f(n) such that f(n) ∈ O(n √ n) and f(n) ∈ Ω(n log n)) but f(n) ∈/ Θ(n √ n) and f(n) ∈/ Θ(n log n)). 2. Prove that if f(n) ∈ O(g(n)) and f(n) ∈ O(h(n)), then f(n) ^2 ∈ O(g(n) × h(n)). 3. By using the definition of Θ prove that 4√ 7n^3 − 6n^2 + 5n − 3 ∈ Θ(n 1.5 )

Q: Find y[n]=x[n]*h[n] x[0] = 2, x[1] = 3, x[2] = 5, x[n] = 0 for all other integers (n). h[0] = 2 ,…

A: Introduction Given x is an array of values and x[0]=2, x[1]=3, x[2]=5, x[n]=0 , for all other…

Q: Find y[n]=x[n]*h[n] x[0] = 2, x[1] = 3, x[2] = 5, x[n] = 0 for all other integers (n). h[0] = 2 ,…

A: Introduction Given x is an array of values and x[0]=2, x[1]=3, x[2]=5, x[n]=0 , for all other…

Q: n-17 .. n. For example, if n = 4 and k = 2, a solution is: [ [2,4], [3,4], [2,3], [1,2], [1,3],…

A: Machine independent language :- > the language which does not depend on the computer or its parts…

Q: Given f(n) ∈ Θ(n),  prove that f(n) ∈ O(n²). Given f(n) ∈ O(n) and g(n) ∈ O(n²), prove that…

A: Asymptotic notations (Θ, O, Ω) compares a time complexity function with another standard function…

Q: • Given f(n) E 0(n), prove that f(n) E O(n²). • Given f(n) E O(n) and g(n) E O(n²), prove that…

A: Task :- Prove given statements about time complexity notations.   Big-Oh notation definition :-…

Q: Prove the following, or give a counter example: (a) f(n) — О(g(n)) and g(n) — - О(h(m)). O(h(n))…

A: Big –O Notation:   Big O notation is the method to describe the rate of growth of a function. This…

Q: Given f(n) E O(n), prove that f(n) E 0(n²). Given f(n) E 0(n) and g(n) E O(n²), prove that f(n)g(n)…

A: Prove the given asymptotic

Q: non that given an integer n, it finds all binary sequences of length 2n such that the sum of the…

A: In this program the if n=3 then the binary sequence first 3 bits and last bit sum should be same.…

Q: For the following, decide whether T(n) = 0(f(n)), T(n) = $2(f(n)), T(n) = 0(f(n)), or none of the…

A: - We have to perform analysis of complexities shown.

Q: (1) (a) Show that n(n2 - 1)(n + 2) is divisible by 4 for all integer n.

A: For any integer n, either (n−1) or n must be even; and, therefore their product is also even. So we…

Q: 3. Put 0, 0, N in the right position, the same as the given example: a) n2 = .-· (n), solution: n2 =…

A:

Q: Let A = {x €N : 3 < x < 13}, B = {x € N : x is even}, and C = {x €N :x is odd}. (a) Find An B. (b)…

A: - The question is to solve the given problem of union and intersection. - Given A = {x ∈ N : 3…

Q: f(n) = log x + n² + 14n + 15 is f(n) in O(n²), O(n³), 2(n²), Q(n), O(n² ), O(n), Q(n³), O(n), or…

A: Hi there, Please find your solution below, I hope you would find my solution useful and helpful.…

Q: (b) For every integer n such that 0sn 3".

A: Given the statement we have to proof using by the exhaustion. For every integer n such that 0&lt;= n…

Q: Prove or disprove the following statements: 1)Suppose that  f(n) = Θ(nlogn) and g(n) = Θ(n2/logn).…

A: 1. The given statement can be prove f(n) = Θ(nlogn) and g(n) = Θ(n2/logn). Then, h(n) = f(n) x g(n)…

Q: (1) (a) Show that n(n2 – 1)(n + 2) is divisible by 4 for all integer n. (b) Prove by contradiction :…

A: n(n²−1)(n+2) is the one of the function and  n²-m² is the another function.

Q: 1. Show that f(n)= n²+2n+1 is O (n²). 2. Show that f(n)= n²+2n+1 is not O(n). Compute c and n.

A:

Q: Powers. For all positive integers k, we have (f(n) + g(n))* e O((S(n))*) + O((g(n))*). (11.8)

A: Claim : For all positive integers k , the claim is that :  P(k) : (f(n) + g(n))k ∈ O(f(n)k) +…

Q: 15- Prove that for every positive integer n: 1·2· 3 + 2·3 - 4+ . +n(n + 1)(n + 2) = n(n + 1)(n +…

A: principle of mathematical induction.    Given p(n) checking for n=1,    LHS:1.2.3=6…

Q: Show that log (n!) is in O(n log n)

A: Remember that log(n!) = log(1) + log(2) + ... + log(n-1) + log(n) You can get the upper bound by…

Q: Give the ordering of these asymptotic complexities in order from biggest to smallest O(n log n)…

A: According to the Question below the solution:

Q: sum1 = 0; for (k=1; k<=n; k*=2) for (j=1; j<=n; j++) sum1++; sum2 = 0; for (k=1; k<=n; k*=2) for…

A: Рrimitive  орerаtiоns  аre  bаsiс  соmрutаtiоns  рerfоrmed  by  аn  аlgоrithm.  Exаmрles  аre…

Q: For each of the following expressions, find if they are O(1), O(2"), O(n²), O(log n), O(n2022), O(n…

A: Below are the answers with reason:

Q: Let f(n) = 2n – n² + 10n – 7. Show that f(n) is O(n³) using specific values of C and no-

A: Let f(n) = 2n3 − n2 + 10n − 7. Prove that f(n) is O(n3). using specific values of C and n0.

