Writing an expression using logic Write an expression using letters $\land, \lor, and$ $\neg$ which has the following truth table: 
$$\begin{array}{ccc|c}
P&Q&R&???\\ \hline
T&T&T&F\\
T&T&F&T\\
T&F&T&F\\
T&F&F&T\\
F&T&T&T\\
F&T&F&F\\
F&F&T&F\\
F&F&F&F
\end{array}
$$
How would you in general solve for ??? I was told there is a mechanical way of doing it but I don't see it.
 A: I’ll get you started. Look at the last column of the truth table: you want your statement to be true when
$$P\text{ is true and }Q\text{ is true and }R\text{ is false}\tag{1}$$
$$\mathbf{OR}$$
$$P\text{ is true and }Q\text{ is false and }R\text{ is false}\tag{2}$$
$$\mathbf{OR}$$
$$P\text{ is false and }Q\text{ is true and }R\text{ is true}\;,\tag{3}$$
and for no other combination of truth values of $P,Q$, and $R$.
Is there a simple statement that’s true exactly when $P$ and $Q$ are true and $R$ is false? Sure: $$P\land Q\land\neg R\;.$$
Now find a similar statement that is true exactly when $(2)$ is true, and another one that’s true exactly when $(3)$ is true. Once you have these three statements, how should you combine them to get what you want?
A: There are exactly three rows which evaluate to "true = $T$".
We can take the first row, and see that $P \land Q \land \lnot R$ evaluates true.
Doing this with each row, we get that the truth table evaluates true when:
$$(P \land Q \land \lnot R) \lor (P \land \lnot Q \land \lnot R) \lor (\lnot P \land Q \land R)$$
This method gives you disjunctive normal form. And it is possible that it might be simplified. But this conveys precisely what the truth table conveys.
A: There are several methods available to solve this.  One method that's fairly easy for smaller circuits is to use a Karnaugh Map.  It requires a bit of study in order to understand what functions will look like on the map, but it makes the task fairly easy after Karnaugh maps are understood.
A second way is using the following trick:
We know that $x \lor y \lor z$, read "x or y or z", will have only one false value, and the rest will be true.  So you can substitute $P$, $Q$, $R$ or their negations, $\overline{P}$, $\overline{Q}$,or $\overline{R}$ into this to get one false value and seven true values.  Then you can essentially "paste" functions like this together to get the result.
For example, "(not $p$) or (not $q$) or (not $r$)" gives the truth table:
T
T
T
T
T
T
T
F
Then, "(not $p$) or (not $q$) or ($r$)" gives the truth table:
T
T
T
T
T
T
F
T
...and combining them gives : "((not $p$) or (not $q$) or (not $r$)) and ((not $p$) or (not $q$) or ($r$))":
T
T
T
T
T
T
F
F
...We can proceed in this fashion mechanically to slowly add one false value until we reach the desired function.
A: Entered by truthtable:
??? = P' Q R + P Q' R' + P Q R';

Minimized:
??? = P R' + P' Q R;

Related Karnaugh-Veitch map as screenshot from KVD Pro:

