# Convert a DNF logic expression into a NOR normal form expression.

I want to convert the logic expression $$(a +b) \cdot c \cdot \overline{d}$$ into a NOR Normalform. I tried to this by first changing the expression into a DNF and then into the NOR Normalform. Here's my attempt:

$$(a +b) \cdot c \cdot \overline{d} \\ \equiv (a \cdot c \cdot \overline{d}) + (b \cdot c \cdot \overline{d}) \\ \equiv \overline{\overline{(a \cdot c \cdot \overline{d})}} + \overline{\overline{(b \cdot c \cdot \overline{d})}} \\ \equiv \overline{(\overline{a} + \overline{c} + d)} + \overline{(\overline{b} + \overline{c} + d)} \\ \equiv \overline{\overline{\overline{(\overline{a} + \overline{c} + d)} + \overline{(\overline{b} + \overline{c} + d)}}}$$

I am not if this is enough because I still don't think that this is in the NOR form, since the double negation will "cancel out" according to DeMorgan's Law. If I simplify the expression by using DeMorgan's Law, I would get a NAND expression and if I simplify it again I would get negated terms that are not in the NOR form. I'm kind of lost at this point.

• Do you have a formal definition of NOR normal form? – Taroccoesbrocco May 23 at 14:48
• @Taroccoesbrocco On my sheet it is defined as as a function that only uses NOT and NOR operations. – Ski Mask May 23 at 15:06
• Does your Nor take arbitrary inputs? (Since you seems using $2$ and $3$ input Nor) – Manx May 23 at 15:10
• @Manx It doesn't say anything the number of inputs it can take. It simply says to convert the given function so that it is only using NOR and NOT gates. As you mentioned, since it is shown to be able to take $2$ and $3$ inputs I assume that the inputs are arbitrary. – Ski Mask May 23 at 15:14
• I see, try use DeMorgan's Law to the whole term – Manx May 23 at 15:15

$$3$$-ary DeMorgan's Laws $$\overline{(P\cdot Q\cdot R)}\equiv\overline{P}+\overline{Q}+\overline{R}\tag{1}$$ $$\overline{(P+Q+R)}\equiv\overline{P}\cdot\overline{Q}\cdot\overline{R}\tag{2}$$ Useful links about $$n$$-ary DeMorgan's Laws:
Apply $$3$$-ary DeMorgan's Law to the whole term we have \begin{align} (a+b)\cdot c\cdot\overline{d}\equiv&\overline{\overline{(a+b)}+\overline{c}+d}\tag*{DeMorgan's Law (1)}\\ \equiv&\overline{\text{Nor}(a,b)+\overline{c}+d}\tag*{Nor definition}\\ \equiv&\text{Nor}(\text{Nor}(a,b),\overline{c},d)\tag*{Nor definition} \end{align} (Using $$2$$ and $$3$$ inputs Nor)