2. Suppose that the functions f1, f2, 91, 92 : N → R20 are such that f₁ = O(91) and ƒ2 € О(92). Prove that (fi + ƒ₂) € ☹(max{91, 92}). Here (f1f2)(n) = fi(n) + f2(n) and max{91, 92}(n) = max{91(n), 92(n)}. 3. Let nЄ N\{0}. Describe the largest set of values n for which you think 2n < n!. Use induction to prove that your description is correct. Here m! stands for m factorial, the product of first m positive integers. 4. Prove that log2 n! = O(n log n).
2. Suppose that the functions f1, f2, 91, 92 : N → R20 are such that f₁ = O(91) and ƒ2 € О(92). Prove that (fi + ƒ₂) € ☹(max{91, 92}). Here (f1f2)(n) = fi(n) + f2(n) and max{91, 92}(n) = max{91(n), 92(n)}. 3. Let nЄ N\{0}. Describe the largest set of values n for which you think 2n < n!. Use induction to prove that your description is correct. Here m! stands for m factorial, the product of first m positive integers. 4. Prove that log2 n! = O(n log n).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Course: Discrete mathematics for CS
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Transcribed Image Text:2. Suppose that the functions f1, f2, 91, 92 : N → R20 are such that f₁ = O(91) and ƒ2 € О(92).
Prove that (fi + ƒ₂) € ☹(max{91, 92}).
Here (f1f2)(n) = fi(n) + f2(n) and max{91, 92}(n) = max{91(n), 92(n)}.
3. Let nЄ N\{0}. Describe the largest set of values n for which you think 2n < n!. Use induction to
prove that your description is correct.
Here m! stands for m factorial, the product of first m positive integers.
4. Prove that log2 n! = O(n log n).
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