# 7 February, 2000

## Logic design -- Getting the homework done

- Truth tables
- Cannonical sum-of-products
- Algebraic simplification

## Everyone's favorite example -- the seven segment display

_ _ _ _ _ _ _ _ _ _ _ _
| | | _| _| |_| |_ |_ | |_| |_| |_| |_ | _| |_ |_
|_| | |_ _| | _| |_| | |_| _| | | |_| |_ |_| |_ |

## Laws of Boolean Algebra

Commutativity |
`x` + `y` = `y` + `x` |
`x` `y` = `y` `x` |

Associativity |
`x` + (`y` + `z`) = (`x` +
`y`) + `z` |
`x` (`y` `z`) = (`x` `y`)
`z` |

Distributivity |
`x` (`y` + `z`) = `x` `y` +
`x` `z` |
`x` + `y` `z` = (`x` + `y`)
(`x` + `z`) |

Identity |
`x` + 0 = `x` |
`x` 1 = `x` |

Idempotency |
`x` + `x` = `x` |
`x` `x` = `x` |

Annihilation |
`x` + 1 = 1 |
`x` 0 = 0 |

Complement |
`x` + `x`' = 1 |
`x` `x`' = 0 |

Involution |
(`x`')' = `x` |

de Morgan's law |
(`x` + `y`)' = `x`' `y`' |
(`x` `y`)' = `x`' + `y`' |

absorption |
`x` + `x` `y` = `x` |
`x` (`x` + `y`) = `x` |