Boolean Algebra and negation

Question

$$ -(A \ast -B) \ast -(-A \ast B) $$

My understanding is that the above logic is equal to $$ (-A \ast B) \ast (A \ast -B) = (-A \ast A) \ast (-B \ast B) = \mathrm{FALSE} $$

but my textbook says that my first manipulation is wrong. According to what laws am I wrong?

Answer 1

$$ -(A \ast -B) \ast -(-A \ast B) $$ Using the rule $ -(X \ast Y) = (-X + -Y) $, dont forget the and becomes an or and vice versa $$ = (-A + --B) \ast (--A + -B) $$ $$ = (-A + B) \ast (A + -B) $$ Multiplying out gives $$ = -A \ast A + -A \ast -B + b \ast A + B \ast -B $$

$$ = FALSE + -A \ast -B + B \ast A + FALSE $$ or just $$ = (-A \ast -B) + (A \ast B) $$ ie XNOR

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$$ -(A \ast -B) \ast -(-A \ast B) $$ From here we could join the negation to give

$$ -((A \ast -B) + (-A \ast B)) $$

which is the negation of XOR or XNOR from the start