Binary Logic Questions for CAT with Answers PDF

In the given statement, Bryan said that he was a Gitty. No truth-teller or liar can make a statement that he is a Gitty (always a lie). Bryan must be an alternator, that is, a Pixie; and his first statement is a lie. Hence, his second statement is true; Avril is an Amora. Edward is a Gitty.

From the given statement, Patricia cannot be an Amora. He cannot be a Gitty, as if he is a Gitty, his second statement will be true, which is a contradiction. Hence, Patricia must be a Pixie and Mary is a Gitty. Helen is an Amora.

Case 1: Assuming Hamish to be Amora
Hamish:
I am an Amora. ---TRUE
Bruce is an Amora. --TRUE
This is a contradiction, as 2 people cannot be from the same community.
Case 2: Assuming Hamish to be Gitty
I am an Amora. ---FALSE
Bruce is an Amora. --FALSE
This means Bruce must be from Pixie, since Bruce is not from Amora and Hamish is already assumed to belong to Gitty.
But the question says that Bruce is not from Pixie. Hence, this case is also disregarded .
Case 3: Assuming Hamish to be Pixie
I am an Amora. ---FALSE
Bruce is an Amora. --TRUE
Thus, Bruce is an Amora, Hamish is Pixie, and Colin is Gitty.
Case 4: Assuming Hamish to be Pixie
I am an Amora. ---TRUE
Bruce is an Amora. --FALSE
This is not possible since then Hamish would again belong to both Pixie and Amora.
Thus, only Case 3 is valid.

(i) Assuming that Thief-1 always speaks the truth:

(ii) Assuming that Thief-2 always speaks the truth:

(iii) Assuming that Thief-3 always speaks the truth:

From case (i), we can conclude that Thief-3 stole Antique Gun.

The only posisble scenario is :
Case 2:
Thief-1:
I stole Bronze Statue. TRUE
Thief-2 stole Rare Painting. TRUE
Thief-2:
I stole Bronze Statue. FALSE
Thief-3 stole Antique Gun. TRUE
Thief-3:
Thief-1 stole Rare Painting. FALSE
Thief-2 stole Bronze Statue. FALSE
Hence, ans is 3.

(i) Assuming that Thief-1 always speaks the truth:

(ii) Assuming that Thief-2 always speaks the truth:

(iii) Assuming that Thief-3 always speaks the truth:

'Thief-3 did not steal Antique Gun' must be a false statement.

It is given that every player spoke exactly two false statements, hence one true statement. Let the first statement of Stanley be true. Hence, the other two must be false. Therefore, Stanley is not playing for Team-A. Owen is not a Mid-fielder and Geoffrey is not playing for Team-C.
Owen's second statement is therefore true. Hence, his first and third statements must be false. Hence, Owen is not a Defender and Stanley is not a Mid-fielder. Owen has to be a Goalkeeper. Stanley should be a Defender. Geoffrey must be a Mid-fielder.
Hence, Geoffrey's last statement is true. But Geoffrey's first statement is also true. Since this is not possible, Stanley's first statement cannot be true.
Let the second statement of Stanley be true. Therefore, Owen is a Mid-fielder. From his other two statements, we can say that Stanley is from Team-A and Geoffrey is from Team-B. Hence, Owen is from Team-C. Owen's second statement is true. His first and third statements are false. Geoffrey's second statement is true. His first and third statements must be false. Therefore, Stanley must be a Goalkeeper and Geoffrey must be a Defender. This is one possible case.
Let Stanley's third statement be true. Geoffrey must be from Team-C. From his first statement, we get that Stanley must be from Team-A. Therefore, Owen must be from Team-B. Owen is not a Mid-fielder.
Owen's second statement and Geoffrey's second statement are false. The only case possible in which both Owen and Geoffrey tell one true statement each is when Owen's first statement and Geoffrey's third statement are true. Hence, Owen is a Defender, Geoffrey is a Mid-fielder and Stanley is a Goalkeeper. This is another possible case.
The possible cases are presented in the following table:

