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Suppose there are signs on the doors to two rooms. The sign on the first door reads “In this room there is a lady, and in the other one there is a tiger”; and the sign on the second door reads “In one of these rooms, there is a lady, and in one of them there is a tiger.” Suppose that you know that one of these signs is true and the other is false.

Behind which door is the lady?

2 Answers

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   p: In the fisrt room there is a lady.
$\neg$p: In the fisrt room there is a tiger.
   q: In the second room there is a lady. 
$\neg$q: In the second room there is a tiger.

 

Translate given English statements into logical statement:

1. (p$\wedge\neg$q)   2. (p$\oplus$q) 

As, given in the question, one of the sign is true and other is false. So, there are two cases possible. 

Case 1: Suppose, First door's sign is True and Second door's is False

1. (p$\wedge\neg$q) is True only if p is True and $\neg$q is True. 

2. (p$\oplus$q) is False if both p & q are True Or False. but it's contradict with 1$^{st}$ statement. So, Case 1 is not valid. 

Case 2: Suppose, First door's sign is False and Second door's is True

1. (p$\wedge\neg$q) is False if atleast one of p & $\neg$q is False. 

2. (p$\oplus$q) is True if either p is True & q is False or p is False & q is True. So, there are two possible subcases. 

Subcase 1 : p is True and q is False. 

but it's contradict with 1$^{st}$ statement. So, it's not valid. 

Subcase 2: p is False and q is True

it's not contradict with 1$^{st}$ statement. So, it's a valid subcase.

Hence, from subcase 2 of Case 2, we can conclude that Behind the Second door is the lady. 

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The lady is behind the first door.

Since the sign on the first door is true, it means that the other sign must be false. This implies that the lady must be in the room behind the first door.

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