Discrete mathematics: logic

salvatorekate95

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There are two types of creatures:
- truthers, who always tell the truth, and
- liars, who always lie.
It is not possible to distinguish them by appearance, but only by the truth or falsity
of the statements they make.

1) A stranger met two inhabitants, A and B, and asked A, "Is any of you a truther?". "If B is a liar, then I am a liar, too.", replied A.

What are A and B?

2) The stranger went on. In the evening, he began to look for a shelter for the night, but was very cautious because he knew that some of the inhabitants were maneaters and it was not possible to recognize them by appearance. Then he met three inhabitants, C, D and E. He asked C, "How many of you are truthers?".
"Flam flim", answered C in her language. "What did she say?", asked the stranger D. "Just one", replied D. "Do not trust D, he is a liar. Come with
me, I'm not a man-eater", said E. "No, come with me, I'm not a man-eater",
countered D.

What should the stranger do?


__________________________

1) I think that A and B are both truthers.

2) I'm absolutely lost. HELP.
 
1. What is B if A is a truth teller? What is B if A is a liar? Does it make any difference what A is?

2. If D lies how many are truth tellers and what are the consequences of that.
For example, assume D lies. Then the are either no truth tellers or both are truth tellers. But they can't both be truth tellers since we have assumed D lies. So what if there are no truth tellers? What did E say?
If D is a truth teller, what are the consequences of that.
 
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1. What is B if A is a truth teller? What is B if A is a liar? Does it make any difference what A is?

2. If D lies how many are truth tellers and what are the consequences of that.
For example, assume D lies. Then the are either no truth tellers or both are truth tellers. But they can't both be truth tellers since we have assumed D lies. So what if there are no truth tellers? What did E say?
If D is a truth teller, what are the consequences of that.


1. If A is truthteller then B is truthteller. If A is liar, then B is truthteller. So B is a truthteller and A is either truthteller or liar. Right?

2. If D lies then E is truthteller.
If D is a truthteller, then E is a liar. C's statement is correct, so there are two truthtellers and there cannot be two truthtellers.

So D lies and E is a truthteller. Right?
 
1. If A is truthteller then B is truthteller. If A is liar, then B is truthteller. So B is a truthteller and A is either truthteller or liar. Right?

2. If D lies then E is truthteller.
If D is a truthteller, then E is a liar. C's statement is correct, so there are two truthtellers and there cannot be two truthtellers.

So D lies and E is a truthteller. Right?

Almost. (1) is correct but lets look at (2) in more detail [actually it looks like you might have gotten turned around there but, just in case]:
Case (2a) D lies: Then there are either no truth tellers or both are truth tellers.
  1. Case (2a-i) No truth tellers: But E said D lies and that is the truth so Case (2a-i) can't be.
  2. Case (2a-ii) 2 truth tellers: But, for this case [Case (2a)], D lies so Case (2a-ii) can't be
  3. Since neither of the (exhaustive) cases [Case (2a-i) and Case (2a-ii)] can be true, D lies is not true and thus D is a truth teller (and since D said E was a liar, E is a liar)
Case (2b) D is a truthteller: Then there is only one truthteller and D is it (and E is a liar)
 
Almost. (1) is correct but lets look at (2) in more detail [actually it looks like you might have gotten turned around there but, just in case]:
Case (2a) D lies: Then there are either no truth tellers or both are truth tellers.
  1. Case (2a-i) No truth tellers: But E said D lies and that is the truth so Case (2a-i) can't be.
  2. Case (2a-ii) 2 truth tellers: But, for this case [Case (2a)], D lies so Case (2a-ii) can't be
  3. Since neither of the (exhaustive) cases [Case (2a-i) and Case (2a-ii)] can be true, D lies is not true and thus D is a truth teller (and since D said E was a liar, E is a liar)
Case (2b) D is a truthteller: Then there is only one truthteller and D is it (and E is a liar)


But if D is a truthteller, then it is true that there is 1 truthteller. If there is 1 truthteller as C said, then it makes C as a truthteller. So we have two truthtellers which cannot be true.

I don't understand why D is not liar. I'm very confused.

I'm very confused now.
 
But if D is a truthteller, then it is true that there is 1 truthteller. If there is 1 truthteller as C said, then it makes C as a truthteller. So we have two truthtellers which cannot be true.

I don't understand why D is not liar. I'm very confused.

I'm very confused now.

Yes, I can see your confusion. I had assumed C wasn't in the problem other than as someone who said something and you only needed to concentrate on D and E. If C is added into the problem (as it should be), the process becomes different:
Case 1: C is a truthteller (tt)
Case 1a: D is a tt - not possible because then there would only be one tt, not at least both C & D
Case 1b: D is a liar - Then there are 0 or 2 tt. Since C is a tt, there are two, C & E.

Case 2: C is a liar
Case 2a: D is a truthteller - Since C is a liar, there are 0 or 2 truthtellers, there must be at least 1 since D is a tt, so D & E are truthtellers
Case 2b: D is a liar - That C did not say there was only one tt means C said there were either 0 or 2 tt. But C lied so there is only 1 and D is not a liar. Not possible.


So we have either C & E are tt or D & E are tt. In both cases E is a tt, so go with E.
 
Yes, I can see your confusion. I had assumed C wasn't in the problem other than as someone who said something and you only needed to concentrate on D and E. If C is added into the problem (as it should be), the process becomes different:
Case 1: C is a truthteller (tt)
Case 1a: D is a tt - not possible because then there would only be one tt, not at least both C & D
Case 1b: D is a liar - Then there are 0 or 2 tt. Since C is a tt, there are two, C & E.

Case 2: C is a liar
Case 2a: D is a truthteller - Since C is a liar, there are 0 or 2 truthtellers, there must be at least 1 since D is a tt, so D & E are truthtellers
Case 2b: D is a liar - That C did not say there was only one tt means C said there were either 0 or 2 tt. But C lied so there is only 1 and D is not a liar. Not possible.


So we have either C & E are tt or D & E are tt. In both cases E is a tt, so go with E.


Thank you very much for your help. :)
 
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