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HW1_Answer - Econ135 Fall 2010 Assignment 1 Set Theory and...

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Econ135 Fall 2010 Assignment 1: Set Theory and the Probability Set Function Answer Keys. 1. 1.2.4: Since we know ( C 1 \ C 2 ) c = ( C c 1 [ C c 2 ) (denote this by d1) ; we can generalize to the case of three sets. Setting C 1 \ C 2 = B 2 ; then ( C 1 \ C 2 \ C 3 ) c = ( B 2 \ C 3 ) c = B c 2 [ C c 3 by applying d1. Then B c 2 = ( C 1 \ C 2 ) c = ( C c 1 [ C c 2 ) by d1, therefore, ( C 1 \ C 2 \ C 3 ) c = ( C c 1 [ C c 2 ) [ C c 3 = C c 1 [ C c 2 [ C c 3 : (Associative law is implictly used already, check this.) For general n sets, we can set C 1 \ C 2 \ ::: \ C n ° 1 = B n ° 1 : Then C 1 \ C 2 \ ::: \ C n ° 1 \ C n = B n ° 1 \ C n ; therefore the same procedure using d1 will lead us to ( C 1 \ C 2 \ ::: \ C n ° 1 \ C n ) c = B c n ° 1 [ C c n : Using d1 to B n ° 1 ; B n ° 2 ; :::; B 2 will give us ( C 1 \ C 2 \ ::: \ C n ) c = C c 1 [ C c 2 [ ::: [ C c n for any n: Similarly we can show, ( C 1 [ C 2 [ ::: [ C n ) c = C c 1 \ C c 2 \ ::: \ C c n : For more rigorous proof, use mathematical induction. 2. 1.2.5: Using venn diagrams, just check corresponding areas. 3. 1.2.6: See Example 1.2.7 4. 1.2.7: See Example 1.2.11 5. 1.3.1: P ( C 1 ) = 4 6 ; P ( C 2 ) = 4 6 ; P ( C 1 \ C 2 ) = 2 6 ; P ( C 1 [ C 2 ) = 1 = P ( C 1 ) + P ( C 2 ) ° P ( C 1 \ C 2 ) : 6. 1.3.7: Recall the property of probability:
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