**1.**

In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0, respectively.

FactSnippet No. 1,656,437 |

In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0, respectively.

FactSnippet No. 1,656,437 |

Boolean equation algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic, and set forth more fully in his An Investigation of the Laws of Thought.

FactSnippet No. 1,656,438 |

Boolean equation algebra has been fundamental in the development of digital electronics, and is provided for in all modern programming languages.

FactSnippet No. 1,656,439 |

Precursor of Boolean equation algebra was Gottfried Wilhelm Leibniz's algebra of concepts.

FactSnippet No. 1,656,440 |

Efficient implementation of Boolean equation functions is a fundamental problem in the design of combinational logic circuits.

FactSnippet No. 1,656,441 |

Boolean equation algebra is not sufficient to capture logic formulas using quantifiers, like those from first order logic.

FactSnippet No. 1,656,442 |

The closely related model of computation known as a Boolean equation circuit relates time complexity to circuit complexity.

FactSnippet No. 1,656,443 |

Three Boolean equation operations described above are referred to as basic, meaning that they can be taken as a basis for other Boolean equation operations that can be built up from them by composition, the manner in which operations are combined or compounded.

FactSnippet No. 1,656,444 |

The Duality Principle, called De Morgan duality, asserts that Boolean equation algebra is unchanged when all dual pairs are interchanged.

FactSnippet No. 1,656,445 |

Certainly any law satisfied by all concrete Boolean equation algebras is satisfied by the prototypical one since it is concrete.

FactSnippet No. 1,656,446 |

However, if we represent each divisor of n by the set of its prime factors, we find that this nonconcrete Boolean algebra is isomorphic to the concrete Boolean algebra consisting of all sets of prime factors of n, with union corresponding to least common multiple, intersection to greatest common divisor, and complement to division into n So this example while not technically concrete is at least "morally" concrete via this representation, called an isomorphism.

FactSnippet No. 1,656,447 |

Boolean equation algebras are special here, for example a relation algebra is a Boolean equation algebra with additional structure but it is not the case that every relation algebra is representable in the sense appropriate to relation algebras.

FactSnippet No. 1,656,448 |

We say that Boolean equation algebra is finitely axiomatizable or finitely based.

FactSnippet No. 1,656,449 |

Many syntactic concepts of Boolean algebra carry over to propositional logic with only minor changes in notation and terminology, while the semantics of propositional logic are defined via Boolean algebras in a way that the tautologies of propositional logic correspond to equational theorems of Boolean algebra.

FactSnippet No. 1,656,450 |

Entailment differs from implication in that whereas the latter is a binary operation that returns a value in a Boolean equation algebra, the former is a binary relation which either holds or does not hold.

FactSnippet No. 1,656,451 |

The natural interpretation of is as = in the partial order of the Boolean algebra defined by x = y just when x?y = y This ability to mix external implication and internal implication ? in the one logic is among the essential differences between sequent calculus and propositional calculus.

FactSnippet No. 1,656,452 |

Claude Shannon formally proved such behavior was logically equivalent to Boolean equation algebra in his 1937 master's thesis, A Symbolic Analysis of Relay and Switching Circuits.

FactSnippet No. 1,656,453 |

Original application for Boolean equation operations was mathematical logic, where it combines the truth values, true or false, of individual formulas.

FactSnippet No. 1,656,454 |

Boolean equation operations are used in digital logic to combine the bits carried on individual wires, thereby interpreting them over {0,1}.

FactSnippet No. 1,656,455 |