# Did I do this CS logic problem right?

1. Sep 14, 2010

### rnj76543

Hi everybody, I just want to check to make sure I did this problem right. I know the answer I just need to make sure I did it formally.

1. The problem statement, all variables and given/known data
The problem goes like this: a logician is on an island with two groups of people truth-tellers and liars. He meets three individuals, A, B and C. A says something he can't understand, B says "A said he is a liar" and C says "B is telling a lie." Truth-tellers only tell the truth and liars only lie. I was asked to answer this question: Which category do B and C belong? Can you tell anything about the category A fits in?

My solution, is this correct?
Here ] = negation/not, -> is implication(if/then), and ^ is logical and.
Propositions I came up with:
A: I am a liar
B: A said he is a liar
C: B is the liar

I thought it out beforehand and C is telling the truth. If A said he is a liar and this is true then he is a truth-teller which means he isn't a liar(a contradiction) and hence B is wrong and therefore C is correct.

I formalized it has such:

( (A -> ]A) ^ (B -> (A ^ ]C)) ^ (C -> (]A ^ ]B))

I did a truth table and besides one impossible answer I got C to work out right and nothing else to be true. But is this the right formula afterall?

I do not want any answers I just want to know if I'm doing it right. And maybe if your feeling kind you can tell me where to look next if I'm doing it wrong.

Again, I know the answer I just need help formalizing it or rather checking my answer.

Thanks for all of your time!

Last edited: Sep 15, 2010
2. Sep 14, 2010

### xcvxcvvc

you posted a setup about truth tellers and liars but you never mentioned what you're trying to find or solve for

3. Sep 15, 2010

### rnj76543

Darn!!! That's what late nights and early mornings do to a person ha. I was asked to answer this question: Which category do B and C belong? Can you tell anything about the category A fits in?

4. Sep 15, 2010

### xcvxcvvc

for this scenario, no one can say, "I am a liar." if someone is a liar and says it, he told the truth, and if someone is a truth teller and says it, he told a lie. Therefore, B is lying. As such, C must be telling the truth. There is no way to say if A is a truthteller or a liar.

5. Sep 15, 2010

### rnj76543

Cool. I think I know how to fix it now. Thanks a bunch.