I am having real trouble getting to the corrects answers when asked to simply Sum of products expressions.
For instance:
Determine whether the left and right hand sides represent the same function:
a) $x_1\bar{x}_3+x_2x_3+\bar{x}_2\bar{x}_3 = (x_1+\bar{x}_2+x_3)(x_1+x_2+\bar{x}_3)(\bar{x}_1+x_2+\bar{x}_3)$
They answer is that they are equivalent. Here is my logic for solving the left hand side, I first expand so each term has 3 variables:
$$=x_1x_2\bar{x}_3+x_1\bar{x}_2\bar{x}_3+x_1x_2x_3+\bar{x}_1x_2x_3+x_1\bar{x}_2\bar{x}_3+\bar{x}_1\bar{x}_2\bar{x}_3$$
combining terms
$$\begin{align*}
&=x_1x_2(\bar{x}_3+x_3)+(x_1+\bar{x}_1)\bar{x}_2\bar{x}_3+x_1\bar{x}_2\bar{x}_3\\
&=x_1x_2+\bar{x}_2\bar{x}_3+x_1\bar{x}_2\bar{x}_3
\end{align*}$$
Then if I apply De Morgan's law in order to get it in Product of sum form, I get nothing close to the equivalent of the right hand side.
Best Answer
Just as when proving trig identities, in general it’s better to start by trying to simplify the more complicated side, which in this case is the right-hand side. If you expand the RHS by brute force and use the identities $xx=x$, $x\bar{x}=0$, $x+\bar{x}=1$, and $x+xy=x$ repeatedly, you shouldn’t have much trouble reducing it to $$x_1x_2+x_1\bar{x}_3+x_2x_3+\bar{x}_2\bar{x}_3,$$ which is almost what you want. Now write $$x_1x_2 = x_1x_2 1 = x_1x_2 (x_3 + \bar{x}_3),$$ expand, and use the absorption identity again to get rid of the extra terms.