Hout1 - Answer key No.1 to selected homework problems:...

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Unformatted text preview: Answer key No.1 to selected homework problems: Math. 61 1.7 1. Suppose the formula for n and try to prove the formula for n + 1: 1(1!) + 2(2!) + + n ( n !) + ( n + 1)(( n + 1)!) = ( n + 1)!- 1 + ( n + 1)(( n + 1)!) = ( n + 1)! + ( n + 1)(( n + 1)!)- 1 = ( n + 1)!(1 + n + 1)- 1 = ( n + 2)!- 1 . 9. First check the equality: 1 a 2- 1 = 1 2( a- 1)- 1 2( a + 1) . And use this formula for a = n + 2 in the induction process. 13. Write S n for the fraction of the LHS (the left hand side). Then S n +1 = S n (2 n + 1) 2( n + 1) ( n + 1)- 1 / 2 (2 n + 1) 2( n + 1) ( n + 2)- 1 / 2 . The last inequality: ( n + 1)- 1 / 2 (2 n +1) 2( n +1) ( n + 2)- 1 / 2 is equivalent to (2 n + 1) 2( n + 1) ( n + 1) 1 / 2 ( n + 2)- 1 / 2 , which in turn equivalent to: (2 n + 1) 2 4( n + 1) 2 = ( n + 1 2 ) 2 ( n + 1) 2 ( n + 1) ( n + 2) , which can be checked by computation. Indeed ( n + 1 2 ) 2 ( n + 1) 2 ( n + 1) ( n + 2) is equivalent to n 3 + 3 n 2 + 9 4 n + 1 2 = ( n + 1 2 ) 2 ( n + 2) ( n + 1) 3 = n 3 + 3 n 2 + 3 n + 1 and RHS-LHS = 3 4 n + 1 2 0 if n 0. 14. Check the starting point of the induction by your self. Now suppose: 2 n + 1 2 n and prove: 2( n + 1) 2 n +1 as follows: 2( n + 1) + 1 = 2 + 2 n + 1 2 + 2 n 2 n + 2 n = 2 2 n = 2 n +1 . 2.1 56. Let P ( X ) denote the set of all subsets in a set X . Then by Theorem 2.1.6, we know that |P ( X ) | = 2 n if | X | = n . The only non-proper subset in P ( X ) is X itself. Thus the answer is 2 n- 1 . 78. Suppose X, A, B, C to be subsets of a universe U . We prove the statement A C = B C A = B . By definition, we have X C = ( X C )- ( X C ) = ( X- C ) ( C- X ) . Pick an arbitrary element a in X . Then there are two possibilities: (i) a C or (ii) a C. 1 2 If a C , then a X C . If a C , then a X C . Since (i) and (ii) never happen at the same time, we see that for a X (a) a C a X C and a C a X C. Pick an arbitrary element x outside X . Then (b) x X C x C and x X C x C. Suppose a A C . Then by (a) applied to X = A , a B C = A C . Then by (b) applied to X = B , a B a C , a contradiction because a A C . Thus a B . This shows that A C B C . Since the roles of A and B are interchangeable, we conclude A C B C . This shows that A C = B C . Now suppose that a A- C . Then by (a) applied to X = A , we see a A C = B C . Since a C , we know a B- C because B C = ( B- C ) ( C- B ). Thus A- C B- C . Interchanging the role of A and B , we get A- C B- C . This shows A- C = B- C ....
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This note was uploaded on 11/28/2009 for the course STATS STATS10 taught by Professor Sugano during the Spring '09 term at UCLA.

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Hout1 - Answer key No.1 to selected homework problems:...

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