If f(n) = theta (g(n) log n) then g(n)=0(f(n)) O True O False Use master theorem to solve the following T(n) = 2"T(n/2)+2" O Oin? log n) O 0(2") O On O Dose not apply let f be: n+V5 +n°-n° + 100 then it is O(n) O True O False let f be: n+V5 +n°–n® + 100 then it is Q(n) O True O False
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- Subject: computer science Show that: ¬q 1) p→¬q 2) (p∧r)∨s 3) s→(t∨u) 4) ¬t∧¬u where ¬ is denied.Show that x log x is O(x2) but that x2 is not O(x log x).Assume that for any integer n is greater than or equal to one prove or disprove the following a)n^2 − n + 1 is O(n) b)5^n is O(4^n) c) n(log(n))^4 is O(n^4/3)
- Prove that 0(n -1) +0(n)= 0(n). Does it follow that 0(n)= 0(n)-0(n -1)? Justify your answerHeuristics Prove or disprove: If h1(n), ..., hk(n) are admissible, so is h(n) = h1(n) + ... + hk(n)Correct answer will be upvoted else downvoted. Computer science. Allow us to signify by d(n) the amount of all divisors of the number n, for example d(n)=∑k|nk. For instance, d(1)=1, d(4)=1+2+4=7, d(6)=1+2+3+6=12. For a given number c, track down the base n to such an extent that d(n)=c. Input The principal line contains one integer t (1≤t≤104). Then, at that point, t experiments follow. Each experiment is characterized by one integer c (1≤c≤107). Output For each experiment, output: "- 1" in case there is no such n that d(n)=c; n, in any case.
- Prove or disprove: (a) 100n 2 − 65n + 10 ∈ O(n 2 ) (b) n 3 log n + 2n 2 + 3 ∈ O(n 3 )A student is interested in computing the area under the function f(x)=x over the interval [0,1] and decides to get an accurate answer using a Monte Carlo experiment (by this we are assuming the student doesn't know the answer to such an elementary mathematical question). Which of the following R codes will give such an approximate answer? M = 10^6x = runif(M)y = runif(M)mean( y <= x ) M = 10^6x = runif(M)mean( x ) M = 10^6x = rnorm(M)mean( x ) M = 10^6x = rnorm(M)y = runif(M)mean( y <= x )Are any of the following implications always true? Prove or give a counter-example.a) f(n) = Θ(g(n)) ⇒ f(n) = cg(n) + o(g(n)), for some real constant c > 0. (Here small o) b) f(n) = Θ(g(n)) ⇒ f(n) = cg(n) + O(g(n)), for some real constant c > 0.(here big O) Please complete both
- Running Time Analysis: Give the tightest possible upper bound for the worst case running time for each of the following in terms of N. You MUST choose your answer from the following (not given in any particular order), each of which could be reused (could be the answer for more than one of a) – f)): O(N2), O(N3 log N), O(N log N), O(N), O(N2 log N), O(N5), O(2N), O(N3), O(log N), O(1), O(N4), O(NN), O(N6)Fermat's "Little" Theorem states that whenever n is prime and a is an integer, a^n−1≡1modn Then 281825≡ 263 mod827. c) If a=146 and n=331, then efficiently compute 146332≡ mod331Use The Extended Euclidean Algorithm to compute281^1≡362 mod827. Then 281^825≡???mod827. c) If a=146 and n=331, then efficiently compute146^332≡ ???mod3311. Let x ∈ Z. Use a direct proof to show that if 5x2 + 8 is odd then x is odd. 2. Show by contraposition that for positive real numbers a and b, if ab − a − b ≥ 0 then a and b are both more than 1. 3. Prove by contradiction that if m and n are integers, then m2- n2 is even or m + n is odd