204ps12006 - Problem Set 1Revised problem 8 Economics 204...

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Problem Set 1–Revised 8/2/06, problem 8 Economics 204 - August 2006 Due Friday, 4 August in Lecture 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. (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. 2. The union of any finite number of B -sets is a B -set. 2. Induction: (a) Write the following expression using summation ( ) notation then find and prove a formula for the sum: 1
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