Prove (using the definition of O) or disprove (via counter-example): If f(n) = O(n)), and g(n)...
Prove or disprove the following: If f(n) =O(g(n)) then nf(n) = O(ng(n))
Prove or disprove by using Definition 2.1.3 for any n E N. Then {ann is a convergent (g) Let an = sequence. (h) Let an sequence." for any n E N. Then {an} is a convergent
8 W 1 n= 7+1 SH Prove disprove Converge or diverge defination. ch#2 fin) = f Sin (1) ocxel prove disprove the Graph of f has content o
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. For each of the following, prove using the definition of O): (a) 7n + log(n) = O(n) (b) n2 + 4n + 7 =0(na) (c) n! = ((n") (d) 21 = 0(221)
4. (15 points) Prove or disprove each one of the following statements: ·0(f(n) + g(n)) = f(n) + 0(g(n))I f(n) and g(n) are strictly positive for all n ·0(f(n) × g(n)) positive for all n f(n) 0(g(n))I f(n) and g(n) are strictly
Prove that f and g are equivalent using both the graphical and algebraic approach. If they are not, provide a counter-example that shows how they are not equivalent. https://i.gyazo.com/df6c283c040522b4baab4e52d0b91104.png
1 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(n) = 5n^2. Prove that f(n) = O(n^3). Let f(n) = 7n^2. Prove that f(n) = Ω(n). Let f(n) = 3n. Prove that f(n) =ꙍ (√n). Let f(n) = 3n+2. Prove that f(n) = Θ (n). Let k > 0 and c > 0 be any positive constants. Prove that (n + k)c = O(nc). Prove that lg(n!) = O(n lg n). Let g(n) = log10(n). Prove that g(n) = Θ(lg n). (hint: ???? ? = ???? ?)???? ?...
Prove that if f (n) = O (g (n)) and g (n) = Ohm (h (n)), it is not necessarily true that f(n) = O (h (n)). You may assume that low degree (i.e., low-exponent) polynomials do not dominate higher degree polynomials, while higher degree polynomials dominate lower ones. For example, n^3 notequalto O (n^2), but n^2 = O (n^3). Prove that if f (n) = O (g (n)) and g (n) = Ohm (h (n)), it is not necessarily...