I previously posted a logic rules cheat sheet and figured it was about time that I do the same for boolean algebra.

ExpressionEquivalent ToName of the Rule
$$ X + Y $$$$ Y + X $$Commutative
$$ X \cdot Y $$$$ Y \cdot X $$Commutative
$$ (X + Y) + Z $$$$ X + (Y + Z) $$Associative
$$ (X \cdot Y) \cdot Z $$$$ X \cdot (y \cdot Z) $$Associative
$$ X + (Y \cdot Z) $$$$ (X + Y) \cdot (Z + Z) $$Distributive
$$ X \cdot (Y + Z) $$$$ (X \cdot Y) + (X \cdot Z) $$Distributive
$$ X + 0 $$$$ X $$Identity
$$ X \cdot 1 $$$$ X $$Identity
$$ X + X' $$$$ 1 $$Complement
$$ X \cdot X' $$$$ 0 $$Complement
$$ X + X $$$$ X $$Idempotence
$$ X \cdot X $$$$ X $$Idempotence