more practice midterm2

more practice midterm2 - 1 c A 2 c A n c whenever A 1;A n...

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More practice problems on Methods of Proof 1. Prove that the square of an even number is an even number by using: a) a direct proof b) a contrapositive proof c) a proof by contradiction 2. Prove that the sum of two odd integers is even. 3. Prove that the sum of two rational numbers is rational. 4. Prove that the sum of an irrational number and a rational number is irrational using proof by contradiction. 5. Prove that the product of two rational numbers is rational. 6. Prove or disprove that the product of two irrational numbers is irrational. 7. Show that 3 p 3 is irrational. 8. Show that p n is irrational if n is a positive integer that is not a perfect square. 9. Prove or disprove that every positive integer can be written as the sum of the squares of two integers. 1

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More practice problems on Induction Use mathematical induction to prove the following statements: 1. P n j =0 ar j = ar n +1 a r 1 when r 6 = 1 . 2. Show that H 2 n 1 + n 2 , where : H k = 1 + 1 2 + 1 3 + ::: + 1 k : 3. ( A 1 \ A 2 \ ::: \ A n ) c = ( A
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Unformatted text preview: 1 ) c [ ( A 2 ) c [ ::: [ ( A n ) c ; whenever A 1 ;:::;A n are subsets of a universal set U , and n & 2 . 4. Prove that: 3 + 3 ± 5 + 3 ± 5 2 + ::: + 3 ± 5 n = 3 & 5 n +1 ² 1 ± 4 whenever n is a nonnegative integer. 5. Show that if h > ² 1 , then 1 + nh ³ (1 + h ) n for all nonnegative integers n . (This is called Bernoulli&s inequality.) 6. Prove that 1 + 1 4 + 1 9 + ::: + 1 n 2 < 2 ² 1 n whenever n is a positive integer greater than 1 : 7. Use mathematical induction to prove that a set with n elements has n ( n ² 1) = 2 subsets containing exactly two elements whenever n is an integer greater than or equal to 2 . 8. Use mathematical induction to show that if A 1 , A 2 ,..., A n and B are sets, then ( A 1 [ A 2 [ ::: [ A n ) \ B = ( A 1 \ B ) [ ( A 2 \ B ) [ ::: [ ( A n \ B ) : 9. Use mathematical induction to show that q ( P 1 _ P 2 _ ::: _ P n ) () q P 1 ^ q P 2 ^ ::: ^ q P n : 2...
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This note was uploaded on 09/08/2011 for the course MATH 21-127 taught by Professor Gheorghiciuc during the Spring '07 term at Carnegie Mellon.

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more practice midterm2 - 1 c A 2 c A n c whenever A 1;A n...

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