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Saumyojit

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Twenty-seven persons attend a party. Which one of the following statements can never be true?

(1)There is a person in the party who is acquainted with all the twenty-six others
.(2)Each person in the party has a different number of acquaintances.
(3)There is a person in the party who has an odd number of acquaintances
(4)In the party, there is no set of three mutual acquaintances.


What is the meaning of this "no set of three mutual acquaintances" ? Suppose A knows D , B also knows D and C also knows D ? This can't happen?
 
Saumyojit said:
What is the meaning of this "no set of three mutual acquaintances" ? Suppose A knows D , B also knows D and C also knows D ? This can't happen?
Click to expand...
A set of mutual acquaintances is a set where each member is an acquaintance with every other member of the set.
 
I think that it means that there doesn’t exist a set of people {A, B, C} such that A knows B, B knows A, A knows C, C knows A, B knows C, C knows B. I don’t see a reason why this couldn’t be true
 
Zermelo said:
I think that it means that there doesn’t exist a set of people {A, B, C} such that A knows B, B knows A, A knows C, C knows A, B knows C, C knows B. I don’t see a reason why this couldn’t be true
Click to expand...
If this is false then other conditions holds true but does it align . I just checked out by hit and trial
We are checking all the 4 conditions for three persons as of now A,B,C

Suppose A knows all 26 others , B knows only these 4 A,C,F,G , C knows A,B,D.

A has 26 acquaintance ,B has 4 and C has only two. (different)

IS there a person in the party who has an odd number of acquaintances? yes , assume C

In the party, there is a set of three mutual acquaintances. (ABC)

Now what to do .
Answer is (2)
 
Saumyojit said:
If this is false then other conditions holds true but does it align . I just checked out by hit and trial
We are checking all the 4 conditions for three persons as of now A,B,C

Suppose A knows all 26 others , B knows only these 4 A,C,F,G , C knows A,B,D.

A has 26 acquaintance ,B has 4 and C has only two. (different)

IS there a person in the party who has an odd number of acquaintances? yes , assume C

In the party, there is a set of three mutual acquaintances. (ABC)

Now what to do .
Answer is (2)
Click to expand...
Don't understand. Which of the 4 possible answers are you addressing here?
 
I am saying option4 if it is never true does the other three conditions holds true?
 
Saumyojit said:
Twenty-seven persons attend a party. Which one of the following statements can never be true?
(1)There is a person in the party who is acquainted with all the twenty-six others
(2)Each person in the party has a different number of acquaintances.
(3)There is a person in the party who has an odd number of acquaintances
(4)In the party, there is no set of three mutual acquaintances.
What is the meaning of this "no set of three mutual,acquaintances" ? Suppose A knows D , B also knows D and C also knows D ? This can't happen?
Click to expand...
Consider the set of partygoers, [imath]\{A,B,C\}[/imath] to say this triad are all three are acquaintances of each other.
Question: Is the relation acquaintance a symmetric relation? i.e If A is an acquaintance of B is it necessary that B is an acquaintance of A?
If the answer to that question is no then what difference does the modifier mutual make?
 
Saumyojit said:
I am saying option4 if it is never true does the other three conditions holds true?
Click to expand...
Sorry, don't understand this either. Out of 4 choices, 3 can be true and 1 can never be true. Which one do you think can never be true?
 
pka said:
Is the relation acquaintance a symmetric relation?
Click to expand...
A knows B maybe he knew him from a distance but B did not know of A

ABC are set of three mutual acquaintances.

But If pka argument is true then it is not true

AFter all, acquaintance means a person one knows slightly, but who is not a close friend
 
pka said:
Consider the set of partygoers, [imath]\{A,B,C\}[/imath] to say this triad are all three are acquaintances of each other.
Question: Is the relation acquaintance a symmetric relation? i.e If A is an acquaintance of B is it necessary that B is an acquaintance of A?
If the answer to that question is no then what difference does the modifier mutual make?
Click to expand...
I think acquaintance is assumed to be a symmetric relation. Otherwise #2 can be true:
Guest 1 doesn't know anybody.
Guest 2 knows guest 1.
Guest 3 knows guests 1-2.
.......
Guest 27 knows guests 1-26.

If it is symmetric #2 can't be true. There are 0 to 26 possible acquaintances for each guest. For everyone to have a different number of acquaintances, all values from 0 to 26 need to be used. But if there is a guest without acquaintances (0), then nobody can have 26 acquaintances.
 
