| Identity |
$$X \cdot 1 = X$$ |
$$X + 0 = X$$ |
| Annulment |
$$X \cdot 0 = 0$$ |
$$X + 1 = 1$$ |
| Complement |
$$X \bar{X} = 0$$ |
$$X + \bar{X} = 1$$ |
| Commutative |
$$X Y = Y X$$ |
$$X + Y = Y + X$$ |
| Idempotent |
$$\bar{\bar{X}} = X X = X$$ |
$$X + X = X$$ |
| Associative |
$$\left( X Y \right) Z = X \left( Y Z \right)$$ |
$$\left( X + Y \right) + Z = X + \left( Y + Z \right)$$ |
| Distributive |
$$X \left( Y + Z \right) = XY + XZ$$ |
$$X + YZ = \left( X + Y \right) \left( X + Z \right)$$ |
| Unity |
$$X Y + \bar{X} Y = Y$$ |
$$\left( X + Y \right) + \left( \bar{X} + Y \right) = Y$$ |
| Absorption |
$$X + X Y = X$$ |
$$X \left( X + Y \right) = X$$ |
| Absorption |
$$X + \bar{X} Y = X + Y$$ |
$$X \left( \bar{X} + Y \right) = XY$$ |
| DeMorgan’s |
$$\overline{ \left( X Y \right) } = \bar{X} + \bar{Y}$$ |
$$\bar{ \left( X + Y \right) } = \bar{X} \bar{Y}$$ |
| Consensus |
$$XY + YZ + \bar{X} Z = XY + \bar{X} Z$$ |
|
| XOR |
$$X \oplus Y = \bar{X} \oplus \bar{Y} = \overline{ \left( \bar{X} \oplus Y \right) } = \overline{ \left( X \oplus \bar{Y} \right) }$$ |
$$\overline{ \left( X \oplus Y \right) } = \overline{ \left( \bar{X} \oplus \bar{Y} \right) } = \bar{X} \oplus Y = X \oplus \bar{Y} $$ |