Propiedades y Teoremas Del Algebra Booleana

ALGEBRA BOOLEANA

* PROPIEDAD CONMUTATIVA:
* A+B = B+A
* A·B = B·A

A | B | A+B | B+A | A·B | B·A |
0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 1 | 1 | 0 | 0 |
1 | 0 | 1 | 1 | 0 | 0|
1 | 1 | 1 | 1 | 1 | 1 |

* PROPIEDAD DISTRIBUTIVA:
* A·(B+C) = A·B + A·C
* A + B·C = (A+B)·(A+C)

A | B | C | B+C | A·(B+C) | A·B | A·C | A·B + A·C |
0 | 0 | 0 | 0 | 0 |0 | 0 | 0 |
0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
1 | 1 | 0 | 1 |1 | 1 | 0 | 1 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

A | B | C | B·C | A + B·C | A+B | A+C | (A+B)·(A+C) |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 |
0 | 1 | 0 | 0 | 0 | 1| 0 | 0 |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

* ELEMENTOSNEUTROS DIFERENTES:
* A+0 = A;
* A·1 = A

A | 0 | 1 | A+0 | A·1 |
0 | 0 | 1 | 0 | 0 |
1 | 0 | 1 | 1 | 1 |

* COMPLEMENTO DE A A:
* A+A’ = 1
* A·A’ = 0

A| A’ | A+A’ | A·A’ |
0 | 1 | 1 | 0 |
1 | 0 | 1 | 0 |

* TEOREMA 1 - El Complemento A es Único:

A | A’ |
0 | 1 |
1 | 0 |

* TEOREMA 2 - Elementos Nulos:
* A+1 = 1
* A·0= 0

A | 0 | 1 | A+1 | A·0 |
0 | 0 | 1 | 1 | 0 |
1 | 0 | 1 | 1 | 0 |

* TEOREMA 3 - Cada Elemento Identidad es el Complemento del Otro:
* 0’ = 1
* 1’ = 0

0 | 1 | 0’ |1’ |
0 | 1 | 1 | 0 |
0 | 1 | 1 | 0 |

* TEOREMA 4 – Idempotencia:
* A+A = A
* A·A = A

A | A+A | A·A |
0 | 0 | 0 |
1 | 1 | 1 |

* TEOREMA 5 – Involución:
* (A’)’ = AA | A’ | (A’)’ |
0 | 1 | 0 |
1 | 0 | 1 |

* TEOREMA 6 – Absorción:
* A+A·B = A
* A·(A+B) = A

A | B | A·B | A+A·B | A+B | A·(A+B) |
0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0...