Solutions to Problem Set 3

Math 310/520


1. You visit an island where each person always tells the truth (a truth-teller) or always lies ( a liar). You meet two residents of the island, Calvin Butterball and Phoebe Small.

Calvin says: "Phoebe and I are both liars."

Determine whether each person is a truth-teller or a liar, or whether the situation is impossible. Explain your reasoning in words.

Suppose that Calvin is a truth-teller. Then his statement is true, and he and Phoebe are both liars. This contradicts the assumption that he's a truth-teller. Hence, this case is ruled out.

Hence, Calvin must be a liar. Therefore, his statement is false, and it's not the case that he and Phoebe are both liars. Since he is a liar, and since both of them can't be liars, Phoebe must be a truth-teller.

Perhaps the most valuable result of all education is the ability to make yourself do the thing you have to do, when it ought to be done, whether you like it or not. - Thomas Huxley

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