As Captain of the USS Enterprise, your mission is to travel at warp speed to the planet Zorg to rescue some Vulcans whose vessel recently crashed. But beware – Romulan spies may be trying to infiltrate your ship! When you arrive, you find four aliens named Aravik, Balev, Chu’lak, and Denak. They offer up the following statements:
• Aravik: Chu’lak is a Vulcan or Denak is a Romulan.
• Balev: Aravik and Chu’lak are Vulcans.
• Chu’lak: Denak and I are Vulcans.
• Denak: Chu’lak is a Romulan.
Knowing as you do that Vulcans always tell the truth and Romulans always lie which of the aliens, if any, do you rescue and which do you leave stranded on the planet? In your answer, clearly demonstrate that there is only one possible solution by enumerating all possible combinations of Romulans and Vulcans. Explain why each combination except one cannot be valid either because a Vulcan is lying or a Romulan is telling the truth.
Answer:
Rescue Denak as it were.
Clarification:
We'll characterize the accompanying propositions, just to make notation simpler:
A: Aravik is a Vulcan.
B: Balev is a Vulcan.
C: Chu'lak is a Vulcan.
D: Denak is a Vulcan.
The statements given by the four people can't be trusted. Be that as it may, every statement is true if and just if the one giving it is a Vulcan. This permits us to outline the proof as follows:
A⇔(C∨¬D)
B⇔(A∧C)
C⇔(D∧C)
D⇔¬C
Denak's statement permits us to substitute ¬C into the framework any place D happens since the two propositions are equivalent.
A⇔(C∨¬¬C)
B⇔(A∧C)
C⇔(¬C∧C)
The suggestion ¬C∧C is contradictory and along these lines bogus. Since C⇔(¬C∧C), the recommendation that Chu'lak is a Vulcan is equivalent to a contradiction. In this way, Chu'lak is a Romulan!
We can rearrange Aravik's adjusted statement:
A⇔(C∨¬¬C)
A⇔(C∨C)
A⇔C
We reason that Aravik is a Vulcan if and just if Chu'lak is a Vulcan. Since we know Chu'lak is a Romulan, so is Aravik.
We can likewise determine what Balev is on the grounds that we have
B⇔(A∧C)
Since Aravik and Chu'lak are not the two Vulcans, nor is Balev.
At long last, the previous statement
D⇔¬C
shows that Denak is a Vulcan. The just one of the bundle.
Truth Table:
The prior was a propositional-rationale verification of the sort a Vulcan would deliver. People are progressively imprudent and prone to attempt each blend of truth and misrepresentation.
On the left, we'll list every single imaginable blend of Vulcanhood for our suspects. On the right, we'll list whether every individual's statement is true right now.
Vulcan? Truthful?
| A | B | C | D | A | B | C | D |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 |
| 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 |
| 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 |
| 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 |
| 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 |
Because of the trustworthiness of Vulcans and the deception of Romulans, the main conceivable outcomes compare to the columns where the sections on the left are equivalent to on the right. There is just one of them, the situation where Denak is the main Vulcan.
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