hw4 - 4. All inexpensive essays are boring. 5. x. ( x B )...

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CSE 20: Discrete Mathematics Fall 2011 Problem Set 4 Instructor: Daniele Micciancio Due on: Thu. Nov. 3, 2011 Problem 1 (6 points) Prove by induction that the sum of the first n odd numbers (1+3+5+7+ ... ) equals n 2 . More specifically, let f ( n ) = n k =1 (2 k - 1) = 1 + 3 + 5 + ··· + (2 n - 1), and define the statement P ( n ) ( f ( n ) = n 2 ). Prove (by induction on n ) that n.P ( n ), where n ranges over the set of all positive integers. Problem 2 (6) Let the following sets be given: B = the set of all boring books H = the set of all historical essays E = the set of all expensive books Express the following statements using set notation. (You can use, beside the name of the sets, the following set operations and relations: , , = , 6 = , , , , 0 , where 0 is the set complement operation.) 1. All historical essays are boring 2. There are some boring books that are not historical essays 3. All boring historical essays are expensive
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Unformatted text preview: 4. All inexpensive essays are boring. 5. x. ( x B ) (( x H ) ( x / E )) 6. x. ( x H x B ) ( x / E ) Problem 3 (5 points) Let O = { 2 k-1 | k Z } be the set of all odd integers. 1. Use set notation to dene the set T of all integers that are the sum of two odd numbers 2. Use set notation to dene the set S of all pairs of integers whose sum is odd 3. Use set notation to dene the set P of all pairs of integers whose product is odd 4. Use set notation and the denition of T,S,P to formulate the statement The product of any two odd integers is odd. 5. Do the same for the statement For any two numbers x and y , the sum x + y is odd if and only if the product x y is odd....
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This note was uploaded on 11/21/2011 for the course CSE 20 taught by Professor Foster during the Spring '08 term at UCSD.

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