Postulates and Theorems of Boolean Algebra

Assume A, B, and C are logical states that can have the values 0 (false) and 1 (true).
"+" means OR, "·" means AND, and NOT[A] means NOT A.


Postulates

(1)   A + 0 = A   A · 1 = A  identity
(2)   A + NOT[A] = 1   A · NOT[A] = 0  complement
(3)   A + B = B + A   A · B = B · A   commutative law
(4)   A + (B + C) = (A + B) + C   A · (B · C) = (A · B) · C   associative law
(5)   A + (B · C) = (A + B) · (A + C)   A · (B + C) = (A · B) + (A · C)   distributive law

Theorems

(6)   A + A = A   A · A = A  
(7)   A + 1 = 1   A · 0 = 0  
(8)   A + (A · B) = A   A · ( A + B) = A  
(9)   A + (NOT[A] · B) = A + B   A · (NOT[A] + B) = A · B  
(10)   (A · B) + (NOT[A] · C) + (B · C) = (A · B) + (NOT[A] · C)   A · (B + C) = (A · B) + (A · C)  
(11)   NOT[A + B] = NOT[A] · NOT[B]   NOT[A · B] = NOT[A] + NOT[B]   de Morgan's theorem

Supplemental Material for ELEN 021, Logic Design