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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science February 8, 2005 Srini Devadas and Eric Lehman Problem Set 2 Solutions Due: Monday, February 14 at 9 PM Problem 1. Use induction to prove that 1 1 1 1 1 = 1 1 1 1 2 3 4 n n for all n 2 . Solution. The proof is by induction on n . Let P ( n ) be the proposition that the equation above holds. Base case. P (2) is true because 1 1 = 1 2 2 Inductive step. Assume P ( n ) is true. Then we can prove P ( n + 1) is also true as follows: 1 1 1 1 1 1 = 1 1 1 1 1 2 3 n n + 1 n n + 1 1 = n + 1 The first step uses the assumption P ( n ) and the second is simplification. Thus, P (2) is true and P ( n ) implies P ( n + 1) for all n 2 . Therefore, P ( n ) is true for all n 2 by the principle of induction. Problem 2. DeMorgans Law for sets says: A ( B C ) = ( A B ) ( A C ) Assume this and prove this extension of DeMorgans Law to n 2 sets: A ( B 1 B 2 . . . B n ) = ( A B 1 ) ( A B 2 ) . . . ( A B n ) Extending formulas to an arbitrary number of terms is a common (if mundane) applica- tion of induction. 2 Problem Set 2 Solution. We use induction. Let P ( n ) be the proposition that A ( B 1 B 2 . . . B n ) = ( A B 1 ) ( A B 2 ) . . . ( A B n ) for all sets A and B 1 , . . . , B n . Base case: P (2) follows from DeMorgans original law with B = B 1 and C = B 2 . Inductive step: Assuming P ( n ) , we can deduce P ( n + 1) as follows: A ( B 1 B 2 B 3 . . . B n ) = A (( B 1 B 2 ) B 3 . . . B n ) = ( A ( B 1 B 2 )) ( A B 3 ) . . . ( A B n ) = ( A B 1 ) ( A B 2 )) ( A B 3 ) . . . ( A B n ) In the first step, we group B 1 and B 2 . This allows us to apply the assumption P ( n ) in the second step. The last step uses DeMorgans original law. Since P (2) is true and P ( n ) P ( n + 1) for all n 2 , the principle of induction implies that P ( n ) is true for all n 2 . Problem 3. Let H n denote the n-th harmonic sum , which is defined by: n 1 H n = k k =1 Harmonic sums come up often. Youll see them again later in 6.042 and also in 6.046. Claim. H 2 m 1 + m/ 2 for all m . (a) The next problem part will ask you to prove this claim by induction. To make this easier, three steps that you may find useful in your proof are listed below. Provide a one sentence justification for each of these steps....
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This note was uploaded on 02/08/2011 for the course EECS 6.042 taught by Professor Dr.ericlehman during the Spring '11 term at MIT.
- Spring '11
- Computer Science