Let f1 and f2 be asymptotically positive non-decreasing functions. Prove or disprove each of the following conjectures. To disprove give a counter example.
If f1(n) = O(g(n)) and f2(n) = O(g(n)) then f1(n)= Θ (f2(n) ).

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Let f1 and f2 be asymptotically positive non-decreasing functions. Prove or disprove each of the following...
Let f1 and f2 be asymptotically positive non-decreasing functions. Prove or disprove each of the following conjectures. To disprove, give a counterexample. a.If f1(n) = Theta(g(n)) and f2(n) = Theta(g(n)) then f1(n) + f2(n) = Theta(g(n)) b.If f1(n) = O(g(n)) and f2(n) = O(g(n))then f1(n) = O(f2(n))
Let f(n) and g(n) be asymptotically positive functions. Prove or disprove each of the following conjectures. f(n) = O(g(n)) implies g(n) = Ω(f(n)) . f(n) = O(g(n)) implies g(n) = O(f(n)). f(n) + g(n) = Θ(min(f(n),g(n))).
4. Let fín) and g(n) be asymptotically positive functions. Prove each of the following statements A. fin)-O(g(n)) if and only if fin) *gn)g(n)) B. fn) - Og(n if and only if fin)2- O(g(n)?)
4. Let fín) and g(n) be asymptotically positive functions. Prove each of the following statements A. fin)-O(g(n)) if and only if fin) *gn)g(n)) B. fn) - Og(n if and only if fin)2- O(g(n)?)
Part I. (30 pts) (10 pts) Let fin) and g(n) be asymptotically positive functions. Prove or disprove each of the following statements T a、 f(n) + g(n)=0(max(f(n), g(n))) 1. b. f(n) = 0(g(n)) implies g(n) = Ω(f(n)) T rc. f(n)- o F d. f(n) o(f(n)) 0(f (n)) f(n)=6((f(n))2)
Need help with 1,2,3 thank you.
1. Order of growth (20 points) Order the following functions according to their order of growth from the lowest to the highest. If you think that two functions are of the same order (Le f(n) E Θ(g(n))), put then in the same group. log(n!), n., log log n, logn, n log(n), n2 V, (1)!, 2", n!, 3", 21 2. Asymptotic Notation (20 points) For each pair of functions in the table below, deternme whether...
Let f (n) and g(n) be asymptotically nonnegative functions. Using the basic definition of _-notation, prove that max( f (n), g(n)) = Θ( f (n) + g(n)).
1. Let a, b,cE Z be positive integers. Prove or disprove each of the following (a) If b | c, then gcd(a, b) gcd(a, c). (b) If b c, then ged(a., b) < gcd(a, c)
Prove (using the definition of O) or disprove (via counter-example): If f(n) = O(n)), and g(n) = O(n2), then f(n) + g(n) = O(n5). Prove (using the definition of O) or disprove (via counter-example): If f(n) = O(n), and g(n) = O(n2), then fin)/g(n) = O(n).
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Prove the following using the definitions of the notations, or disprove with a specific counterexample: Theta(g(n)) = O(g(n)) Ohm(g(n)) Theta(alpha g(n) = Theta(g(n)), alpha > 0 If f(n) O(g(n)), then g(n) Ohm(f(n)). For any two non-negative functions f(n) and g(n), either f(n) Ohm(g(n)), or f(n) < O(g(n))
Let f : B → A and g : A → B be functions. (a) Prove or disprove the following statement: If g ◦ f is an injection, then f is also an injection. (b) Prove or disprove the following statement: If g ◦ f is a surjection, then f is also a surjection.