0
$\begingroup$

I've been learning about propositions and truth tables recently and been given examples like "If it is raining I will take my umbrella." P=it is raining Q=i take my umbrella. It's very easy to understand that the compound proposition here is that P=>Q. However, I just got a new one that I don't understand how to turn into a compound proposition because the first part has nothing to do with the second... The riddle goes:

There are three paths into a city, each guarded by a soldier. Be careful because they're all liars.

The first says: "The left-side path will take you to the city. Moreover, if the middle path takes you to the city, so will the right-side path"

The second says: "Neither the left-side path or the right-side path will let you into the city"

The third says: "The left-side path will take you to the city, the middle path will not."

So if there are propositions g,m,s and g is the proposition "the left-side path takes you to the city" (same for m and s). Then how do you write compound propositions for each of them?

I had a shot at the first: g^s=>m but then what do I do with this. If the centaur is lying does that mean (g^s=>m) is FALSE?

I don't know why they're asking us for truth tables and compound statements! Logic alone says it's the right-side path!

PS. So to get the right path, I just have to get a truth table where (c1^c2^c3) is true right? And then going back to the far left column where it's just g,s,m - one of them should be T and that's the right path?

$\endgroup$
2
$\begingroup$

I'm a little confused by your naming of the statements, so let's call $l$, $m$, $r$ the statements "The left/middle/right path will take you to the city". Then the first guard says $l \wedge (m\Rightarrow r)$, which is the same as $l\wedge ((\neg m)\vee r)$. The negation of this statement is $$\neg(l\wedge ((\neg m)\vee r)) = (\neg l)\vee \neg((\neg m)\vee r) = (\neg l)\vee (m \wedge \neg r).$$ The second guard's statement is $(\neg l) \wedge \neg r$, and the third's is $l \wedge \neg m$. You can take the negations of each of these and simplify.

To determine which is the right path, you can for example determine which of $l$, $m$, and $r$ makes all three (negated) statements true. To do so you could construct a truth table.

$\endgroup$
  • $\begingroup$ Thank you very much! May I ask, does the bracket you have around (¬l) have any significance? $\endgroup$ – hchenn Oct 14 '14 at 13:47
  • $\begingroup$ So I went and had a go with the truth table but I have three answers that are all true...Could you tell me what's wrong with the table linked here please? docs.google.com/document/d/… $\endgroup$ – hchenn Oct 14 '14 at 14:44
  • $\begingroup$ From those answers, I agree that you cannot tell which is the right path, unless you know that only one path is correct. If you do know that, then the answer must be path $s$, since in the other cases where you get $T$, either of two paths works. $\endgroup$ – rogerl Oct 14 '14 at 16:57
  • $\begingroup$ thanks for your help! I noticed I made an error in the wording of the question but it's been updated here. docs.google.com/document/d/… $\endgroup$ – hchenn Oct 15 '14 at 3:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.