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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 21127 taught by Professor Gheorghiciuc during the Spring '07 term at Carnegie Mellon.
 Spring '07
 GHEORGHICIUC
 Math, Integers

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