# How was the result of this SOP (sum of products) expression reached?

I have a boolean expression for which I have to get the truth table and standard SOP expression without using a Karnaugh map.

Here is what I got:

I do have the final answer of the SOP expression, but what are the steps that were done to reach it?

I tried solving it but I'm not sure of my answer:

Are the steps I took correct?

I'm really confused.

Well you are making a mistake $$\(\overline{A\overline{B}}) (C+\overline{C})\$$ is not eaqual to what you just wrote. You can't just ignore the complement that is sitting right there. It would be equal to $$\\overline{A\overline{B}}C + \overline{A\overline{B}}(\overline{C})\$$

Anyway the easier thing to do this, if you already have truth table, look for the values of F when it's 1. As you can see there are 7 terms in your SOP expression and there are 7 true (or one) values in your F column.

Let's start with the first row as you can see it's $$\ A= 0, B=0, C=0\$$ and the value is true. We could write that as $$\\overline{A}*\overline{B}*\overline{C} \$$.

Why I wrote it this way? Well since it is AND operation the only way that expression can be true is when A B C are zero. Same goes for every row except 6th one.

Let's do one more example, 7th row, $$\ A= 1, B=1, C=0\$$, we could write that as $$\{A}{B}\overline{C} \$$.

Do that for every row and you have complete SOP expression.

• Got it! Thank you! Apr 1, 2020 at 15:19