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Unformatted text preview: Problem Set 1 Solutions Economics 204  August 2006 1. Set Theory (a) Determine whether this formula is always right or sometimes wrong. Prove it if it is right. Otherwise give both an example and a counterexample and state (but dont prove) an additional neccesary and sufficient condition for it to always be right: A \ ( B \ C ) = ( A \ B ) C. Sometimes wrong. Let A be the set of firstyear econ grad students at Berkeley, B be the set of international econ grad students at Berkeley, and C be the set of male econ grad students at Berkeley. Then your GSI, Zack, is an element of ( A \ B ) C , but not of A \ ( B \ C ). He is a member of the set { domestic firstyears } { male econ grads } but not of the set { firstyear international female econ grads } . It can be correct though. Let A be the set of firstyears, B be the set of interna tional econ students, and C be the set { Econ 204 students with econ.berkeley.edu email addresses } . In this case,both the left and righthand sides consist of { firstyear econ grad students except international students who are not in Econ 204 } . An additional necessary and sufficient condition for the formula to be always cor rect would be that C A . While not a proof, Figure 1 illustrates why this is true. Observe that the lefthand side is contained in the righthand side. The set C \ A is included on the right, but not the left. C A is precisely the condition needed for this to be empty and render the two sides equal. (b) Certain subsets of a given set S are called Asets and others are called Bsets. Suppose that these subsets are chosen in such a way that the following properties are satisfied: The union of any collection of Asets is and Aset. The intersection of any finite number of Asets is an Aset. The complement of an Aset is a Bset and the complement of a Bset is an Aset. Prove the following: 1. The intersection of any collection of Bsets is a Bset. Proof: Let I index B I , a collection of Bsets, and let x T i I B i . Now x T i I B i if and only if x B i i I , which is true if and only if x / B C i for all i I , which is true if and only if x / S i I B C i . This means that T i I B i = ( S i I B C I ) C . Because B i is a Bset for all i , the complement B C i is 1 an Aset. The union of these Asets is an Aset, and its complement is, in turn, a Bset. Thus T i I B i is a Bset. 2. The union of any finite number of Bsets is a Bset. Proof: Now let I be finite. x S i I B i x B i for some i I . This is true if and only if x / B C i for some i I which is true if and only if x / T i I B C i . Thus, S i I B i = ( T i I B C i ) C . B C i is an Aset for all i so the intersection of B i is also an Aset and the complement of this intersection is a Bset. Thus, S i I B i is a Bset....
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 Summer '08
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 Economics

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