Liar Puzzle Reasoning
Let $A$, $B$, and $C$ represent the statements "A is a truth-teller", "B is a truth-teller", and "C is a truth-teller", respectively. Correspondingly, $¬A$, $¬B$, and $¬C$ represent "A is a liar", "B is a liar", and "C is a liar". We assume each person is either always a truth-teller or always a liar.
The given statements are:
- A claims: "$B$ is a liar and $C$ is a liar" (i.e., $¬B \land ¬C$)
- B claims: "$C$ is a liar" (i.e., $¬C$)
Analyzing A's Status
We test the possibilities for A:
- Case 1: Assume A is a truth-teller ($A$).
- If $A$ is true, A's statement ($¬B \land ¬C$) must be true. This means $B$ is a liar ($¬B$) and $C$ is a liar ($¬C$).
- Since $B$ is a liar ($¬B$), B's statement ($¬C$) must be false.
- If B's statement ($¬C$) is false, it implies $C$ is true (C is a truth-teller).
- This leads to a contradiction: $C$ must be a liar (from A's statement) and $C$ must be a truth-teller (from B's lie).
- Therefore, the assumption that A is a truth-teller is false.
- Conclusion on A: A must be a liar ($¬A$).
Analyzing B's and C's Status
Since A is a liar ($¬A$), A's statement ($¬B \land ¬C$) is false. The negation, $¬(¬B \land ¬C)$, must be true. By De Morgan's laws, this is equivalent to $B \lor C$, meaning at least one of B or C is a truth-teller.
Now, let's consider B's statement ($¬C$) based on whether B is a truth-teller or a liar:
- Possibility 1: B is a truth-teller ($B$).
- B's statement ($¬C$) is true. So, C is a liar ($¬C$).
- Current status: $¬A$, $B$, $¬C$. (A is liar, B is truth-teller, C is liar).
- This scenario is consistent: A is a liar, B tells the truth (about C being a liar), and C is a liar. There are exactly two liars (A and C).
- Possibility 2: B is a liar ($¬B$).
- B's statement ($¬C$) is false. So, C is a truth-teller ($C$).
- Current status: $¬A$, $¬B$, $C$. (A is liar, B is liar, C is truth-teller).
- This scenario is consistent: A is a liar, B lies (about C being a liar, which is true), and C is a truth-teller. There are exactly two liars (A and B).
Determining the False Statement
We have two possible valid scenarios:
- A is Liar, B is Truth-teller, C is Liar (Total 2 liars)
- A is Liar, B is Liar, C is Truth-teller (Total 2 liars)
Let's evaluate the given options based on these scenarios:
- Option 1: A is a liar ($¬A$)
- This is true in both scenarios.
- Option 2: Exactly one of B and C is a liar
- Scenario 1: C is a liar, B is not. (True)
- Scenario 2: B is a liar, C is not. (True)
- This statement is true in both scenarios.
- Option 3: Exactly one among A, B and C is a liar
- Scenario 1: A and C are liars (2 liars). (False)
- Scenario 2: A and B are liars (2 liars). (False)
- This statement is false in both scenarios.
- Option 4: Exactly two among A, B and C are liars
- Scenario 1: A and C are liars. (True)
- Scenario 2: A and B are liars. (True)
- This statement is true in both scenarios.
The question asks for the statement that is FALSE. Based on the analysis, the statement "Exactly one among A, B and C is a liar" is false.
Important Questions from Analytical Decision Making (Notes)
_________________ psychology is a branch in which the findings of psychology are applied in the field of education.
- Prof. Murthy likes to let her students choose who their partners will be; however, no pair of students may work together for more than seven class periods in a row. Alice and Bob have worked together for seven class periods in a row. Calvin and Denny have worked together for three class periods in a row. Calvin does not want to work with Alice. Who should be assigned to work with Bob?
- Of four agents Alpha, Beta, Gamma and Delta, three have to be sent together on a mission. If Alpha and Beta cannot go together, Beta and Gamma cannot go together and Gamma and Delta cannot go together, then which of the following holds?
- In an experiment one group of subjects was asked to estimate the product of $8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ in 5 seconds. The other group was asked to estimate the product of $1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8$ in the same time. The former group attained a larger estimate than the later group. The biasing effect involved in this case is because of
Match List-I with List-II and select the correct answer by choosing from the codes given below :
List – I
(Phenomenon)List – II
(Explanation)a. Thinking i. Process of choosing between two or more alternatives on the basis of information about them b. Reasoning ii. Processing information in various ways to move towards a desired goal c. Decision making iii. Mental activity through which we transform available information in order to reach conclusion d. Problem solving iv. An activity that involves the manipulation of mental representation of various features of the external world
