in Mathematical Logic
2,403 views
0 votes
0 votes

The police have three suspects for the murder of Mr. Cooper: Mr. Smith, Mr Jones, Mr. Williams. Smith Jones, and Williams each declare that they did not kill Cooper. Smith also states that Cooper was friend of Jones and that Williams disliked him. Jones also states that he did not know Cooper and that he was out of town the day Cooper was killed. Williams also states that he saw both Smith and Jones with Cooper the day of the killing and that either Smith or Jones must have killed him. Can you determine who the murderer was if

  1. one of the three men is guilt, the two innocent men are telling the truth, but the statements of the guilty man may or may not b true?
  2. innocent men do not lie?
in Mathematical Logic
2.4k views

1 Answer

2 votes
2 votes

$a)$ There are $3$ possible cases :-

Case $1)$ When Williams is guilty but Jones and Smith are innocent

It means Jones and Smith are telling the truth but what William is saying may be true or may be false. Now, statements from Jones and Smith are contradicting because Smith says Cooper was a friend of Jones but Jones says he did not know Cooper. So, It is not a possible case.

Case $2)$ When Smith is guilty but Jones and Williams are innocent

It means Jones and Williams are telling the truth but what Smith is saying may be true or may be false. Now, statements from Jones and Williams are contradicting because Jones says he was out of town when Cooper was killed and William says he saw Jones with Cooper. So, It is also not a possible case.

Case $3)$ When Jones is guilty but Smith and Williams are innocent

It means Smith and Williams are telling the truth but what Jones is saying may be true or may be false. Here, nothing is contradicting from the statements of Smith and Williams. So, It is a possible case. It means Jones is guilty.

So, Answer :- Jones was the murderer.

$(b)$ Here, Total $8$ cases are possible as :-

Let, J = Jones, S = Smith , W = Williams

  and I = Innocent, G = Guilty

J S W
I I I
I I G
I G I
I G G
G I I
G I G
G G I
G G G

Since, Innocent men always tell the truth. Now, According case $(a)$ , when both Jones and Smith are innocent or when Jones and Williams are innocent then contradiction will be arised. So, first $3$ cases are not possible.

$\rightarrow$ , Now , observe $4^{th}$ case, i.e.  J=I, S=G, W=G, It is a possible case because when Jones is telling truth , It means Smith and Williams are lying. Since, No information is given for Guilty people. So, They may tell truth or lie. So, It is a possibility that Jones is innocent and telling truth whereas Smith and Williams are guilty and telling lie.

$\rightarrow$ , Now , observe $5^{th}$ case, i.e.  J=G, S=I, W=I, It is also possible because if Jones is lying then nothing is contradicting. So, It is a possible case.

$\rightarrow$ , Now , observe $6^{th}$ case, i.e.  J=G, S=I, W=G, If J is lying but W and S are telling truth. So, nothing will be contradicting. So, It is also a possible case.

$\rightarrow$ , Now , observe $7^{th}$ case, i.e.  J=G, S=G, W=I, If S & W are telling truth but J is lying. So, It is also a possible case.

  $\rightarrow$ , Now, observe $8^{th}$ case, i.e.  J=G, S=G, W=G, If S & W are telling truth but J is lying. So, It is also a possible case.

1 comment

in a) case 3) nothing is contradicting from the statements of Smith and Williams if Jones's statement is false.
0
0

Related questions