IIITH-PGEE 2016

Question

An anthropologist is visiting the island of knights and knaves one after the other.. This particular island is a very peculiar place, for it has only two kinds of inhabitants, namely, knights and knaves! Now, knights always tell the truth and knaves always lie! There are cluster of island on which one is named ' Maya'

Question 1
On first island, the anthropologist encountered two natives, who made these statements.

A: B is a knight, and this is the island of Maya.
B: A is a knave, and this is the island of Maya.

Is this the island of Maya? What are A and B ?

Answer 1

A            B

T             T

T             F

F             T

F             F

Case 1:

T          T   -  then both are telling truth.

A: B is a knight, and this is the island of Maya. -  True

B: A is a knave, and this is the island of Maya.  -  True

But both cannot be true .

Because if B is knight , B is telling truth 

then A is knave - True

this is the island Maya - True

then A must be telling False

And

if "this is the island of Maya." - True

then "B is a knight" - False

So, there is a contradiction So, both cannot say true statement

Now, Case 2 : 

T    F

A: B is a knight, and this is the island of Maya. - True

B: A is a knave, and this is the island of Maya. - False

But A is true and A telling B is Knight So,B's statement cannot be false.

So, here is the contradiction .

A-True and B- False not possible

Case 3:

F    T

A: B is a knight, and this is the island of Maya. - False

B: A is a knave, and this is the island of Maya. - True

B is a knight - True

this is the island of Maya -True

But if A is Knave he cannot tell true.

Here is the contradiction

Case 4:

F      F

A: B is a knight, and this is the island of Maya.- False
B: A is a knave, and this is the island of Maya.- False

if both are telling False then both are Knave ,i.e. obvious

So, in B's statement "A is knave" is true

"this is the island of Maya" -False

means we can say this island is not Maya

A's statement both statement are False

"B is a knight" -False

" this is the island of Maya." -False

False ^ False = False

So, finally A is knave , B is knave , this is not island Maya

Comments

According  to case 3 can nt be the solution is B knight and A knave and this is not island of Maya plzz correct if I am wrong

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Can someone check the case where A is Knave, B is knight and the island is Maya. I think this is also a correct case.

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Let Q - This is the island of Maya. 

Now 

A : "B is telling the truth AND Q".

B : "A is telling the lie AND Q".

Now, if we assume Q is true, i.e. "This is the island of Maya", then the problem reduces to

A : "B is telling truth AND True".

B : "A is telling lie AND True". 

(since $P \wedge T = P$)

which leads to a variation of liars' paradox. 

So, now let's assume Q is false, i.e. "This is NOT the island of Maya". Now, the problem reduces to,

A : "B is telling the truth AND False".

B : "A is telling the lie AND False". 

Anything ANDed with False is False, i.e. $P \wedge F = F$, irrespective of what $P$ is. 

So, the problem now reduces to, 

A:"False"

B: "False"

So, by definition, since both are telling lies, there's no liar's paradox, and we assumed Q to be false, therefore, 

A, B are knaves, and "This is NOT the island of Maya."

 

@commenter commenter

 

 

Answer 2

check for all the possibilities of A, B and Maya Island.

A	B	Maya
0	0	0
0	0	1
0	1	0
0	1	1
1	0	0
1	0	1
1	1	0
1	1	1

I've considered 0 as knave and 1 as knight.

For a Knave either of any part can be false but for knight both must be true.

we'll check all 8 conditions.

1. Both knave and this is not island of maya.

A said: B is knight and this is island of maya. (no contradction as both statements are wrong).
B said: A is knave and this is island of maya. (no contradiction here also as "This is island of maya" statement is wrong").

similarly we can check for all other possibilities which will hardly take 1 minute and come to conclude that Both are knave and this isn't island of maya.
 

Comments

You are taking the statements individually, but they must be taken separately- i.e., only the whole statement is false in case of knave, one part can be true.

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previous was what i did in paper and marked none of these... 
edited my answer now.

Answer 3

A: B is a knight, and this is the island of Maya

B: A is a knave, and this is the island of Maya.

Say, B is a knight=P

this is the island of Maya=Q

A is a knave=R

A: B is a knight, and this is the island of Maya.

Say A is telling truth

Now according to A: (P⋀Q)

or alternatively we can also say ~(P⋀Q)= ~P⋁~Q=P⟶~Q

means if B is a knight, then this is not the island of Maya.

So, According to A : B is naive and island is Maya

then definitely we can say B is telling false

B: A is a knave, and this is the island of Maya.

here atleast one statement to be false for the statement to be false

means say A is a knave = False

this is the island of Maya= True

then False ⋀True=False . right?

According to statement A island is Maya =true

then A is a knave =False

So,A is knight

Now, we can conclude

A=Knight

B=Knave

island is Maya

Similarly can do cross verification

Comments

@Arjun sir I think ur answer is correct. Plz write a detail solution for it

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@srestha

if u will come , I shall go there ?// positive sentence

if u will not come , I shall not go there // negation is there but it is same statement as positive

These two are not equivalent statements. The second one should be

"If I don't go there, then you haven't come". 

$(a \implies b) = (\neg a \vee b) = (\neg b \implies \neg a).$

 

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yes, yes I know there is a miss from me , which I was confused about. Thank u sir. I will try again . But ur logic  is too good. I am trying to understand it first correctly