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204ps12006sols - Problem Set 1 Solutions Economics 204...

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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 don’t 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 first-year 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 first-years }∪ { male econ grads } but not of the set { first-year international female econ grads } . It can be correct though. Let A be the set of first-years, 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 right-hand sides consist of { first-year 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 left-hand side is contained in the right-hand 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 A -sets and others are called B -sets. Suppose that these subsets are chosen in such a way that the following properties are satisfied: The union of any collection of A -sets is and A -set. The intersection of any finite number of A -sets is an A -set. The complement of an A -set is a B -set and the complement of a B -set is an A -set. Prove the following: 1. The intersection of any collection of B -sets is a B -set. Proof: Let I index B I , a collection of B -sets, and let x i I B i . Now x 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 / i I B C i . This means that i I B i = ( i I B C I ) C . Because B i is a B -set for all i , the complement B C i is 1
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an A -set. The union of these A -sets is an A -set, and its complement is, in turn, a B -set. Thus i I B i is a B -set. 2. The union of any finite number of B -sets is a B -set. Proof: Now let I be finite. x 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 / i I B C i . Thus, i I B i = ( i I B C i ) C . B C i is an A -set for all i so the intersection of B i is also an A -set and the complement of this intersection is a B -set. Thus, i I B i is a B -set.
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