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Homework 1-Solutions copy

Homework 1-Solutions copy - HOMEWORK 1 SOLUTIONS(1 Let A =...

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HOMEWORK 1 SOLUTIONS (1) Let A = { 3 , 4 , 5 } , B = { 3 , 4 } , C = { 4 } . Find D = A 4 B 4 C . Solution: Recall from class that A 4 B = ( A \ B ) ( B \ A ). That is, A 4 B contains all elements that lie in A but not in B and all elements that lie in B but not in A . Note that we also have A 4 B = ( A B ) \ ( A B ). Thus ( A 4 B ) 4 C = h ( A 4 B ) \ C i h C \ ( A 4 B ) i = h ( A \ B ) ( B \ A ) \ C i h C \ ( A B ) \ ( A B ) i = A \ ( B C ) B \ ( A C ) C \ ( A B ) C A B . On the other hand A 4 ( B 4 C ) = h A \ ( B 4 C ) i h ( B 4 C ) \ A i = h A \ ( B C ) \ ( B C ) i h ( B \ C ) ( C \ B ) \ A i = A \ ( B C ) A B C B \ ( C A ) C \ ( B A ) . A close inspection shows that the last term in both of the above equations are the same. (2) Suppose 70% of Californians like cheese, 80% like apples and 10% like neither. What percentage of Californians like both cheese and apples? Solution: First let’s ask a different question. How many Californians like apples or like cheese? If we add the percentages together we get 70%+80% = 150%. This is clearly an over-counting. The problem is that we have counted those who like both apples and cheese twice. Since 10% of Californians like neither apples nor cheese, we know that 90% either like apples or like cheese (including those who like both). Hence, the difference between the percentages, 150% - 90% = 60% answers the original question.
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