Section 1.2: Problem 12 Solution
Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
There are three suspects for a murder: Adams, Brown, and Clark. Adams says “I didn’t do it. The victim was an old acquaintance of Brown’s. But Clark hated him.” Brown states “I didn’t do it. I didn’t even know the guy. Besides I was out of town all that week.” Clark says “I didn’t do it. I saw both Adams and Brown downtown with the victim that day; one of them must have done it.” Assume that the two innocent men are telling the truth, but that the guilty man might not be. Who did it?
There are at least two ways to solve this. One is just to assume in order that Adams, Brown or Clark did it, and then check whether their claims satisfy the conditions stated in the problem. Another way is, I think, more related approach, which we take. So, consider the following translations of some statements.
=”Adams did not do it”,
=”Adams knew the victim”,
=”Brown did not do it”,
=”Brown knew the victim”,
=”Brown was out of town”,
=”Clark did not do it”,
=”Clark hated the victim”. Then, what they say can be translated (roughly with some preassumptions) as
,
,
. Now, simply note that both pairs
and
and
and
are mutually exclusive. Therefore, neither can be a pair of true statements, and
is false. Overall, we have the following true propositions:
(Adams did not do it),
(Adams knew the victim),
(Brown did it),
(Brown knew the victim),
(Brown was in town),
(Clark did not do it), and
(Clark hated the victim).