Let f(n) and g(n) be positive functions over the natural numbers. For each of the following claims either prove formally that the claim is correct, or disprove it by giving a counter example. a) f(n) is e(f(n/2)) . b) f(n) + g(n) is E(min(f(n),g(n)). c) f(n) + g(n) is E(max(f(n),g(n)). d) if f(n)f(n) is O(n) then f(n) is O(n).

Let f(n) and g(n) be positive functions over the natural numbers. For each of the following claims either prove formally that the claim is correct, or disprove it by giving a counter example. a) f(n) is e(f(n/2)) . b) f(n) + g(n) is E(min(f(n),g(n)). c) f(n) + g(n) is E(max(f(n),g(n)). d) if f(n)f(n) is O(n) then f(n) is O(n).

Programming Logic & Design Comprehensive

9th Edition

ISBN:9781337669405

Author:FARRELL

Publisher:FARRELL

Chapter4: Making Decisions

Section: Chapter Questions

Problem 9RQ

See similar textbooks

Similar questions

Prove using the concept of interpretations and the value of a formula vI under an interpretation I that [∀x p(x) ∨ ∃x q(x)] → ∃x[p(x) ∨ q(x)] is valid.
Prove that it is logically equal using common equivalent proposition. 1. ¬[p v (¬p ˄ q) ≡ (¬p ˄ ¬q) 2. p → (q ˄ ¬r)
1. Show that (p ∧ q) → r and (p → r) ∧ (q → r) is logically equivalent.
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)}
Let A = {x ∈ Z : x ≤ 3} and let B = {x ∈ Q : x2 = 9}. Is B ⊆ A? Give a brief reason for your answer.
Let R=ABCDEGHK and F= {ABK→C, A→DG, B→K, K→ADH, H→GE} . Is it in BCNF? Prove your answer.
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.
Rank the following functions based on their asymptotic value in the increasing order, i.e., list themas functions f1, f2, f3, . . . , f9 such that f1 = O(f2), f2 = O(f3), . . . , f8 = O(f9). Remember to writedown your proof for each equation fi = O(fi+1) in the sequence above. Functions: √n, log n, nlog n, 100n, 2n, n!, 9n, 33^3^3, n/log^2 n
Give a proof by contradiction that given integers j and k where j ≥ 2 that thenj is not divisible by k or(∨) j is not divisible by (k + 1).
1. The Fibonacci numbers F(0), F(1), F(2), are defined by the recurrence relation:F(0) = 0,F(1) = 1,F(n) = F(n-1) + F(n-2) for n >= (greater than or equal to) 2.The first few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, and 21. Prove by induction that for all n >= (greater than or equal to) 1.[F(n-1)][F(n+1)] - F(n)2 = (-1)n.Hint: Express F(n-1) in terms of F(n)and F(n-2) and express F(n+1) in terms of F(n) and F(n-1). Expand and simplify, then apply the inductive hypothesis. Handwritten only
Answer the following: This problem exercises the basic concepts of game playing, using tic-tac-toe (noughts and crosses) as an example. We define Xn as the number of rows, columns, or diagonals with exactly n X’s and no O’s. Similarly, On is the number of rows, columns, or diagonals with just n O’s. The utility function assigns +1 to any position with X3=1 and −1 to any position with O3=1. All other terminal positions have utility 0. For nonterminal positions, we use a linear evaluation function defined as Eval(s)=3X2(s)+X1(s)−(3O2(s)+O1(s)). a. Show the whole game tree starting from an empty board down to depth 2 (i.e., one X and one O on the board), taking symmetry into account. b. Mark on your tree the evaluations of all the positions at depth 2. c .Using the minimax algorithm, mark on your tree the backed-up values for the positions at depths 1 and 0, and use those values to choose the best starting move. Provide original solutions including original diagram for part a!
Answer the following: This problem exercises the basic concepts of game playing, using tic-tac-toe (noughts and crosses) as an example. We define Xn as the number of rows, columns, or diagonals with exactly n X’s and no O’s. Similarly, On is the number of rows, columns, or diagonals with just n O’s. The utility function assigns +1 to any position with X3=1 and −1 to any position with O3=1. All other terminal positions have utility 0. For nonterminal positions, we use a linear evaluation function defined as Eval(s)=3X2(s)+X1(s)−(3O2(s)+O1(s)). a. Show the whole game tree starting from an empty board down to depth 2 (i.e., one X and one O on the board), taking symmetry into account. b. Mark on your tree the evaluations of all the positions at depth 2. c .Using the minimax algorithm, mark on your tree the backed-up values for the positions at depths 1 and 0, and use those values to choose the best starting move. Provide original solution!