- Two elements: 0 and 1
- Two binary operators: AND and OR
- One unary operator: NOT

- George
Boole, 1815-1864
- first job was an elementary school teacher
- last job was Professor of Mathematics at Queen's College Cork
- Logic as mathematics, not "metaphysics"

- Edward V. Huntington
- Developed the textbook (pp. 41-42) axioms in 1904
- Developed a three axiom set in 1933
- Commutativity
- Associativity
- Huntington equation: (
`x`' +`y`)' + (`x`' +`y`')' =`x`

- Marshall Stone
- In 1936, represented Boolean algebras as "compact zero-dimensional Hausdorff space"
- In 1937, represented Boolean algebras as a field of sets

- Claude Shannon
- In 1938, showed that Boolean algebra can describe logic circuits
- Later developed information theory

- William McCune
- Works with automatic theorem proving at Argonne National Lab
- Used AI program EQP to prove the Robbin's conjecture in 1996

Identity | α + 0 = α | α 1 = α |

Annihilation | α + 1 = 1 | α 0 = 0 |

Idempotency | α + α = α | α α = α |

Involution | (α')' = α | |

Complementarity | α + α' = 1 | α α' = 0 |

Commutativity | α + β = β + α | α β = β α |

Associativity | α + (β + γ) = (α + β) + γ | α (β γ) = (α β) γ |

Distributivity | α (β + γ) = α β + α γ | α + β γ = (α + β) (α + γ) |

de Morgan's law | (α + β)' = α' β' | (α β)' = α' + β' |

absorption | α + α β = α | α (α + β) = α |

α + α' β = α + β | α (α' + β) = α β |

Take a Boolean equation. Change AND's to OR's, OR's to AND's, 0's to 1's, and 1's to 0's. The equation still holds.

Prove the *uniting theorem*, the equivalence of the following

`x``y`+`x``y`'`x`

Show the equivalence of the following two Boolean equations

`x`'`y`'`z`' +`x`'`y`'`z`+`x`'`y``z`' +`x`'`y``z`+`x``y`'`z`' +`x``y``z``x`' +`y`'`z`' +`y``z`