Prove or disprove the following:
If f(n) =O(g(n)) then nf(n) = O(ng(n))
Since f(n) = O(f(n)), this means that as per Big-O definition, there exists a positive constant c such that for n >= n0, we have
0 <= f(n) <= c*g(n)
Raising to the power of n, we get
0 <= n^f(n) <= n^(c * g(n))
=> 0 <= n^f(n) <= n^(c * g(n))
And, as per Big-O definition,
nf(n) = O(ng(n))
Prove or disprove the following: If f(n) =O(g(n)) then nf(n) = O(ng(n))
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).
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
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))
Assume it is given that T1(n) = O(g1(n)) and T2(n) = O(g2(n)). Prove or disprove each one of the following claims T1(n)/T2(n) = O(g1(n)/g2(n))
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
Suppose f(n) = O(s(n)), and g(n) = O(r(n)). All four functions are positive-valued and monotonically increasing. Prove (using the formal definitions of asymptotic notations) or disprove (by counterexample) each of the following claims: (a) f(n) − g(n) = O(s(n) − r(n)) (b) if s(n) = O(g(n)), then f(n) = O(r(n)) (c) if r(n) = O(s(n)), then g(n) = O(f(n)) (d) if s(n) + g(n) = O(f(n)), then f(n) = Θ(s(n))
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...
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 : 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.
Prove or find a counterexample for the following. Assume that f (n) and g (n) are monotonically increasing functions that are always larger than 1. f (n) = o (g (n)) rightarrow log (f (n)) = o (log (g (n))) f (n) = O (g (n)) rightarrow log (f (n)) = O (log (g (n))) f (n) = o (g (n)) rightarrow 2^f (n) = o (2^g (n)) f (n) = O (g (n)) rightarrow 2^f (n) = O (2^g...