Answer to Question #131737 in Discrete Mathematics for sudhanshu kumar

Question

The following exercises relate to inhabitants of the island of knights and knaves created by
Smullyan, where knights always tell the truth and knaves 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 cannot determine what these two people are, can you draw any conclusions?
(a)A says “At least one of us is a knave” and B says nothing.
(b)A says “If I am a knight, then so is B” and B says nothing.(c)A says “We are both knights” and B says “Either A is a knight, or I am a knight, but not both”.

Answer 1

(a) Let's assume A is lying. Then no one among A and B are knaves, so they have to be knights, but knights never lie. We came up with contradiction, so A is telling truth. A is a knight. Since one of them is a knave, the knave must be B. (A - knight, B - knave)

(b) Let's assume A is knight, then B is a knight too. Since that is just what A said, and if he is a knight, his statement must be true. But look what we have here: we just showed that if he is a knight, then so is B.This is exactly what he claimed, and we’ve just seen this statement is true. Since he said a true statement, he must be a knight, and so therefore B must be too. (A - knight, B -knight)

(c) Assume A is a knight, then A and B should be knights. B as a knight must tell truth, but it is not consistent with what B said. Let's assume A is a knave, then there B can be either a knight or a knave. If B is telling truth, then he is a knight and they are indeed knave and knight as B claims. If B is lying, then negation of "either A is a knight, or I am a knight" gives "A is not a knight and B is not a knight", that is consistent with what B claims. (A - knave, B - knight or knave)