CP1300 Boolean Algebra Answers

1. What is boolean Algebra?

   A formal mathematics based system, for describing logic
   and logical relationships.
   

2. Write out the truth table for F = (X'*Y') + (X*Y). What boolean operator is it?

   X	Y	F
   ---------------------
   0	0	1
   0	1	0
   1	0	0
   1	1	1
   
   
   F is the inverse of XOR, called XNOR
   

3. Write out the truth table for F = (X * (Y + Z)) * (Y * (X' + Z)). What boolean operator is it?

   X	Y	Z	F
   --------------------------
   0	0	0	0
   0	0	1	0
   0	1	0	0
   0	1	1	0
   1	0	0	0
   1	0	1	0
   1	1	0	0
   1	1	1	1
   
   
   F is AND for 3 inputs
   

4. Simplify the following functions (make use of the laws of boolean algebra)
   1. F = (X * (Y + Z)) * (Y * (X' + Z))

      = (XY + XZ) * (YX' + YZ)
      
      = ((XY + XZ) * YX') + ((XY + XZ) * YZ)
      
      = ((XY * YX') + (XZ * YX')) + ((XY * YZ) + (XZ * YZ))
      
      = (0 + 0) + (XYZ + XYZ)
      
      = XYZ
      

   2. F = ((X' * Y') + (X * Y)) * (X * Y)

      = ((X'Y' + XY) * X) * Y
      
      = (X * (X'Y' + XY)) * Y
      
      = (XX'Y' + XXY) * Y
      
      = (0 + XY) * Y
      
      = XYY
      
      = XY
      
      = X * Y
      

   3. F = (X' * (Y' + Z'))'

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

   4. F = Z + ((Z * (Y + (X * Y))) + Z)

      = Z + ((Z * (Y + XY)) + Z)
      
      = Z + (Z + (Z * (Y + XY)))
      
      = Z + (Z + (Z * (Y + YX)))
      
      = Z + (Z + (Z * (Y)))
      
      = Z + (Z + ZY)
      
      = Z + (Z)
      
      = Z + Z
      
      = Z
      

   Laws:

   X + 0 = X X + X' = 1 X + X = X X + 1 = 1 (X')' = X X + Y = Y + X X + (Y + Z) = (X + Y) + Z X * (Y + Z) = (X * Y) + (X * Z) (X + Y)' = X' * Y' X + (X * Y) = X

   X * 1 = X X * X' = 0 X * X = X X * 0 = 0 X * Y = Y * X X * (Y * Z) = (X * Y) * Z X + (Y * Z) = (X + Y) * (X + Z) (X * Y)' = X' + Y' X * (X + Y) = X