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In a certain country every inhabitant is either a truth teller (who always tells the truth) or a liar (who always lies).

Traveling in this country you meet two of the inhabitants, Pat and Mel. Pat says, “If I am a truth teller, then Mel is a truth teller.”

(a) Is Pat a truth teller or a liar?

(b) Is Mel a truth teller or a liar?

Provide mathematical justification for your answers.

Kenta
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5 Answers5

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An 'If.. Then' statement is false ONLY when first statement is true and the second statement is false.

Let Pat be lying. Then by the above, statement 1 is true. Which means he is a truth teller. A contradiction!!!

Thus, Pat is a truth teller. So, by the statements, statement 1 is true. This implies Statement 2 is true as well.

Thus, Both are truth tellers.

Win Vineeth
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Let $P$ = Pat is a truth teller and $M$ = Mel is a truth teller.

Either $P$ and $P \implies M$. or $\lnot P$ and $\lnot(P \implies M)$

$(P \land P \implies M) \lor (\lnot P \land (\lnot (P \implies M)))$

Now $(P \land P \implies M) \approx (P \land M)$

So $(P \land M) \lor (\lnot P \land (\lnot (P \implies M)))$.

Now $(P \implies M) \approx (M \lor \lnot P)$

So $\lnot(P \implies M) \approx \lnot(M \lor \lnot P)\approx (\lnot M \land \lnot \lnot P) \approx (\lnot M \land P)$

So $(P \land M) \lor (\lnot P \land (\lnot M \land P))$.

But $(\lnot P \land (\lnot M \land P)) \approx ((\lnot P \land P) \land M)\approx (FALSE \land M) \approx FALSE$

So $(P \land M) \lor FALSE$.

So $P \land M$.

So Pat and Mel are both truth tellers.

fleablood
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1

Let $A = \text{'Pat is a truth teller'},\; B = \text{'Mel is a truth teller'}$. Then the statement is $S = A \to B$, that's equal to $S = \neg A \vee B$.

  1. $A=1 \Rightarrow S = 1 \Rightarrow B = 1$
  2. $A=0\Rightarrow S = 0 \Rightarrow S = 1 \vee B = 1$. Contradiction. So the are both truth tellers
Denis Korzhenkov
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If Pat is a liar, then his implication is false. But $A\implies B$ is short-hand for $\neg A \vee B$, hence its negation is $A\wedge \neg B$. Which would mean that $A$ is Pat is a truth teller (and Mel a liar) but that contradicts the hypothesis.

Hence Pat is a truth teller and what he says is correct, hence also Mel is a truth teller.

Angelo Rendina
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It can be rephrased as: "I am not a truth teller OR Mell is a truth teller.

If Pat speaks the truth then he is a truth teller so the first part of the sentence is not true. Then the second part must be true. In that case both are truth telllers.

If Pat lies then the sentence is not true which means both parts of sentence are not true. But then Pat is supposed to be a truth teller and a contradiction is found.

Final conclusion: both are truth tellers.

drhab
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