For the following palrs of functions f(n) and g(n), f(n) = n10 9(n) = 2n/2 which of the following statement(s) Is correct? (A) f(n) = 0(g(n)) (B) g(n) = 0(f(n)) (C) f(n) = 2(g(n)) (D) g(n) = (f(n)) (A) (B). (C) O (A). (D) (B) none of the above
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- The question describes a function S(k) which is defined as the sum of the positive divisors of a positive integer k, minus k itself. The function S(1) is defined as 1, and for any positive integer k greater than 1, S(k) is calculated as S(k) = σ(k) - k, where σ(k) is the sum of all positive divisors of k. Some examples of S(k) are given: S(1) = 1 S(2) = 1 S(3) = 1 S(4) = 3 S(5) = 1 S(6) = 6 S(7) = 1 S(8) = 7 S(9) = 4 The question then introduces a recursive sequence a_n with the following rules: a_1 = 12 For n ≥ 2, a_n = S(a_(n-1)) Part (a) of the question asks to calculate the values of a_2, a_3, a_4, a_5, a_6, a_7, and a_8 for the sequence. Part (b) modifies the sequence to start with a_1 = k, where k is any positive integer, and the same recursion formula applies: for n ≥ 2, a_n = S(a_(n-1)). The question notes that for many choices of k, the sequence a_n will eventually reach and remain at 1, but this is not always the case. It asks to find, with an explanation, two specific…The binomial coefficient C(N,k) can be defined recursively as follows: C(N,0) = 1, C(N,N) = 1, and for 0 < k < N, C(N,k) = C(N-1,k) + C(N - 1,k - 1). Write a function and give an analysis of the running time to compute the binomial coefficients as follows: A. The function is written using dynamic programming.% Define the function f(x) f = @(x) -1*(x < 0) + 1*(x >= 0 & x <= 2); % Set the maximum value of N Nmax = 512; % Initialize the x-axis values for plotting x = linspace(-1,3,1000); % Compute the Fejér sums for various values of N F = zeros(length(x), Nmax); for N = 1:Nmax % Compute the Nth Fejér sum Sn = @(x) 0; for n = 1:N bn = (-1)^(n+1) / (n*pi); Sn = @(x) Sn(x) + bn*sin(n*pi*x); end FN = @(x) (1/N) * Sn(x); % Evaluate the Fejér sum at each point on the x-axis I tried to submit this to MATLAB, but can't seem to get it right.
- Please help me with these question. SHow all you work. Thank you 1. Prove that∀k ∈ N, 1k + 2k + · · · + nk ∈ Θ(nk+1). 2. Suppose that the functions f1, f2, g1, g2 : N → R≥0 are such that f1 ∈ Θ(g1) and f2 ∈ Θ(g2).Prove that (f1 + f2) ∈ Θ(max{g1, g2}).Here (f1 + f2)(n) = f1(n) + f2(n) and max{g1, g2}(n) = max{g1(n), g2(n)}Given g = {(1,c),(2,a),(3,d)}, a function from X = {1,2,3} to Y = {a,b,c,d}, and f = {(a,r),(b,p),(c,δ),(d,r)}, a function from Y to Z = {p, β, r, δ}, write f o g as a set of ordered pairs.The binomial coefficient C(N,k) can be defined recursively as follows: C(N,0) = 1, C(N,N) = 1, and for 0 < k < N, C(N,k) = C(N-1,k) + C(N - 1,k - 1). Write a function and give an analysis of the running time to compute the binomial coefficients as follows: A. The function is written recursively.
- Transform the following sentences to CNF. 1. ꓯ x (P(x) ꓦ Q(x)) → R(x) 2. ꓯ x ꓯ y ꓯ z (A(x,y) ꓥ A(y,z)) → A(x,z) 3. ꓯ x ꓯ y ꓱ z P(x,z) ꓥ Q(y,z) 4. ꓯ x ⌐ [(P(x) ꓥ Q(x)) ꓦ (R(x) ꓥ S(x))]Consider the function f : N × N → N given byf(m, n) = 2m-1(2n − 1), (m, n) ∈ N × NShow that f is bijectiveWhich functions are one-to-one? Which functions are onto? Describe the inversefunction for any bijective function.(a) f : Z → N where f is defined by f (x) = x4 + 1(b) f : N → N where f is defined by f (x) = { x/2 if x is even, x + 1 if x is odd}(c) f : N → N where f is defined by f (x) = { x + 1 if x is even, x − 1 if x is odd}
- The sequence of values: n, c(n), c(c(n)), c(c(c(n))), c(c(c(c(n)))),... is known as the hailstone sequence starting at n. Implement c(n) as a function. Use predicates to test the parity of each input n.How do we define that a function f(n) has an upper bound g(n), i.e., f(n) ∈ O(g(n))?Give an example of a function from Z to N that is one-to-one, but not onto.