Given that every player spoke exactly two false statements, hence they all spoke one true statement. Let the first statement of Stanley be true. Hence, the other two must be false. Therefore, Stanley is not playing for Team-A. Owen is not a mid-fielder and Geoffrey is not playing for Team-C.
Owen's second statement is, therefore, true. Hence, his first and third statements must be false, which means that Owen is not a defender and Stanley is not a mid-fielder. Owen has to be a goalkeeper. Stanley should be a defender. Geoffrey must be a mid-fielder.
Hence, Geoffrey's last statement is true. But Geoffrey's first statement is also true. Since this is not possible, Stanley's first statement cannot be true.
Let the second statement of Stanley be true. Therefore. Owen is a mid-fielder. From his other two statements, we can say that Stanley is from Team-A and Geoffrey is from the Team-B. Hence, Owen is from the Team-C. Owen's second statement is true. His first and third statements are false. Geoffrey's second statement is true. His first and third statements must be false. Therefore, Stanley must be a goalkeeper and Geoffrey must be a defender. This is one possible case.
Let Stanley's third statement be true. Geoffrey must be from Team-C. According to his first statement, Stanley must be from Team-A. Therefore, Owen must be from Team-B. Owen is not a mid-fielder.
Owen's second statement and Geoffrey's second statement are false. The only case possible in which both Owen and Geoffrey tell one true statement each is when Owen's first statement and Geoffrey's third statement are true. Hence, Owen is a defender, Geoffrey is a mid-fielder and Stanley is a goalkeeper. This is another possible case.

Given that every player spoke exactly two false statements, hence they all spoke one true statement. Let the first statement of Stanley be true. Hence, the other two must be false. Therefore, Stanley is not playing for Team-A. Owen is not a mid-fielder and Geoffrey is not playing for Team-C.
Owen's second statement is, therefore, true. Hence, his first and third statements must be false, which means that Owen is not a defender and Stanley is not a mid-fielder. Owen has to be a goalkeeper. Stanley should be a defender. Geoffrey must be a mid-fielder.
Hence, Geoffrey's last statement is true. But Geoffrey's first statement is also true. Since this is not possible, Stanley's first statement cannot be true.
Let the second statement of Stanley be true. Therefore. Owen is a mid-fielder. From his other two statements, we can say that Stanley is from Team-A and Geoffrey is from the Team-B. Hence, Owen is from the Team-C. Owen's second statement is true. His first and third statements are false. Geoffrey's second statement is true. His first and third statements must be false. Therefore, Stanley must be a goalkeeper and Geoffrey must be a defender. This is one possible case.
Let Stanley's third statement be true. Geoffrey must be from Team-C. According to his first statement, Stanley must be from Team-A. Therefore, Owen must be from Team-B. Owen is not a mid-fielder.
Owen's second statement and Geoffrey's second statement are false. The only case possible in which both Owen and Geoffrey tell one true statement each is when Owen's first statement and Geoffrey's third statement are true. Hence, Owen is a defender, Geoffrey is a mid-fielder and Stanley is a goalkeeper. This is another possible case.
The possible cases are presented in the following table:

Hence, Owen can be from Team-C or Team-B, which means that it cannot be determined.

Given that every player spoke exactly two false statements, hence they all spoke one true statement. Let the first statement of Stanley be true. Hence, the other two must be false. Therefore, Stanley is not playing for Team-A. Owen is not a mid-fielder and Geoffrey is not playing for Team-C.
Owen's second statement is, therefore, true. Hence, his first and third statements must be false, which means that Owen is not a defender and Stanley is not a mid-fielder. Owen has to be a goalkeeper. Stanley should be a defender. Geoffrey must be a mid-fielder.
Hence, Geoffrey's last statement is true. But Geoffrey's first statement is also true. Since this is not possible, Stanley's first statement cannot be true.
Let the second statement of Stanley be true. Therefore. Owen is a mid-fielder. From his other two statements, we can say that Stanley is from Team-A and Geoffrey is from the Team-B. Hence, Owen is from the Team-C. Owen's second statement is true. His first and third statements are false. Geoffrey's second statement is true. His first and third statements must be false. Therefore, Stanley must be a goalkeeper and Geoffrey must be a defender. This is one possible case.
Let Stanley's third statement be true. Geoffrey must be from Team-C. According to his first statement, Stanley must be from Team-A. Therefore, Owen must be from Team-B. Owen is not a mid-fielder.
Owen's second statement and Geoffrey's second statement are false. The only case possible in which both Owen and Geoffrey tell one true statement each is when Owen's first statement and Geoffrey's third statement are true. Hence, Owen is a defender, Geoffrey is a mid-fielder and Stanley is a goalkeeper. This is another possible case.
The possible cases are presented in the following table:

The name of the goalkeeper is Stanley.