Saumyojit said:
How can there be a guest with zero acquaintances?
Why he is in the party? who has called him ? At least one acquaintance has to be there.
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Let's say I know you but not your brother. I invited you to a party. You said you would bring your brother. At the last moment you decided not to go and study for a math test instead. But your brother went. As a result he doesn't know anybody at the party.
 
lev888 said:
I think acquaintance is assumed to be a symmetric relation. Otherwise #2 can be true:
Click to expand...
Here is a counter-example. The governor of our state some thirty plus years ago had been my father's law school room-mate. It would be fair to say the he was an acquaintance of mine but he had no idea who I was. In fact at my father's funeral I had to remind him who I was. I do not consider that a symmetric relationship.
 
pka said:
Here is a counter-example. The governor of our state some thirty plus years ago had been my father's law school room-mate. It would be fair to say the he was an acquaintance of mine but he had no idea who I was. In fact at my father's funeral I had to remind him who I was. I do not consider that a symmetric relationship.
Click to expand...
I am talking about this problem, not general usage.
 
Saumyojit said:
Twenty-seven persons attend a party. Which one of the following statements can never be true?

(1)There is a person in the party who is acquainted with all the twenty-six others
.(2)Each person in the party has a different number of acquaintances.
(3)There is a person in the party who has an odd number of acquaintances
(4)In the party, there is no set of three mutual acquaintances.

What is the meaning of this "no set of three mutual acquaintances" ? Suppose A knows D , B also knows D and C also knows D ? This can't happen?
Click to expand...
The problem is ambiguous as to the exact meaning of "acquainted", so it may be necessary to try solving using one definition, and if it turns out that there are more than one statement, or none, that can never be true, then try the other. Also, since this is a logic puzzle, we have to set aside real-life considerations such as fuzzy definitions of "acquainted", or why someone with no acquaintances would be there. We have to assume it's all-or-nothing.

But my sense is that if we took an asymmetrical definition, we would then face an additional problem: Is an "acquaintance" someone you know, or someone who knows you? And the use of "mutual" as a modifier doesn't mean "bidirectional", but "each of the three knows each of the others". So I would at least start with the symmetrical definition. That can be modeled simply as an undirected graph with 27 nodes.

Now, I see this problem in many places (some of which declare that the official answer is (2) as Saumyojit said). That gives a further check on the interpretation.
  1. Is it possible to have one node connected to all the others? Of course.
  2. Is it possible for each node to have a different degree? This is the one to think about; there is a very simple answer.
  3. Is it possible for one node to have an odd degree? Of course.
  4. Is it possible to have no triangles in the graph? This may take a little more thought, but not much.
So this interpretation does work. I'll let someone else check out the symmetric version.
 
lev888 said:
I think acquaintance is assumed to be a symmetric relation. Otherwise #2 can be true:
Guest 1 doesn't know anybody.
Guest 2 knows guest 1.
Guest 3 knows guests 1-2.
.......
Guest 27 knows guests 1-26.
Click to expand...
Suppose acquaintance is not symmetric .
Guest 4 knows 1-3 guests . but it is given in the condition There is a person in the party who has an odd number of acquaintances . Guest 2 has already no of odd acquaintance.

lev888 said:
If it is symmetric #2 can't be true. There are 0 to 26 possible acquaintances for each guest. For everyone to have a different number of acquaintances, all values from 0 to 26 need to be used.
Click to expand...
YEs understood.
But if it is symmetric #2 is false BUT how can you say that ABC does not have three mutual acquaintances.
 
Dr.Peterson said:
Now, I see this problem in many places (some of which declare that the official answer is (2) as Saumyojit said). That gives a further check on the interpretation.
  1. Is it possible to have one node connected to all the others? Of course.
  2. Is it possible for each node to have a different degree? This is the one to think about; there is a very simple answer.
  3. Is it possible for one node to have an odd degree? Of course.
  4. Is it possible to have no triangles in the graph? This may take a little more thought, but not much.
So this interpretation doe
Click to expand...
I understand if it is symmetric it is undirected graph and nodes are the persons .
What about the conditions?
 
Saumyojit said:
Suppose acquaintance is not symmetric .
Guest 4 knows 1-3 guests . but it is given in the condition There is a person in the party who has an odd number of acquaintances . Guest 2 has already no of odd acquaintance.


YEs understood.
But if it is symmetric #2 is false BUT how can you say that ABC does not have three mutual acquaintances.
Click to expand...
It seems you don't understand the problem. The 4 choices are independent. The situation described in #2 does NOT include conditions from #1, #3, #4.
 
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