Q: Prove that n 2  + 1  2n, where n is a positive integer with 1  n  4.

A:

Q: Find and show the complexity of these questions : 1- T(n) = T(n-1) + T(n-2) + c T(0) = 1 T(1)= 1

A: The time complexity can be calculated by different ways: Substitution method Recursive tree method…

Q: Find and show the complexity of these questions : 1- T(n) = T(n-1) + T(n-2) + c T(0) = 1 T(1)= 1

A:

Q: Put 0, 0, N in the right position, the same as the given example a) n² = .- (n), solution: n2 =…

A: First of all, we need to define the symbols as follows. θ = equal function(average case) O = worst…

Q: Running Time Analysis: Give the tightest possible upper bound for the worst case running time for…

A: The Question is incomplete: Here is the complete question below:

Q: Match the following: A) f(n) = 3n+5,g(n)=n² B) f(n)=n®".g(n)= 3@=) f(n) = \n,g(n)=5'og S(n) – \n.…

A: According to the information given:- we have to match the correct answer...

Q: Put 0, 0, N in the right position, the same as the given example a) n² = ..· (n), solution: n² =…

A: here have to determine about big o, omega, theta of function.

Q: Running Time Analysis: Give the tightest possible upper bound for the worst case running time for…

A: Stack: Stack is a linear data structure and it follows a particular order in which the operations…

Q: 2. The derivation of Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, ... (FIB(N – 1) + FIB(N – 2) If N>…

A: Fibonacci series is a series in which the nth element is the sum of previous two elements given that…

Q: .Show that f(n)= n2 +2n+1 is not O(n). Compute c and no. 2.Show that f(n)= n2+2n+1 is O (n2).

A: Given that 1.Show that f(n)= n2 +2n+1 is not O(n). Compute c and no. 2.Show that f(n)= n2+2n+1 is O…

Q: Show that if f(x) is O(x), then f(x) is O(x2).

A:

Q: Prove that for any integer n, n6k-1 is divisible by 7 if gcd(n,7)=1 and k is a positive integer.

A: Given: Prove that for any integer n, n6k-1 is divisible by 7 if gcd(n,7)=1 and k is a positive…

Q: Problem 5. Prove that n³ e O(1.2"), but n³ ¢ O(1.2"). You may use the facts that log(n) E O(n) but n…

A: Proved the given asymptotic function

Q: Is f (n) = 13n² + 6nyn log(3n²) – 4n logź(5n*) + 2n – 1 in N(n³)? Is f(n) in O(n²)? Is f (n) in…

A: Time complexity means, the time taking to execute the code. There are 3 types of Time complexities,…

Q: Given the functions |log n if n even n' otherwise. f,(n) = f2(n) = min {5000 log n² ,m } > 20000 log…

A: option:ALets assume f1(n)=0f2n is correctcase 1: n is evenf1(n)=log nf2(n)=min5000log n2,n320000if…

Q: Explain, with an example why the following definition, would not be suitable or useful: f(n) is Ω(…

A: This is about time complexity.

Q: Prove that if n is an integer, then ⌊n/2⌋ = n/2 if n is even and (n − 1)/2 if n is odd.

A: Actually, the answer has given below;

Q: 2. The derivation of Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, .. (FIB(N – 1) + FIB(N – 2) If N> 2…

A: We write function named fibonacciRecursion according to the conditions provided and return the…

Q: Show that T(n) = 0(n²) where T(n) = 1+ 2 + 3+. .. +n

A: We have to show that T(n) = O(n2)  Where T(n) = 1+2+3+.......+n

Q: If a is a symbol and n > 0, then an denotes a···a, or in words, the string of length n consisting of…

A: If a is a symbol and n &gt; 0, then an denotes a···a, or in words, the string of length n consisting…

Q: Choose the appropriate portion to complete the algorithm written to find this summation x2 x+ x6 x8…

A: The correct option is given in step 2.

Q: Log base 10 of x + log base 10 of x+3 =1  x?

A: first we solve the problem applying log formula  1. log(ab)=log(a)+log(b) 2. log1010 = 1 Then we…

Q: Consider the following code fragment. Compute the worst-case time complexity in terms of n. for…

A: Kindly note, there are two separate questions posted. As per our guidelines, we are allowed to one…

Q: Prove that any positive integer N is divisible by 11 if and only if the difference between the sum…

A: Let n = n1 n2 ...nk be athe given integer.Then n = n1 x10k + n2x10k-1+.. . + nkIf n is divisible by…

Q: Consider T(n) = 6T(n/2) + n³; then T(n) is equal to %3D Select one: O a. (n² log n) O b. O(n³) O c.…

A: Given that, T(n) = 6T(n/2) +n3

Q: Given a positive integer, N, the '3N+1' sequence starting from N is defined as follows: If N is an…

A: Introduction of the Program: The C++ program takes the number from the user as input and then the…