a)
f(n) = (n2 - n)/2, g(n) = 3n
bigger term in f(n) is n2 where as the biggest term in g(n) is n
we can see that f(n) >= g(n)
so, g(n) = O(f(n))
b)
f(n) = n log n, g(n) = n√n/2
bigger term in f(n) is n log n where as the biggest term in g(n) is n1.5 (because √n is n0.5)
we can see that f(n) <= g(n)
so, f(n) = O(g(n))
(g(n)), g(n) is (f(n)), or both. 1. For each of the following pairs of functions determine...
1. For each of the following pairs of functions, prove that f(n)-O(g(n)), and / or that g(n) O(f(n)), or explain why one or the other is not true. (a) 2"+1 vs 2 (b) 22n vs 2" VS (c) 4" vs 22n (d) 2" vs 4" (e) loga n vs log, n - where a and b are constants greater than 1. Show that you understand why this restriction on a and b was given. f) log(0(1) n) vs log n....
For each pair of functions determine if f(n) ? ?(g(n)) or f(n) ? ?(g(n)) or f(n) ? O(g(n)) and provide a proof as specified. For each of the following, give a proof using the definitions. 1. f(n) = log(n), g(n) = log(n + 1) 2. f(n) = n3 + nlog(n) ? n, g(n) = n4 + n 3. f(n) = log(n!), g(n) = nlog(n) 4. f(n) = log3(n), g(n) = log2(n) 5. f(n) = log(n), g(n) = log(log(n))
Compare the following pairs of functions f, g. In each case, say whether f- o(g) f-w(g), or f = Θ(g), and prove your claim. 157. f(n) -100n+logn, gn) (logn)2. 158,介f(n) = logn, g(n) = log log(n2). 159. . f(n)-n2/log n, g(n) = n(log n)2. 160·介介f(n)-(log n)106.9(n)-n10-6 . 161. (n)logn, g(n) (log nlog n 162. f(n) n2, gn) 3. Compare the following pairs of functions f, g. In each case, say whether f- o(g) f-w(g), or f = Θ(g), and prove...
1. (10 pts) For each of the following pairs of functions, indicate whether f = 0(g), f = Ω(g), or both (in which case f-6(1). You do not need to explain your answer. f(n) (n) a) n (b) n-1n+1 (c) 1000n 0.01n2 (d) 10n2 n (lg n)2 21 е) n (f) 3" (g) 4" rl. 72 i-0 2. (12 pts) Sort the following functions by increasing order of growth. For every pair of consecutive functions f(n) and g(n) in the...
For each pair of functions f(n) and g(n), indicate whether f(n) = O(g(n)), f(n) = Ω(g(n)), and/or f(n) = Θ(g(n)), and provide a brief explanation of your reasoning. (Your explanation can be the same for all three; for example, “the two functions differ by only a multiplicative constant” could justify why f(n) = n, g(n) = 2n are related by big-O, big-Omega, and big-Theta.) i. f(n) = n^2 log n, g(n) = 100n^2 ii. f(n) = 100, g(n) = log(log(log...
*9. For each of the following pairs of functions, determine the highest order of contact between the two functions at the indicated point xo: (e) f,g : R-R given by f(x)and g(x) 1+2r ro0 (f) f, g : (0, oo) → R given by f(r) = In(2) and g(z) = (z-1)3 + In(z): zo = 1. (g) f.g: (0, oo) -R given by f(x)-In(x) and g(x)-(x 1)200 +ln(x); ro 1 x-1)200 *9. For each of the following pairs of functions,...
7. We list several pairs of functions f and g. For each pair, please do the following: Determine which of go f and fog is defined, and find the resulting function(s) in case if they are defined. In case both are defined, determine whether or not go f = fog. (a) f = {(1,2), (2,3), (3, 4)} and g = {(2,1),(3,1),(4,1)). (b) f = {(1,4), (2, 2), (3, 3), (4,1)} and g = {(1, 1), (2, 1), (3, 4),(4,4)}. (c)...
Compare the asymptotic orders of growth of the following pairs of functions. log2 n and . n (n+1)/2 and n2. 2n and 3n
7. Determine whether each of these functions is one-to-one or onto. (a) f:Z + Z, f(n) 3n +1.
(1 point) Determine which of the following pairs of functions are linearly independent. NO_ANSWER 1. f(t) = 5t? + 35t, g(t) = 5t2 – 35t NO_ANSWER 2. f(t) = edt cos(ut), g(t) = edt sin(ut) ,70 NO_ANSWER 3. f(x) = 51, g(x) = 5(2-3) NO_ANSWER 4. f(t) = 3t , g(t) = 1t|