Fundamental Theorems & Postulates of Boolean Algebra
Identities: (1) X + 0 = X (1D) X•1 = X
Null Elements: (2) X + 1 = 1 (2D) X•0 = 0
Indempotency: (3) X + X = X (3D) X•X = X
Involution (Double Negation): (4) (X')' = X
Complements: (5) X + X' = 1 (5D) X•X' = 0
Commutativity: (6) X + Y = Y + X (6D) X•Y = Y•X
Associativity: (7) (X+Y)+Z = X+(Y+Z) (7D) (X•Y)•Z = X•(Y•Z)
Distributivity: (8) X•Y+X•Z = X•(Y+Z) (8D) (X+Y)•(X+Z) = X+Y•Z
Combining: (9) X•Y + X•Y' = X (9D) (X+Y)•(X+Y') = X
Covering: (10) X + X•Y = X (10D) X•(X+Y) = X
DeMorgan's Laws: (12) (X•Y•Z)' = X'+Y'+Z' (12D) (X+Y+Z)' = X'•Y'•Z'
Consensus: (17) X•Y+X'•Z+Y•Z = X•Y+X'•Z (17D) (X+Y)•(X'+Z)•(Y+Z) =
Shannon Expansion: (18) F(X,Y,Z) = X•F(1,Y,Z) + X'•F(0,Y,Z)
(18D) F(X,Y,Z) = (X+F(0,Y,Z))•(X'+F(1,Y,Z))