Question

ONLY THE LAST ONE (4) . DISCRETE MATH

Problem 1: Show that f(n) = (n + 2) log2(n+ 1) + log2 (n3 + 1) is O(n log2 n). Problem 2: Prove that x? + 7x + 2 is 12(x°). Problem 3: Prove that 5x4 + 2x} – 1 is ©(x4). Problem 4: Find all pairs of functions in the following list that are of the same order: n2 + logn, 21 + 31, 100n3 +n2, n2 + 21, n? + n, 3n3 + 2n

Answer 1

We have classes of function in the increasing order below 1 < logn < du <n<nlogn <n² 203 2.. .. L223 ... ano what this means is say o (n) > Cloon on >ollogn) s impleis) that rate of growth of n is nights that rate of growth of logn. but note & Olnek) = 0(0) in fact o lunec) = 0 (n) constants dont affect no alla O rder of growth. also if we have two functions: Like & fin² and fantens and a nen=fith then 0(f) = 0/2 only highes so Erishna 0(f₂) = 0 (02) des function - Olnen) = 0 (0²) is considered now lets find order o of each functions, then we will compare them. f = n2e logn now no> logn. so o(fi) = 0.0²)

t2 = 2 + 30 now 3 > 2 SO OFy) - 003") d f3= 100 n3 + n² now toon²>n2 erosa - 100 n3 >n² so Of 3) = 0 (loons) = 0(m3). fu= n2t2h now an>n2 cordes of growth) elfu & Olaf stone fg = n²en3 пошо о? > - so Olfg) = 0 ( 13 +12) = 0 (13) of = 303 420 now 2 > n3 and an > 303 what this means is that sufficiently large values of 2 will enceed 3. So Off6) = 0(3 n3 + 2) = 0 (22) for n

Offy) - 01F6). 0 (F1) = 0 In?) (F2) 2 0(3) 0(63) = 0 (13) Olky) - 012") Olfg) = O(n) 0166) = 0(2") and we have O(fi) < 0(43) - 065) Offy) - Ole) < 0(f2).