Which one is the right answer in this boolean expression?

Question

(A+B)' (A'+B')' =(A'B')+(AB)= A'B'+AB (Ans)

 OR

(A+B)' (A'+B')' =(A'+B')(A+B)

                = A'A+ A'B+ AB'+ BB' 

                = 0+ A'B+ AB'+ 0

                = A'B+ AB'  (Ans)

I get two different answers. I think the procedures I used in both are right. So which one is the right solution or are both of them wrong? The solution to the expression in my lecture has been given as 0 which really doesn't add up.

Answer 1

$(A+B)'=(A'B')$ and $(A'+B')'=(AB)$. This fact does not affect the multiplication between these two terms. So you should start $$ (A+B)'(A'+B')'=(A'B')(AB) $$