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In a room there are only two types of people, namely $\text{Type 1}$ and $\text{Type 2}$. $\text{Type 1}$ people always tell the truth and $\text{Type 2}$ people always lie. You give a fair coin to a person in that room, without knowing which type he is from and tell him to toss it and hide the result from you till you ask for it. Upon asking the person replies the following

"The result of the toss is head if and only if I am telling the truth"

Which of the following options is correct?

  1. The result is head
  2. The result is tail
  3. If the person is of $\text{Type 2}$, then the result is tail
  4. If the person is of $\text{Type 1}$, then the result is tail
in Mathematical Logic
edited by
6.6k views
26
Actually, options B, C and D can be eliminated easily.
1. If option B is correct then either option (C) or option (D) should also be correct.
2. If option (C) is correct then option (B) is also correct.
3. If option (D) is correct then Option (B) is also correct.

But, more than one option is not correct, hence eliminate options B, C and D.
0

@Manu Thakur sir 👏. Take a bow.

0

Can we answer it in the following way

Type1 - always tells truth

Type2 - always tells false

We need to find $P\left(\dfrac{Head}{Truth}\right)$. So using Baye's Theorem

$P\left(\dfrac{Head}{Truth}\right)  = \dfrac{ P\left( \dfrac{Truth}{Head}\right)}{P\left( \dfrac{Truth}{Head}\right) + P\left( \dfrac{Truth}{\overline{Head}}\right)}$

                          $   =  \dfrac{ P\left( \dfrac{Truth}{Head}\right)}{P\left( \dfrac{Truth}{Head}\right) + P\left( \dfrac{Truth}{Tail}\right)}$

So, only Type1 person always tells truth when Head comes. Hence the result must be Head

Please check if I am making any mistake.

0

wow @Manu Thakur sir what a great Logic it is ( specially when time is less ).

2

Type 1: Truth

Type 2 : false

The result of the toss is head $\leftrightarrow$ I am telling the truth


  • T$\leftrightarrow$T $\equiv$ T

Truth spoken then surely the result is head

  • F$\leftrightarrow$F $\equiv$ T

False spoken then the result is tail

(But think when we are getting a tail, when the person is lying it means actually the result is head only )

1

If the person is of Type 1, then

"The result of the toss is head if and only if I am telling the truth"

this sentence is true.

And, he's telling the truth. So, head.


If the person is of Type 2, then

"The result of the toss is head if and only if I am telling the truth"

this sentence is false.

 

"The result of the toss is head if and only if I am lying" would be true.

Since the given person is lying; head.


Option A

0
superb explanation
1
Assume Type 1 persom is K(always Telling truth), Type 2 person is N(Always lying).

Statement is $H \leftrightarrow P(K)$ (where H means Head, P(K) means person is K i.e. telling truth )

Case 1 : Assume the person is K.

then $H \leftrightarrow P(K)$ is true, and since P(K) is true in this case, so H is also true. So, Head in this case.

Case 2 : Assume the person is N.

then $H \leftrightarrow P(K)$ is False, and since P(K) is false in this case, so H is true, so, Head in this case also.

Hence, answer A.

9 Answers

104 votes
 
Best answer

We do not know the type of the person from whom those words are coming from and so can have two cases :

  1. Truth-teller : definitely implies that result of toss is Head.
  2. Liar : the reality will be the negation of the statement.

The negation of $(x\iff y)$ is exactly one of $x$ or $y$ holds. So, we negate the statement : "The result of the toss is head if and only if I am telling the truth". This give rise to two possibilities

  • it is head and lie spoken
  • it is not head and truth spoken

Clearly the second one cannot be true because it cannot be a reality that the liar speaks the truth.

So, this implies that even if we negate the statement to see the reality or don't do that; The reality is that the toss yielded a Head.

Answer = option (A).


edited by
7

correct and perfect explanation among all solutions available here..........

2
nice explanation.
2

Thanks a lot.Realy nice explanation.

