How to simplify $\overline{A\oplus \overline{AB}}$ correctly?

Question

My attempt:

$$\overline{A\oplus \overline{AB}}$$

Let $\overline{AB}=C$. Now,

$$\overline{A\oplus C}$$

$$=AC+\overline{AC}$$

$$=AC+\overline{A}+\overline{C}$$

$$=A(\overline{AB})+\overline{A}+AB$$

$$=A(\overline{A}+\overline{B})+\overline{A}+AB$$

$$=(0+A\overline{B})+\overline{A}+AB$$

$$=A\overline{B}+\overline{A}+AB$$

$$=A(B+\overline{B})+\overline{A}$$

$$=A.1+\overline{A}$$

$$=A+\overline{A}$$

$$=1$$

However, this is the wrong answer. According to my book, the correct answer is $A\overline{B}$.

My book's attempt:

$$\overline{A\oplus \overline{AB}}$$

$$=\overline{A}.\overline{\overline{AB}}+A.\overline{AB}\tag{1}$$

$$=\overline{A}.AB+A(\overline{A}+\overline{B})$$

$$=0+(0+A\overline{B})$$

$$=A\overline{B}$$

My questions:

1. Why did I not get the correct answer in my attempt?
2. In $(1)$, shouldn't it have been $\overline{A.\overline{AB}}+A.\overline{AB}$ instead of what it is?

Answer 1

You made a subtle error:

$\overline{A\oplus C} \not= AC +\color{red}{\overline{AC}}$

$\overline{A\oplus C} = AC +\color{blue}{\overline{A}.\overline{C}}$

When you back substitute $C=\overline {AB}$, you should get a sum whose terms simplify to $0$ and $A\overline{B}$.