GATE ME 2017 Set 2 — Question 64
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General Aptitude → Analytical Aptitude → Deduction & Induction
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Question
All people in a certain island are either 'Knights' or 'Knaves' and each person knows every other person's identity. Knights NEVER lie, and knaves ALWAYS lie.
P says "Both of us are knights". Q says "None of us are knaves".
Which one of the following can be logically inferred from the above?
P says "Both of us are knights". Q says "None of us are knaves".
Which one of the following can be logically inferred from the above?
Correct answer
(D) The identities of P, Q cannot be determined
Solution
Let's analyze the statements made by P and Q.P's statement: "Both of us are knights"
Q's statement: "None of us are knaves" (This is equivalent to "Both of us are knights")So, both P and Q are essentially making the same statement: "Both P and Q are knights."Let's consider two cases:Case 1: P is a Knight.
If P is a Knight, then P's statement must be true. Therefore, "Both P and Q are knights" is true. This implies that P is a Knight AND Q is a Knight.
Now, let's check if this is consistent with Q's nature:
If Q is a Knight, then Q's statement "Both P and Q are knights" must be true. This is consistent with our deduction that P is a Knight and Q is a Knight.
So, the scenario (P is Knight, Q is Knight) is logically consistent.Case 2: P is a Knave.
If P is a Knave, then P's statement must be false. Therefore, "Both P and Q are knights" is false. This implies that it is NOT true that both P and Q are knights. In other words, at least one of P or Q is a Knave.
Since we assumed P is a Knave, this condition is satisfied.
Now, let's consider Q's nature:
If Q were a Knight, then Q's statement "Both P and Q are knights" would have to be true. But this contradicts our assumption that P is a Knave. So, Q cannot be a Knight if P is a Knave.
Therefore, Q must be a Knave.
If Q is a Knave, then Q's statement "Both P and Q are knights" must be false. This is consistent with P being a Knave and Q being a Knave.
So, the scenario (P is Knave, Q is Knave) is logically consistent.Since both (P is Knight, Q is Knight) and (P is Knave, Q is Knave) are logically consistent possibilities, we cannot uniquely determine the identities of P and Q.The final answer is
Q's statement: "None of us are knaves" (This is equivalent to "Both of us are knights")So, both P and Q are essentially making the same statement: "Both P and Q are knights."Let's consider two cases:Case 1: P is a Knight.
If P is a Knight, then P's statement must be true. Therefore, "Both P and Q are knights" is true. This implies that P is a Knight AND Q is a Knight.
Now, let's check if this is consistent with Q's nature:
If Q is a Knight, then Q's statement "Both P and Q are knights" must be true. This is consistent with our deduction that P is a Knight and Q is a Knight.
So, the scenario (P is Knight, Q is Knight) is logically consistent.Case 2: P is a Knave.
If P is a Knave, then P's statement must be false. Therefore, "Both P and Q are knights" is false. This implies that it is NOT true that both P and Q are knights. In other words, at least one of P or Q is a Knave.
Since we assumed P is a Knave, this condition is satisfied.
Now, let's consider Q's nature:
If Q were a Knight, then Q's statement "Both P and Q are knights" would have to be true. But this contradicts our assumption that P is a Knave. So, Q cannot be a Knight if P is a Knave.
Therefore, Q must be a Knave.
If Q is a Knave, then Q's statement "Both P and Q are knights" must be false. This is consistent with P being a Knave and Q being a Knave.
So, the scenario (P is Knave, Q is Knave) is logically consistent.Since both (P is Knight, Q is Knight) and (P is Knave, Q is Knave) are logically consistent possibilities, we cannot uniquely determine the identities of P and Q.The final answer is
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