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relate to inhabitants of the island of knights and knaves created by Smullyan, where knights always tell the truth always lie. You encounter two people. A and B. Determine, if possible, what A and B are if they address you in the ways described. If you can not determine what these people are, can you draw any conclusions?

A says “The two of us are both knights ” and B says “A is knave.”

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There are $4$ Possible cases :-

$1)$ $A$ is knight and $B$ is knight

So, statements “The two of us are both knights ” and “A is knave” both should be true simultaneously which is not possible since $1^{st}$ statement says $A$ is knight and $2^{nd}$ statement says $A$ is knave which is contradicting what we assumed.So, It is not a possible case.

$2)$ $A$ is knight and $B$ is knave

So, statement “The two of us are both knights” is true and statement “A is knave” is false. Since, $1^{st}$ statement is saying $B$ Is knight but $B$ is actually knave. So, It is not a possible case.

$3)$ $A$ is knave and $B$ is knight

So, statement “The two of us are both knights” is false and statement “A is knave” is true. Since, “The two of us are both knights” is false which means atleast one should be knave. Since, $A$ is knave. $B$ may be knave or knight according to this statement but if B is knave then $A$ must be knight because in that case “A is knave” will be false which is contradicting what we assumed.So, $B$ must be knight. So, $A$ is knave and $B$ is knight is a possible case.

$4)$ $A$ is knave and $B$ is knave

So, statement “The two of us are both knights” is false and statement “A is knave” is also false. which means $A$ is knight which is contradicting what we assumed. So, It is not a possible case.

So, Answer :- $A$ is knave and $B$ is knight.

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