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How should I create the truth table to solve each of these questions? Maybe can give me an example?
$Smoke \implies Smoke$
$Smoke \implies Fire$
$(Smoke \implies Fire) \implies (\neg Smoke \implies \neg Fire)$
$Smoke \lor Fire \lor \neg Fire$
$((Smoke \land Heat) \implies ((Smoke \implies Fire) \lor (Heat \implies Fire)) $
$(Smoke \implies Fire) \implies ((Smoke \land Heat) \implies Fire))$
$Big \lor Dumb \lor (Big \implies Dumb)$
$(Big \land Dumb) \lor \neg Dumb$
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You would first set up a table with all variables being assigned each possible combination of truth values. You then propagate those truth values to the literals and next smallest subformulae, then continue propagation until you conclude the entire formula.
Example:
$$F: (P \land Q) \implies \neg (P \land \neg Q)$$
$$\begin{array}{|c|c|c|c|c|c|c|c|} \hline P & Q & \neg Q & P \land Q & P \land \neg Q & \neg(P \land \neg Q) & F \\ \hline F & F & T & F & F & T & T \\ \hline F & T & F & F & F & T & T \\ \hline T & F & T & F & T & F & T \\ \hline T & T & F & T & F & T & T \\ \hline \end{array}$$
