I am trying to simplify a logic expression but I think I simplified it too much. The expression is as follows:
$$\overline{\overline{(A \cdot B)} \cdot C \cdot (\overline{A}+\overline{(B+C)})}$$
This is what I got after I did the simplification:
But Logic Friday says that the answer should be:
$$A \cdot B+A \cdot C+\overline{C}$$
Which one is correct? Am I allowed to do the manipulations I did in this case?
Best Answer
Both answers are correct.
Let: $$ f_1(A,B,C) = AB+AC+\overline{C}\\ f_2(A,B,C) = A+\overline{C} $$ Let's build the thruth table: $$\begin{array}{|c|c|c|c|c|} \hline A & B & C & f_1 & f_2 \\ \hline 0 & 0 & 0 & 1 & 1\\ \hline 0 & 0 & 1 & 0 & 0\\ \hline 0 & 1 & 0 & 1 & 1\\ \hline 0 & 1 & 1 & 0 & 0\\ \hline 1 & 0 & 0 & 1 & 1\\ \hline 1 & 0 & 1 & 1 & 1\\ \hline 1 & 1 & 0 & 1 & 1\\ \hline 1 & 1 & 1 & 1 & 1\\ \hline \end{array}$$
As you can see the two functions correspond.
Please note that this tabular method of proving that two functions are the same is perfectly valid and is called Proof by exhaustion.