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Problem 1 :Prove that the following two sets A and B are equinumerous: (a) A = {n EN: n is divisible by 3} and B = {n EN: n is divisible by 4} (b) A={n EN:n is not divisible by 3} and B = {neN: n is not divisible by 4}

Problem 1 :Prove that the following two sets A and B are equinumerous: (a) A = {n EN: n is divisible by 3} and B = {n EN: n is divisible by 4} (b) A={n EN:n is not divisible by 3} and B = {neN: n is not divisible by 4}

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Problem 1 : Prove that the following two sets A and B are equinumerous: (a) A = {n EN: n is divisible by 3) and B = {n EN:n is divisible by 4} (b) A={n EN:n is not divisible by 3} and B = {neN: n is not divisible by 4} Problem 2 A consists of all functions f: N→ N, and B consists of all functions g: NN that are monotone (that is, for every m, n E N, if m ≤ n then g(m) ≤ g(n)). Problem 31 : Prove that these two sets A, and B of functions are equinumerous, where Prove that the set of functions g: NN that are monotone is not countable. Problem 4 3: Prove that the following set A is countable: A consists of all infinite sequences ao, a1,... that are monotone and such that for every i = 0, 1,..., a € {0, 1, 2}. Problem 5- : Find the limits of the following sequences (show your work, explain your reasoning: you may use properties discussed in class) Problem 6 3/2 +1 (a) limn-+00 (b) lim-00 log. (3n²+2) n+1 n²-2n+4 (c) limn-+00 n+1 n²-2n+4 (d) limn-+on+1 (e) limn-+00 (f) limpin-on²+1 n+log(n+1) n+1 2logy n : Use mathematical induction to prove that n² <n! holds for every n > 4.
Transcribed Image Text:Problem 1 : Prove that the following two sets A and B are equinumerous: (a) A = {n EN: n is divisible by 3) and B = {n EN:n is divisible by 4} (b) A={n EN:n is not divisible by 3} and B = {neN: n is not divisible by 4} Problem 2 A consists of all functions f: N→ N, and B consists of all functions g: NN that are monotone (that is, for every m, n E N, if m ≤ n then g(m) ≤ g(n)). Problem 31 : Prove that these two sets A, and B of functions are equinumerous, where Prove that the set of functions g: NN that are monotone is not countable. Problem 4 3: Prove that the following set A is countable: A consists of all infinite sequences ao, a1,... that are monotone and such that for every i = 0, 1,..., a € {0, 1, 2}. Problem 5- : Find the limits of the following sequences (show your work, explain your reasoning: you may use properties discussed in class) Problem 6 3/2 +1 (a) limn-+00 (b) lim-00 log. (3n²+2) n+1 n²-2n+4 (c) limn-+00 n+1 n²-2n+4 (d) limn-+on+1 (e) limn-+00 (f) limpin-on²+1 n+log(n+1) n+1 2logy n : Use mathematical induction to prove that n² <n! holds for every n > 4.
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