1
The "If I'm telling the truth" part is confusing me.

Suppose we have liar, and the result is tails. Then also he can say the same statement.
The result is head iff he's telling the truth, but he isn't telling it, so the result is tails.

Please clarify.
2

The negation of (x⇔y) is (x⇔¬y) so, we negate the statement : "The result of the toss is head if and only if I am telling the truth" as "The result of the toss is head if and only if I am not telling the truth" or  "The result of the toss is head if and only if I am telling lie". 

Now we can interpret from Type2 as "The result of the toss is head if and only if I am telling lie" and he is lying so the result of toss is head...

0

Thanks a lot.Realy nice explanation.

0
Very well explained thankyou
48 votes
Person 1 (truth teller), result is head. No doubt here as he is a truth teller. 
Person 2(lie teller). result is head if and only if he is telling truth. He is telling lies. So, the truth is the opposite of his statement. We can analyze his statement as two
  1. If I'm telling the truth result is head 
  2. If result is head I'm telling the truth

Both these are of the form $A \rightarrow B = \neg A \vee B$. Now, the truth will be the negation of these statements which will be $A \wedge \neg B$ which can be expressed as

  1. I'm telling the truth and result is not head
  2. Result is head and I'm telling false

Both of these means, the result is head as the person is lie teller. (Actually even if iff is replaced with if, answer would be A)

So, option A.

2

Arjun sir kidnly explain how u conclude ?But he is a liar and hence his statement is false, which means, result is head if and only if he is lying, and since he is lying result is head. bcz if lier then how can half part can be true??

1
Is it clear now?
1
Thank you sir :)
8

I solve this question as

S1:"The result of the toss is head if and only if I am telling the truth"

Let person is a truth teller.so the statement is also true.

so result is head.

Let person is telling lies.so statement itself is false.

so result is head 

2
@arjun sir in Person 2 part, there needs some editing. After negating his statements, point 2 will be:

 "Result is head and  i am telling false".
2
Thanks for that :)
1
@Arjun SIr...After negating the statements, the first one becomes result is "not" head(1.), Then how can we conclude Result is head?
0
The second person is lie teller.
0

@Arjun Can you please tell me if the below approach is correct?

Let h: result of the toss is head

t: the person is telling truth

h$\Leftrightarrow$t = h't' + ht

Type 1: t is true, so h't' + ht = h

Type 2: t is false, so h't' + ht = h'

Since Type 1 says truth always the answer will be h = head.

Also as Type 2 lies always the answer will not be h' = not not head = head.

14 votes

A. The result is head
p : "The result of the toss is head"
q : "I am telling the truth"
    p <-> q
    
Type 1:-
    Pq + p'q' = True
Head is answer when type 1.

Type 2:-
    p <-> q = False
Note: Do not take negation on both side i.e. ~(p<->q) = True is logically incorrect because RHS part is true means Type 2 person is saying True.

    please check the truth table for Ex-NOr
                 P    Ex-NOr q      Result
    case1    F                T        F
    case2    T                F        F

    There are only 2 case when exNor becomes false
    to p <-> q become False this 2 cases both need to be false.
Case1    when ~p then q = F
                i.e. "when result is Tail then I am telling Truth" this is False statement
                i.e. "the result is Tail then I am saying Lie"
                i.e. "The result is Head"

Case2    when p then ~q = F  
                 i.e. "when result is Head then I am telling Lie" this is False statement
                 i.e."when result is Head then I am telling Truth.
                 i.e. "the result is Head"
Head is answer when Type 2.


edited by
0

Thank you, Best Answer according to me.

your answers this part is really awesome...

Type 2:- 
    p <-> q = False 
Note: Do not take negation on both side i.e. ~(p<->q) = True is logically incorrect because RHS part is true means Type 2 person is saying True. 

    please check the truth table for Ex-NOr 
                 P    Ex-NOr q      Result 
    case1    F                T        F 
    case2    T                F        F 

0
13 votes

P:The result of the toss is head

Q:The person is telling the truth

Statement by the person:P$\Leftrightarrow$Q

Truth table for P$\Leftrightarrow$Q :

P Q P$\Leftrightarrow$Q
T T T
T F F
F T F
F F T

Since the statement is true for either both P and Q are true or both are false,so there are two possibilities:

1.The person tells truth that the result is head.

