# Boolean Algebra

Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false

## Expressions

**Consistent:**if it cannot be both true and false**Complete:**if every fully instantiated expression if true or false**Tautology:**if it evaluates to true for every combination of its propositional variables**Contradiction:**if it evaluates to false for every combination of its propositional variables

## Operators

### Exclusive Or

P or Q, but not both: .

F | F | F | F | F |

F | T | T | F | T |

T | F | T | T | F |

T | T | F | F | F |

### Nor

Neither P nor Q: .

F | F | T | T | T |

F | T | F | T | F |

T | F | F | F | T |

T | T | F | F | F |

### Negative And (NAND)

P and Q are not both true: .

F | F | T | T | T |

F | T | T | T | F |

T | F | T | F | T |

T | T | F | F | F |

### Conditional

If P, then Q: . It is sometimes described like this:

- P only if Q
- P is a sufficient condition of Q
- Q is a necessary condition for P

F | F | T | T |

F | T | T | T |

T | F | F | F |

T | T | T | F |

A conditional can also be expressed in the following form, called
*contrapositive*: .

F | F | T | T | T |

F | T | T | F | T |

T | F | F | T | F |

T | T | T | F | F |

Proof:

### Biconditional (iff)

P if and only Q: .

F | F | T | T | T |

F | T | F | T | F |

T | F | F | F | T |

T | T | T | T | T |

## Laws

### DeMorganâ€™s Laws

### Commutative Laws

### Associative Laws

### Idempotence Laws

### Unit Element Laws

### Zero Element Laws

### Complement Laws

### Distributive Laws

### Absorption Laws

### Double Negation Law

## Truth Sets

The truth set of a statement is the set of all values of that make the statement true.

- The truth set of :
- The truth set of :