Simplify X = A(AB)' + A'B'C + ABC

Question

Why is this way of simplifying the Boolean expression wrong?

X = A(AB)' + A'B'C + ABC
  = A(AB)' + C(A'B'+AB) //A'+A = 1
  = A(AB)' + C
  = A(A'+B')+ C
  = (AA')+AB'+ C
  = AB'+ C

Is it wrong because C(A'B'+AB) ≠ C(1)? Can someone explain to me why they are not the same? Thank you.

Answer 1

$A'B'\ne(AB)'$

By de Morgan: $$A'B'=(A+B)'$$

Answer 2

Well, why should $A'B' + AB = 1$? The information $A' + A = 1$ only helps if you get to that in some way, but you can't pull out $B$ or $B'.$

If you want to see why they are not the same, just write down the truth tables and compare them.