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184a_05f_1e

# 184a_05f_1e - Math 184A First Exam(40 points 19 October...

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Math 184A First Exam (40 points) 19 October 2005 Please put your name and ID number on your blue book. The exam is CLOSED BOOK except for one page of notes. Calculators are NOT allowed. You must show your work to receive credit. 1. (10 pts.) Prove that the number of ordered lists without repeats (including the empty list) that can be constructed from an n -set is nearly n ! e . Hint : By Taylor’s theorem, e is nearly 1 + 1 / 1! + 1 / 2! + 1 / 3! + · · · + 1 /n !. 2. (10 pts.) For each of the following, EITHER give an example of the thing that is described OR explain why none exists. (a) A surjection from { 1 , 2 , 3 } to { a,b,c,d } . (b) An injection from { 1 , 2 , 3 } to { a,b,c,d } . (c) A permutation f of { 1 , 2 , 3 , 4 , 5 } such that f 40 is NOT the identity function and f 40 negationslash = f . Also, if you find such an f , compute f 40 . Remember that the identity function is the function g such that g ( x ) = x for all x and f 40 ( x ) is f ( f ( · · · f ( x ))), not ( f ( x )) 40 . 3. (10 pts.) The local description of a decision tree for constructing sequences of A’s and B’s is given below. The notation BA S ( n 2) means place BA in front of each
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