Discrete Mathematics and Its Applications, Seventh Edition

Published by McGraw-Hill Education
ISBN 10: 0073383090
ISBN 13: 978-0-07338-309-5

Chapter 1 - Section 1.8 - Proof Methods and Strategy - Supplementary Exercises - Page 112: 13

Answer

$Aaron$ must be a $knave$, because a $knight$ would never make the false statement that all of them are $knaves$. If $Bohan$ is a $knight$, then he would be speaking the truth. If $Crystal$ is a $knight$, so that is one possibility. On the other hand, $Bohan$ might be a $knave$, in which case his statement is already false, regardless of$Crystal$'s identity. In this case, if $Crystal$ were also a $knave$, then $Aaron$ would have told the truth, which is impossible. So there are two possibilities for the ordered triple ($Aaron$, $Bohan$, $Crystal$), namely ($knave$, $knight$ , $knight$ ) and ($knave$, $knave$, $knight$ )

Work Step by Step

$Aaron$ must be a $knave$, because a $knight$ would never make the false statement that all of them are $knaves$. If $Bohan$ is a $knight$, then he would be speaking the truth. If $Crystal$ is a $knight$, so that is one possibility. On the other hand, $Bohan$ might be a $knave$, in which case his statement is already false, regardless of$Crystal$'s identity. In this case, if $Crystal$ were also a $knave$, then $Aaron$ would have told the truth, which is impossible. So there are two possibilities for the ordered triple ($Aaron$, $Bohan$, $Crystal$), namely ($knave$, $knight$ , $knight$ ) and ($knave$, $knave$, $knight$ )
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