now that f(n)-11n²+9 is in (1). se C=121.
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- Show that 5n+5 is O(n). (pick c and n0)Show that if f(x) is O(x), then f(x) is O(x2).What are the complexities of the following code segments in terms of n? Give an upper bound. a) int i=1;while (i<= n) { int j = i; while (j > 0) j = j/2;i++; } b) int i,j s=0; for (i=0; i<n; i++) { i--; s++; if (s == n) { i++; s = 0; } } c) while (n > 0) { for (int i=0; i<n; i++) sum++; n = n/2; }
- Prove the following: a. [(a mod n) - (b mod n)] mod n = (a - b) mod n b. [(a mod n) * (b mod n)] mod n = (a * b) mod nDescribe each of the following sets.a. {x ∈ R|−5 < x < 1}b. {x ∈ Z|−5 < x < 1}c. {x ∈ Z−|−5 < x < 1}In n=4, what set(s) of cube(s) is(are) equivalent to the set {0000, 0010, 01x1, 0110, 1000, 1010}? (equivalent= they cover the same set of minterms)
- Prove or disprove "Suppose that m and n are integers. If m > n ≥ 0, then gcd (m, n) = gcd (m − n, n)."Given a collection of numbers that might contain duplicates,return all possible unique permutations.For example,[1,1,2] have the following unique permutations:[ [1,1,2], [1,2,1], [2,1,1]]""" def permute_unique(nums): perms = [[]] for n in nums: new_perms = [] for l in perms: for i in range(len(l)+1): new_perms.append(l[:i]+[n]+l[i:]).Consider f(n) = 3n^2+2n-1, mathematically show that f(n) is O(n²), (n²), and O(n²).
- Show that (p ∧ q) → r and (p → r) ∧ (q → r) is logically equivalent.Let, a1 = 3, a2 = 4 and for n ≥ 3, an = 2an−1 + an−2 + n2, express an in terms of n.Explain, with an example why the following definition, would not be suitable or useful: f(n) is Ω( g(n) ) if and only if there exists n0, such that:forall n ≥ n0, there exists c > 0 such that,f(n) ≥ c g(n)