Question

Simplify the following Boolean expressions to the minimum number of terms using the properties of Boolean algebra (show your work and write the property you are applying). State if they cannot be simplified

A. X’Y + XY

B. (X + Y)(X + Y’)

C. (A’ + B’) (A + B)’

D. ABC + A’B + A’BC’

E. XY + X(WZ + WZ’)

Answer 1

A.
   X'Y + XY

   =>   X'Y + XY
   =>   Y(X' + X)           [X' X = 1]
   =>   Y.1
   =>   Y

B.
   (X + Y)(X + Y’)

   =>   (X + Y)(X + Y')
   =>   X(X + Y') + Y(X + Y')       [Distributive Law]
   =>   X + XY' + XY + XY'
   =>   X + XY' + XY
   =>   X(1 + Y' + Y)           [1 + X = X]
   =>   X

C.
   (A' + B') (A + B)'

   =>   (A' + B') (A + B)'
   =>   (A' + B') (A'B')       [DeMorgan's Law]
   =>   A'A'B' + B'A'B'           [Distributive Law]
   =>   A'B' + A'B'
   =>   A'B'  

D.
   ABC + A'B + A'BC'

   =>   ABC + A'B + A'BC'
   =>   ABC + A'B(1 + C')       [Taking common]
   =>   ABC + A'B

E.
   XY + X(WZ + WZ')

   =>   XY + X(WZ + WZ')
   =>   XY + X(W(Z + Z'))
   =>   XY + X(W.1)           [Z + Z' = 1]
   =>   XY + XW
   =>   X(Y + W)