Determine if ∀a, b ∈ N, f(n) = a^n , g(n) = b^n , then it follows that f ∈ Θ(g).
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Determine if ∀a, b ∈ N, f(n) = a^n , g(n) = b^n , then it follows that f ∈ Θ(g).
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- Please answer the following question in depth with full detail. Suppose that we are given an admissible heuristic function h. Consider the following function: 1-h'(n) = h(n) if n is the initial state s. 2-h'(n) = max{h(n),h'(n')−c(n',n)} where n' is the predecessor node of n. where c(n',n) min_a c(n',a,n). Prove that h' is consistent.PLEASE HELP ME. kindly show all your work 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)}. 3. Let n ∈ N \ {0}. Describe the largest set of values n for which you think 2n < n!. Use induction toprove that your description is correct.Here m! stands for m factorial, the product of first m positive integers. 4. Prove that log2 n! ∈ O(n log2 n). Thank you. But please show all work and all stepsGiven 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.
- For f (a, b) = (a | b) | b(a) Simplify f (a, b).(b) Find DNF for f (a, b).(c) Is f (a, b) satisfiable?Let P2(x) be the least squares interpolating polynomial for f(x) := sin(πx) on the interval [0,1] (with weight function w(x) = 1). Determine nodes (x0,x1,x2) for the second-order Lagrange interpolating polynomial Pˆ2(x) so that P2 = Pˆ2. You are welcome to proceed theoretically or numerically using Python.Prove or disprove "Suppose that m and n are integers. If m > n ≥ 0, then gcd (m, n) = gcd (m − n, n)."
- Let, a1 = 3, a2 = 4 and for n ≥ 3, an = 2an−1 + an−2 + n2, express an in terms of n.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)}According to the rule of sums: Suppose $$ g_{1} \in O(f_{1}), \space{} and \space{} g_{2} \in O(f_{2}),$$ then $$g_{1} + g_{2} \in O(\max{(f_{1},f_{2})}).$$ Now, if $$h \in O(n \log n)\space{} and\space{g \in O(n)},$$ then, what is the order of growth of $$g + h \space{} ?$$
- Let A = {1, 2, 3, 4, 5} and B = {2, 3, 4, 5, 6, 7} and C = {a, b, c, d, e} 17. Give an example of f: A -> B, g: B -> C such that (g o f) is 1-1 but g is not 1-1.Given an n×n matrix M in which every entry is either a 0 or 1. Present an algorithm that determines if ∃i, 1 ≤ i ≤ n, such that M[i, j] = 0 and M[j, i] = 1, ∀j, 1 ≤ j ≤ n ∧ j 6= i, using examining an entry of M as the key operation. Your algorithm must examine at most 3n − ⌊lg n⌋ − 3 entries of M.Give an example of a function in n that is in O(√n) but not in Ω(√n). Briefly explain