2.The person tells lie that the result is not head i.e the result is tail,

Both possibilities shows the result is head.

So, the answer is A.

0
in your table 3rd row also should be true.. becoz in biconditional only P Q = F  rest of all are true
0
Awsm explaination
10 votes

The question involves understanding the bi-implication completely.

The key statement is

The result of the toss is head if and only if I am telling the truth.

Now let us denote two propositional variables p and q where

p: "The result of the toss is head"

q: "I am telling the truth"

and hence the statement is $p\Leftrightarrow q$

Now we have two cases

Case 1: The person is of Type 1: Here it is very simple case.Since Type 1 person always tells the truth and hence according to him the result of the toss is Head.

Case 2: If the person if Type 2, then he always tells lies. Means, the statement $p \Leftrightarrow q$ given by this person is actually the negation of the original statement which he means 

i.e. $\sim (p \leftrightarrow q) = (p \leftrightarrow \sim q)$

so $p \leftrightarrow \sim q$ is The result of the toss is head if and only if I am not telling the truth.

And, yes this bi-implication is true for this Type 2 person because he never tells truth. So, here in this case also the result is Head.

And hence, the result Head is consistent irrespective of the type of person we choose.

Answer (A)

4 votes
whatever Type 2 person says should be false.

So if Result is head, then the statement "The result of the toss is head if and only if I am telling the truth" is false is consistent with lying.

So i think answer is A

if a person is Type 2 we know he lying -> he is not telling truth -> Result is head
0
the answer is c
3

We know that

1)type 2 person always says false statement

which implies "The result of the toss is head if and only if I am telling the truth" is false

The statement "The result of the toss is head if and only if I am telling the truth"

becomes false if result is head and type2 person is not telling the truth.

This is negation of P->Q where P=Result is head and Q=I am(type2 person) telling the truth.

so !(P->Q) = P^!Q which is result is head and i am telling false.

4 votes

"The result of the toss is head if and only if I am telling the truth"

say type1 people telling truth

putting negation

The result of toss is not head if and only if I am not telling truth // now is this statement false ? No it is true . right?

So, result must head as Type1 people is telling truth


Now, type2 people is telling lie

but he also telling "The result of the toss is head if and only if I am telling the truth"

Now, our question is how the same statement could be FALSE?


here is the logic

The result of the toss is head = P

I am telling the truth= Q


According to lie teller P =TRUE (means the result is true)

Q=FALSE (as FALSE teller must speaking FALSE )

Now according to the statement P⟷Q = (P⟶Q) ⋀(Q⟶P).....................(i)

Now, for being it false either P⟶ Q = FALSE or Q⟶P is FALSE. right?

Now, P⟶Q means (~P ⋁Q) i.e. The result of the toss is not head or I am telling truth = False ⋁ False = FALSE // here Q is  false as he is a lie teller

Q⟶P means(~Q⋁P) i.e. I am telling FALSE or the result of the toss is head = TRUE⋁TRUE = TRUE

Now, putting it in equation (i) we get  FALSE⋀TRUE =FALSE

So, the lier also telling same statement , but he is lying.

So,  the result is head :)


edited by
0 votes

"head if and only if he is telling the truth

h: head ; tr: telling the truth

$(\neg tr \lor h)\wedge(tr \lor \neg h)$

negation oof the above will give $( tr \wedge \neg h)\lor(\neg tr \wedge h)$ qhwich means :

  1. Either it is not head and he is telling the truth
  2. Either it is a head and he is not telling truth

The 1st one is not true as liar cannot tell the truth so 2nd one is valid.

so, answer is (A)

Answer:

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