Given that every player spoke exactly two false statements, hence they all spoke one true statement. Let the first statement of Stanley be true. Hence, the other two must be false. Therefore, Stanley is not playing for Team-A. Owen is not a mid-fielder and Geoffrey is not playing for Team-C.
Owen's second statement is, therefore, true. Hence, his first and third statements must be false, which means that Owen is not a defender and Stanley is not a mid-fielder. Owen has to be a goalkeeper. Stanley should be a defender. Geoffrey must be a mid-fielder.
Hence, Geoffrey's last statement is true. But Geoffrey's first statement is also true. Since this is not possible, Stanley's first statement cannot be true.
Let the second statement of Stanley be true. Therefore. Owen is a mid-fielder. From his other two statements, we can say that Stanley is from Team-A and Geoffrey is from the Team-B. Hence, Owen is from the Team-C. Owen's second statement is true. His first and third statements are false. Geoffrey's second statement is true. His first and third statements must be false. Therefore, Stanley must be a goalkeeper and Geoffrey must be a defender. This is one possible case.
Let Stanley's third statement be true. Geoffrey must be from Team-C. According to his first statement, Stanley must be from Team-A. Therefore, Owen must be from Team-B. Owen is not a mid-fielder.
Owen's second statement and Geoffrey's second statement are false. The only case possible in which both Owen and Geoffrey tell one true statement each is when Owen's first statement and Geoffrey's third statement are true. Hence, Owen is a defender, Geoffrey is a mid-fielder and Stanley is a goalkeeper. This is another possible case.
The possible cases are presented in the following table:

Hence, all are true. 

After going through all statements made by Graeme, Harry, and Mathew, one can make out that Mathew's second statement is either a true statement of a truth-teller or a false statement of an alternator, irrespective of which Mathew's first and third statements have to be true. This gives us the following result:

Based on above, we can determine that Graeme's first statement is false and second one is true. As the three of them have to be either a truth-teller, a liar, or an alternator, we can conclude that Graeme is an alternator and her third statement is hence false (which means Mathew is not an alternator). This leads us to the conclusion that Mathew is a truth-teller. Thus, Mathew's second statement can also be marked as true.
Hereafter, when we analyse Harry's statements, we realise that they all are false; so Harry must be a liar. With 'T' standing for true and 'F' for false, the nature of the statements will be as follows:

Hence, Harry supports Barcelona.

After going through all statements made by Graeme, Harry, and Mathew, one can make out that Mathew's second statement is either a true statement of a truth-teller or a false statement of an alternator, irrespective of which Mathew's first and third statements have to be true. This gives us the following result:

Based on above, we can determine that Graeme's first statement is false and second one is true. As the three of them have to be either a truth-teller, a liar, or an alternator, we can conclude that Graeme is an alternator and her third statement is hence false (which means Mathew is not an alternator). This leads us to the conclusion that Mathew is a truth-teller. Thus, Mathew's second statement can also be marked as true.
Hereafter, when we analyse Harry's statements, we realise that they all are false; so Harry must be a liar. With 'T' standing for true and 'F' for false, the nature of the statements will be as follows:

Harry is a liar.

After going through all statements made by Graeme, Harry, and Mathew, one can make out that Mathew's second statement is either a true statement of a truth-teller or a false statement of an alternator, irrespective of which Mathew's first and third statements have to be true. This gives us the following result:

Based on above, we can determine that Graeme's first statement is false and second one is true. As the three of them have to be either a truth-teller, a liar, or an alternator, we can conclude that Graeme is an alternator and her third statement is hence false (which means Mathew is not an alternator). This leads us to the conclusion that Mathew is a truth-teller. Thus, Mathew's second statement can also be marked as true.
Hereafter, when we analyse Harry's statements, we realise that they all are false; so Harry must be a liar. With 'T' standing for true and 'F' for false, the nature of the statements will be as follows:
All the above statements are false.

After going through all statements made by Graeme, Harry, and Mathew, one can make out that Mathew's second statement is either a true statement of a truth-teller or a false statement of an alternator, irrespective of which Mathew's first and third statements have to be true. This gives us the following result:

Based on above, we can determine that Graeme's first statement is false and second one is true. As the three of them have to be either a truth-teller, a liar, or an alternator, we can conclude that Graeme is an alternator and her third statement is hence false (which means Mathew is not an alternator). This leads us to the conclusion that Mathew is a truth-teller. Thus, Mathew's second statement can also be marked as true.
Hereafter, when we analyse Harry's statements, we realise that they all are false; so Harry must be a liar. With 'T' standing for true and 'F' for false, the nature of the statements will be as follows:

Hence, Mathew supports Arsenal.