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Five friends have access to a chat room. Is it possible to determine who is chatting if the following information is known? Either Kevin or Heather, or both, are chatting. Either Randy or Vijay, but not both, are chatting. If Abby is chatting, so is Randy. Vijay and Kevin are either both chatting or neither is. If Heather is chatting, then so are Abby and Kevin. Explain your reasoning.

4 Answers

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Let,

$1)$ Either Kevin or Heather, or both, are chatting : $K \;\vee\; H$

$2)$ Either Randy or Vijay, but not both, are chatting : $R\oplus V$

$3)$ If Abby is chatting, so is Randy : $A \rightarrow R$

$4)$ Vijay and Kevin are either both chatting or neither is : $(V \wedge K) \vee (\sim V \wedge \sim K)$ i.e. $V \Leftrightarrow K$

$5)$ If Heather is chatting, then so are Abby and Kevin : $H \rightarrow A\;\wedge \;K$

Now, we have to find the assignment for $K,H,R,V$ and $A$ so that whole system will be consistent means some assignment of truth values for these variables so that all the statements must be true.

Now, Let's start with $5)$  $H \rightarrow A\;\wedge \;K$ and $1)$ $K \;\vee\; H,$

For this, Possible assignments of truth values($T$ = True and $F$ =  False)  are :-

$1) A= F , K = T , H = F$ 

$2) A= T , K = T , H = F$

$3) A= T , K = T , H = T$

Now, statement $3)$ $A \rightarrow R$ will be true when :-

$1) A= F , K = T , H = F , R = T$ 

$2) A= F , K = T , H = F , R = F$ 

$3) A= T , K = T , H = F , R = T$

$4) A= T , K = T , H = T , R = T$

Now,  statement $2)$ $R\oplus V$ will be true when either $R=T, V = F$ (or) $V=T , R = F$ , So, possible assignments will be :-

$1) A= F , K = T , H = F , R = T , V = F$ 

$2) A= F , K = T , H = F , R = F , V = T$ 

$3) A= T , K = T , H = F , R = T , V = F$

$4) A= T , K = T , H = T , R = T , V = F$

Now, last statement $4)$ $(V \wedge K) \vee (\sim V \wedge \sim K)$ i.e. $V \Leftrightarrow K$ will be true when both $K$ and $V$ are True or False.

So, in above assignments $1), 3)$ and $4)$ are not possible. Only possibility is assignment $2)$ i.e. $A= F , K = T , H = F , R = F , V = T.$ It means only Kevin and Vijay are chatting.

So, Answer is :- Only Kevin and Vijay are chatting. 

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Suppose, K:Kevin, H:Heather, V:Vijay, R: Randy, A: Abby. 

Translate all given statements into logical statements, 

1)  (K$\vee$H) 
2)  (R$\oplus$V) 
3)  (A$\rightarrow$R) 
4)  (V$\odot$K) 
5)  (H$\rightarrow$(A$\wedge$K)) 

 

Now, we are trying to make all above logical statements true by assigning truth values. 

First, we assign truth value to either 2nd Or 4th logical statement because $\oplus$ and $\odot$ both have only two cases for True. 

So, we assign truth value to 4th statement:(V$\odot$K)

for Case 1: (T, T) 

That means V=T, K=T now, 2nd statement is True if R=F. then 3rd statement is True if A=F then 5th statement is True if H=F then 1st statement is True because K=T, H=F. 

So, we can make all statements true by Case1. if  we can't make then go for Case 2.

Hence, in Case 1 only V and K are true. So, we can conclude only Vijay and Kevin are chatting. 

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It is possible to determine who is chatting in the chat room based on the information provided.

From the first piece of information, we know that either Kevin or Heather, or both, are chatting. From the second piece of information, we know that either Randy or Vijay, but not both, are chatting. From the third piece of information, we know that if Abby is chatting, Randy is also chatting. From the fourth piece of information, we know that Vijay and Kevin are either both chatting or neither is. And from the last piece of information, we know that if Heather is chatting, then so are Abby and Kevin.

Given this information we can conclude that if Heather is chatting, then Abby and Kevin are also chatting, because we know that if Heather is chatting, then so are Abby and Kevin. Also, if Abby is chatting, Randy is also chatting. Vijay and Kevin are either both chatting or neither is.

However, it is not possible to determine whether Randy or Vijay is chatting, because we know that either Randy or Vijay, but not both, are chatting.

Therefore, It is possible to determine who is chatting in the chat room based on the information provided.
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The previous answers have already come up with the 5 expressions represented by the problem. They also proposed the idea of assigning values to the expressions.  

Another way to approach this problem is to do AND of all 5 expressions. On simplification, it will get reduced to R’VKA’H’, which clearly says that only V and K are talking.

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