Do we have to expand the brackets before applying De Morgan's Laws?

Question

If we have the example:

$$\overline{(A + B + C)D}$$

Then can we apply De Morgan's Law as is, or do we first need to expand out the brackets?

If I expand the terms first, I get:

$$original: \overline{(A + B + C)D}$$ $$expanded: \overline{AD + BD + CD}$$ $$applying De Morgan's: \overline{AD}.\overline{BD}.\overline{CD}$$ $$simplifying: \overline{A}\overline{B}\overline{C}\overline{D}$$

In contrast, if I don't expand the brackets, I get something like:

$$original: \overline{(A + B + C)D}$$ $$applying De Morgan's: \overline{(A + B + C)}+\overline{D}$$ $$applying De Morgan's: (\overline{A} . \overline{B + C})+\overline{D}$$ $$applying De Morgan's: (\overline{A} . \overline{B} . \overline{C})+\overline{D}$$ $$simplifying: \overline{A} . \overline{B} . \overline{C}+\overline{D}$$

So you can see, I'm getting different answers. I'm curious which method is correct - I think the first seems correct, but perhaps the 2nd is correct or both are wrong.

If anyone is able to explain why one particular method is wrong, I'd appreciate it too.

Answer 1

Both lead to the same result; your error is in the first one when you simplify from $$\overline{AD}.\overline{BD}.\overline{CD}$$ to $$\overline{A}\overline{B}\overline{C}\overline{D}$$.

If you apply De Morgan's law to each pair, you get: $$(\overline{A}+\overline{D})(\overline{B}+\overline{D})(\overline{C}+\overline{D}).$$ From here, you can apply the distributive law to get what you got by the second method in the next to last line. The simplifcation there is also incorrect because you cannot simply drop the parenthesis; you must apply the distributive property.