Digital design之Boolean Algebra

1. 0 and 1 (duality: 0 -- 1, · -- +)

  X + 0 = X,  X · 1 = X

  X + 1 = 1,  X · 0 = 0

2. Idempotent

  X + X = X,  X · X = X

3. Involution

  (X')' = X

4. Complementarity

  X + X' = 1,  X · X' = 0

5. Commutative

  X + Y = Y + X,  X · Y = Y · X

6. Associative

  (X + Y) + Z = X + (Y + Z) = X + Y + Z

  (X · Y) · Z = X · (Y · Z) = X · Y · Z

7. Distributive

  X · (Y + Z) = X · Y + X · Z

  X + (Y · Z) = (X + Y) · (X +Z)

8. Simplification

  X · Y + X · Y' = X, (X + Y) · (X + Y') = X

  X + X · Y = X, X · (X + Y) = X  

9. Multiplying and factoring

  (X + Y) · (X' + Z) = X · Z + X' · Y

  X · Y + X' · Z = (X + Z) · (X' + Y)

10. Consensus

  X · Y + Y · Z + X · Z = X · Y + X' · Z

  (X + Y) · (Y + Z) · (X' + Z) = (X + Y) · (X' + Z)

11. DeMorgan's law

  (X + Y + Z + ...)' = X' · Y' · Z' · ...

  (X · Y · Z · ...)' = X' + Y' + Z' + ...

  {f(X1, X2, ..., Xn, 0, 1, +, ·)}' = {f(X1', X2', ..., Xn', 1, 0, ·, +)}

12. Duality

  (X + Y + Z + ...)D = X · Y · Z · ...

  (X · Y · Z)D = X + Y + Z + ...

  {f(X1, X2, ..., Xn, 0, 1, +, ·)}D = f(X1', X2', ..., Xn', 1, 0, ·, +)

 

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