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Unformatted text preview: 2. a variation on an extra practice problem and on a lookat problem 3. from another textbook 4. from another textbook Numbers: • real numbers = R • natural numbers = N = { 1 , 2 , 3 , 4 , . . . } • integers = Z = { , ± 1 , ± 2 , ± 3 , ± 4 , . . . } • rational numbers = Q = n p q ∈ R : p, q ∈ Z and q 6 = 0 o 1 1. Let A and B be sets. 1a. By deﬁnition, the power set of A is: P ( A ) = { C : } . 1b. Let A ⊆ B . Show that P ( A ) ⊆ P ( B ) . 2. Let A = { A i : i ∈ N } be an indexed family of sets. Let k, m ∈ N with k ≤ m . Show that m \ i =1 A i ⊆ k \ i =1 A i . 3. Let a 1 = 1 and a 2 = 3 and a n +2 = 3 a n +12 a n for each n ∈ N . 3a. Calculate a 1 through a 5 . 3b. Show that a n =1 + 2 n ( * ) for each n ∈ N . 4. Let a 1 = 1, a 2 = 1, and a 3 = 1 and a n +3 = a n +2 + a n +1 + a n for each n ∈ N . 4a. Calculate a 1 through a 6 . 4b. Show that a n ≤ 2 n2 ( * ) for each n ∈ N with n > 1 . 2...
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 Fall '11
 Staff
 Math, Advanced Math, extra practice problem, neat formal proof, Prof. Girardi, mark box, Calculate a1

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