleova algebra

  Boolean algebra /bˈuːliən ˈaldʒɪbɹə/
  Bulova algebra

  Boolean algebra /bˈuːliən ˈaldʒɪbɹə/ 
  ブール代数
  algebraic structure

  Boolean algebra /bˈuːliən ˈaldʒɪbɹə/ 
  Boolesk algebra
  algebraic structure

  boolean algebra /bˈuːliən ˈaldʒɪbɹə/
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