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Four friends have been identified as suspects for an unauthorized access into a computer system. They have made statements to the investigating authorities. Alice said“Carlos did it.” John said “I did not do it.” Carlos said“Diana did it.” Diana said “Carlos lied when he said that I did it.”

a) If the authorities also know that exactly one of the four suspects is telling the truth, who did it? Explain your reasoning.

b) If the authorities also know that exactly one is lying, who did it? Explain your reasoning.

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Assume, = telling the truth. = telling the lie. 

a) Condition: Exactly one is telling the truth. 

Case 1.a: Suppose, Alice is telling the truth. 

If Alice is T then John is also T. So, it's contradict with given condition. 

1.b: Suppose, Alice is telling the lie. 

If Alice is F then we can't say who's telling the truth. 

So, Case 1 is not valid. 

Case 2.a: Suppose, John is telling the truth. 

If John is T then we can't say about other. 

2.b: Suppose, John is telling the lie. 

If John is F then A is F then C is F then D is T. So, it's satisfy given condition. 

Hence, from Case 2.b we can conclude that if only Diana is telling the truth then John did it. 

b) Condition: Exactly one is lying. 

Case 1.a: Suppose, Alice is telling the truth. 

If Alice is T then John is T then Carlos is F then Diana is T. So, it's satisfy given condition. 

Hence, from Case 1.a we can conclude that if only Carlos is lying then Carlos did it. 

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a) If the authorities also know that exactly one of the four suspects is telling the truth, then Carlos did it. Alice and Diana are both pointing the finger at Carlos, and John is the only one who is not accusing anyone else. This means that John must be telling the truth when he says he didn't do it, so Carlos is the only one left who could have done it.

b) If the authorities also know that exactly one is lying, then Diana did it. Since Alice is the only one accusing someone other than herself, she is likely to be telling the truth. Since John is the only one not accusing anyone else, he is likely to be telling the truth as well. This leaves Carlos and Diana, and since

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