Here's my problem: I understand how to create the sum of products (SOP) and product of sums (POS) forms of boolean functions, but I don't understand why we do it the way we do it. And I haven't found an answer anywhere online. Literally every source I've read simply tells you how to form these expressions but never explains the intuition.
Example: Suppose we have a truth table for some boolean function $F$.
The rules for forming a sum-of-products (SOP) expression say to:
- look at only the rows where $F = 1$
- create product terms out of $A$, $B$, and $C$, inverting any of these three variables that are $0$ and leaving the remaining variables that equal $1$ as they are
- sum all the product terms
So I would do:
$F = A'BC + AB'C+ABC'$
For the three rows where $F = 1$.
Question: why exactly do I ignore rows where $F=0$, and why do I invert any variables that are $0$?