Basic Boolean Algebra Multiplication Question

Question

I have the following term

$$ t1: \overline {\overline{x1x2\Leftarrow\Rightarrow x1x3}\Leftarrow\Rightarrow x2x3} $$

which I already converted to this:

$$ t2: ((x1x2\overline{x1x3} + \overline{x1x2}x1x3) \overline{x2x3}) + ((\overline{x1x2}x1x3 + x1x2\overline{x1x3})x2x3) $$

The final result should look like this: $$ t3: x2x3 + x1\overline{x2}x3 + x1x2\overline{x3} $$

Unfortunately I have no idea how to get from $t2$ to $t3$. If somebody could explain the neccessary steps that would be appreciated.

Thanks in advance.

Answer 1

This answer answers the original version of the question; an edit while I was typing removed the part I'm replying to here :-(

I think the overbar notation is making you confused, because you don't have both $x_1$ and $\overline{x_1}$ in the multiplication $x_1 x_2 \overline{x_1 x_3}$.

Note that $\overline{x_1 x_3}$ is not the same as $\overline{x_1}\cdot \overline{x_3}$, but rather (by De Morgan) $\overline{x_1}+\overline{x_3}$.

So when you have $$x_1 x_2 \overline{x_1 x_3} = x_1 x_2 (\overline{x_1}+\overline{x_3})$$ first use the distributive law to get $$ x_1 x_2 \overline{x_1} + x_1 x_2 \overline{x_3} $$

Then in the first of these terms you do have both $x_1$ and $\overline{x_1}$, so that term disappears and you're left with just $x_1 x_2 \overline{x_3}$.