M122F07A4 - n is O n(Hint induction(b Prove without using...

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Math 122 F01 and F02 2007 Assignment 4 Due : Tuesday, November 27, 2007, before the lecture starts. The point of this assignment is to practice writing proofs, and in particular to practice with the concepts big-O, big-Omega and big-Theta. In order to do these questions, it is necessary to understand all of the relevant definitions and their immediate consequences. 1. [5] Let a, b Z . Prove that if a | b then b a | b . (Note that b a Z .) 2. Let a, b, m Z . (a) [5] Prove that if a b (mod m ), then k Z , ak bk (mod m ). (b) [2] Give a counterexample to show that ak bk (mod m ), where k Z , does not imply that a b (mod m ). 3. [5] Let a, b Z . Suppose that there exist integers x and y so that ax + by = 1. Prove that gcd ( a, b ) = 1. 4. [5] Prove that the positive integer n is a perfect square if and only if it has an odd number of positive divisors. (Hint: use the fact that n is a square of and only if every prime appears to an even power in the prime factorization of n .) 5. (a) [5] Use the definition (no limits) to prove that 4
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Unformatted text preview: n is O ( n !). (Hint: induction.) (b) Prove without using limits that n 2 is not Ω( n 4 ). (c) [5] Is it true that 2 n is Θ(3 n )? Justify your answer with an appropriate argument that does not use limits. 6. The definition of the limit of a function f : Z + → R is: lim n →∞ f ( n ) = L ∈ R if and only if for every real number ± > there exists N (which depends on f and ± ) such that, L-± < f ( n ) < L + ± for all n ≥ N . (a) [4] Suppose f : Z + → R + ∪ { } (so f takes only non-negative values). Use the definition above to prove that if lim n →∞ f ( n ) = L then there exists N such that f ( n ) ≤ L + 1 for all n ≥ N . (b) [6] Suppose f, g : Z + → R + ∪ { } (so f and g take only non-negative values). Prove that if lim n →∞ f ( n ) /g ( n ) = L , then f ( n ) is O ( g ( n )). Prove further that if L > 0 then f ( n ) is Θ( g ( n ))....
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This note was uploaded on 02/05/2011 for the course MATH 377 taught by Professor Stephenlang during the Spring '11 term at University of Victoria.

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