# Exam1Spring08 - Math 213 Exam 1(Solutions Prof I.Kapovich...

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Math 213 Exam 1 (Solutions) Prof. I.Kapovich February 22, 2008 Problem 1 [20 points] For each of the following statements indicate whether it is true or false. You do not have to explain your answers. (1) For the set S = {∅ , {∅} , {{∅}}} we have | S | = 1. (2) If | A | = m and | B | = n , then the number of all functions from A to B is m n . (3) For every n r 1 we have C ( n, r ) P ( n, r ). (4) Whenever E, F are events such that p ( E ) = 0, then E and F are independent. (5) For every n 1 the number n i =0 ( n i ) is even. Solution. (1) FALSE. In fact, we have | S | = 3. (2) FALSE. In fact, he number of all functions from A to B is n m . (3) TRUE. Indeed, C ( n, r ) = P ( n, r ) /r ! and so C ( n, r ) P ( n, r ). (4) TRUE. Indeed, in this case p ( E ) = 0 and 0 p ( E F ) p ( E ) = 0, so that p ( E F ) = 0. Thus 0 = 0 · p ( F ), that is p ( E F ) = p ( E ) p ( F ). (5) TRUE. Indeed, n i =0 ( n i ) = 2 n is even. Problem 2 [20 points] Prove that for every integer n 1 we have: 1 2 - 2 2 + 3 2 - · · · + ( - 1) n - 1 n 2 = ( - 1) n - 1 n ( n + 1) 2 . Give all the details of your work.

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