sol.assign3 - pirates on the boat, however our result...

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Solutions for Assignment 3 Discrete Mathematics II Macm 201 (Fall 2010) Section 8.2 Section 8.3
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Sections 8.4 and 8.5
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faculty of science department of mathematics MACM 201 - D100A SSIGNMENT #3 Solution to the instructor question Let the condition c i be that rule R i applies. Let A i be the set of pirates to whom rule R i applies ( i =1 .. 10 ). We have for any 1 k 10 and any 1 i 1 <i 2 < · · · <i k 10 , N ( c i 1 c i 2 ...c i k )= | A i 1 A i 2 ∩···∩ A i k | . By the principle of inclusion/exclusion and (e), we have N c 1 ¯ c 2 · · · ¯ c 10 )= N - ± 1 i 10 | A i | + ± 1 i<j 10 | A i A j | - ± 1 i<j<k 10 | A i A j A k | . By (a), at least one rule applies to each pirate, hence N c 1 ¯ c 2 · · · ¯ c 10 )=0 , hence N = ± 1 i 10 | A i | - ± 1 i<j 10 | A i A j | + ± 1 i<j<k 10 | A i A j A k | . By applying remaining knowledge about rules, we get N = 10 · 20 - ² 10 2 ³ 4+ ² 10 3 ³ 2 = 260 . So there were 260 pirates on the boat who wanted to give up their job. NOTE: Something strange happened in this question. Indeed, by (a) and (b) there are at most 200
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Unformatted text preview: pirates on the boat, however our result proves that there are 260 pirates on the boat. What is going on? The problem is that the system of sets that is implied by the rule conditions cannot exists - is incon-sistent, so we get inconsistent result. I did not check whether such system indeed can exists when I made up this question. For the future use I have changed the question from Determine how many pirates are on the boat. to Determine whether this story is made up or it could possibly happen. So the answer with this extra reasoning is The story could not possibly happen and is made up. DR. L. STACHO, FALL 2010 1...
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sol.assign3 - pirates on the boat, however our result...

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