# riddles and compound proposition

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?

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.

• Thank you very much! May I ask, does the bracket you have around (¬l) have any significance? – hchenn Oct 14 '14 at 13:47
• 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/… – hchenn Oct 14 '14 at 14:44
• 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. – rogerl Oct 14 '14 at 16:57
• 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/… – hchenn Oct 15 '14 at 3:17