Convert a SOP equation to POS using algebraic manipulation

Question

A’BC’ + AB’D + AC’(B’+D’)

Usually I would use distributive laws but I do not know how it would apply in this case. How can I manipulate this question so I can get it into POS form?

Answer 1

One mechanical approach starts by multiplying out and taking the complement. If we call your expression $\mathscr{E}$, then

$$\begin{align*} \mathscr{E}'&=\big(A'BC'+AB'D+AC'(B'+D')\big)'\\ &=(A'BC'+AB'D+AB'C'+AC'D')'\\ &=(A'BC')'(AB'D)'(AB'C')'(AC'D')'\\ &=(A+B'+C)(A'+B+D')(A'+B+C)(A'+C+D)\;. \end{align*}$$

Now you can multiply this out and repeat the process to get the POS form of $(\mathscr{E}')'=\mathscr{E}$. This works because the De Morgan law makes it easy to write the complement of an SOP form as a POS form.