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, ·, +)