How should I create the truth table to solve each of these questions? Maybe can give me an example?

  1. $Smoke \implies Smoke$

  2. $Smoke \implies Fire$

  3. $(Smoke \implies Fire) \implies (\neg Smoke \implies \neg Fire)$

  4. $Smoke \lor Fire \lor \neg Fire$

  5. $((Smoke \land Heat) \implies ((Smoke \implies Fire) \lor (Heat \implies Fire)) $

  6. $(Smoke \implies Fire) \implies ((Smoke \land Heat) \implies Fire))$

  7. $Big \lor Dumb \lor (Big \implies Dumb)$

  8. $(Big \land Dumb) \lor \neg Dumb$

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    $\begingroup$ Could you replace the image with text? Also, did you try to solve it yourself? $\endgroup$ – fade2black Sep 30 '17 at 11:23
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    $\begingroup$ If you know how to construct a truth-table, this is not hard: each word is a variable and you just need to calculate what each operation does. Though, all of them are easily solved without it. $\endgroup$ – rus9384 Sep 30 '17 at 11:40
  • $\begingroup$ We won't do your homework for you. Ryan's answer gives you a general method, which is actually more than you should have expected from just posting your homework online. $\endgroup$ – Raphael Oct 2 '17 at 10:07

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.


$$F: (P \land Q) \implies \neg (P \land \neg Q)$$

$$\begin{array}{|c|c|c|c|c|c|c|c|} \hline P & Q & \neg Q & P \land Q & P \land \neg Q & \neg(P \land \neg Q) & F \\ \hline F & F & T & F & F & T & T \\ \hline F & T & F & F & F & T & T \\ \hline T & F & T & F & T & F & T \\ \hline T & T & F & T & F & T & T \\ \hline \end{array}$$


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