Boolean Theorems

( 1)  X0 = 0

( 2)  X1 = X

( 3)  XX = X

       -
( 4)  XX = 0


( 5)  X + 0 = X

( 6)  X + 1 = 1

( 7)  X + X = X

          -
( 8)  X + X = 1

( 9)  X + Y = Y + X  (communitative law)

(10)  XY = YX  (communitative law)

(11)  X + (Y + Z) = (X + Y) + Z = X + Y + Z  (associative law)

(12)  X(YZ) = (XY)Z = XYZ  (associative law)

(13a) X (Y + Z) = XY + XZ

(13b) (W + X)(Y + Z) = WY + XY + WZ + XZ

(14)  X + XY = X

      Proof: X + XY = X (1 + Y)
                    = X1 (using theorem 6)
                    = X (using theorem 2)

          -
(15)  X + XY = X + Y

DeMorgan's Theorems:

      -------   --
(16)  (X + Y) = XY

      ----   -   -
(17)  (XY) = X + Y

      -----------   ---
(18)  (X + Y + Z) = XYZ

      -----   -   -   -
(19)  (XYZ) = X + Y + Z


          -
          -
(?)   X = X