Answer the following short questions: a) Is 21 = O(2")? b) Is 22n = O(2")? c) f(n)=O(g(n)) implies g(n)=O(f(n)) d) f(n) = O((f(n))²) %3D
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: Match the following: A) f(n)= 3n+5,g(n)=n² B) f(n) =n,g (n)= 31) C) f(n) = Jn,g(n)=5t% S(n) =…
A: Correct Option is A A-> IV B-> II C->I D-> V E->III
Q: 4. Let A = {a, b, c, d}, B = {a, y} and C = {m, n, 0, p}. Find a. A × B = b. В2 — |A x B x C| = с.
A:
Q: Problem 8. Use the Principle of Mathematical Induction to prove that 2" < n! if n is an integer…
A: hand written solution to the induction hypothesis of 2n < n! is proved in step 2.
Q: 5. Order Notation. Consider each pair of functions below. Give a formal using a direct proof…
A: Here in this multi parts question we have have two function f(n) and g(n).so using some formal proof…
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: Use limits to prove the following asymptotic statements. 1. V4n² + 1 = O(n). 2. n" = Q(n!). 3. Vn +…
A: 1. 4n2+1=Θn Limit method We have two functions f(n) ang g(n), then we setup limit quotient between f…
Q: The binomial coefficients are the integers Prove the recursion k(n) k! n! n! (kn)! k (k+ (* + 1) =…
A: Define function factorial to return the factorial of the number. Input k and n values. Print the…
Q: Question 3: prove the followings a). A U (B U C) = (A U B) U C b). A N (B N C)= (A N B) N C c).…
A: - Due to the restrictions we have on our guideline , we are allowed to answer the first three…
Q: 1. If n is any natural with n 2 3, there are at least two numbers in the set {0,.., n-1} whose…
A: It is necessary to find multiplicative inverse modulo n in the set. if n is any natural with n>=3…
Q: QUESTION 8 Show by induction that 21"-8" is divisible by 13 for all n20
A: This is a question on mathematical induction.
Q: Find f (1), f (2), f (3), and f (4) if f(n)-f(n+1)-f(n)2 +f(n)+1 where f(0)-1.
A: The function can be evaluated by consider the previous function. The type of the function in the…
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: T (n) is taken as the ground state T (1) = 10. For numbers back to T (n + 1) = 2n + T (n), n> = 1.…
A: T (n) is taken as the ground state T (1) = 10. For numbers back to T (n + 1) = 2n + T (n), n> =…
Q: Find a recurrence relation that represents the following recursive function: int rec( int n) { if(n…
A: - The question wants to know the correct choice for the provided recursive code. - The provided code…
Q: A. Prove the following: 1. n(n – 1) = 0(n²) 2. 10 log, n+ 4 = ®(log, n) 3. 3n + 5 = 0(n²) 4. 100n =…
A:
Q: Show that if f(n) is O(g(n)) and g(n) is e(h(n)) then f(n) is O(h(n))
A: θ-notation denotes tight bound. O-notation denotes upper bound. For a given function g(n), we denote…
Q: Find f(1), ƒ (2), ƒ (3), and f (4) if ƒ (n) is defined recur- sively by f(0) = 1 and for n = 0, 1,…
A: As per our guidelines, only one question or three sub parts will be answered. So, please repost the…
Q: Show that the following argument is valid for any natural number n ∈N. p1 →p2 p2 →p3 ... pn−1 →pn…
A: Argument given is, p1 -> p2, p2 -> p3,…
Q: Given f(n) E O(n), prove that f(n) E O(n²). Given f(n) E O(n) and g(n) E O(n²), prove that f(n) g(n)…
A:
Q: (b) Prove: max(f(n), g(n)) E O(f(n)g(n)), i.e., s(n) E O(f(n)g(n)). Assume for all natural n, f(n) >…
A: Given that, f(n)>=1 and g(n)>=1 s(n)= max(f(n), g(n))
Q: 4. T(n) = 15n² – 9 log n is O(n²) %3D 5. T(n) = 4n log n – 17 is O(n log n)
A: Given expressions: T(n)=15n2-9log n T(n)=4nlog n -17 To prove: T(n)=15n2-9log n is θ(n2)…
Q: QUESTION 12 Show by induction that 21" – 8" is divisible by 13 for all n
A: Mathematical induction is a mechanism for confirming natural-number solutions or establishing facts.…
Q: Question 3. Show that n³ +4n² + 1 n+3 = O(n*).
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: For each of the following pairs of functie ff and gg, circle one of the answers f \in o(g), f \in…
A: Please refer below for your reference:
Q: Consider the function f:N×N¬N defined recursively by: 1) Base case: Let meN and define 0,m) = 0 2)…
A: ANSWER: Leave the assertion alone P(n): f(n,3*m)= 3*f(n,m) given f(0,m)=0 and f(x,m)= f(x-1,m)+m.…
Q: numerous bouquet combinations, including two 5-rose bouquets (total profit of $70), and a 4-rose…
A: Roses 1 2 3 4 5 Profit $5 $15 $24 $30…
Q: answers following Prove that n + n² +n = 0(n*). Prove that n+n² + 1 = N(n*). s). Prove that n² +n+1…
A:
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: b) Prove by mathematical induction that n'-n is divisible by 3 for all n>1.
A: Here, we are going to prove n3-n is divisible by 3 for all n>1. We will prove using mathematical…
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: Consider the following sequence: aj = 2, and an = 2an-1 + 1, for n > 1. What is the value of a6?…
A: According to the information given:- We have to find out the value of a6.
Q: Solve the recurrence relation: T (n) = T (n/2) + T (n/4) + T (n/8) + n. Use the substitution method,…
A: 1.
Q: Sort these functions in an increasing order: b^n, n^b, log n, n, n'log n, n! O n, log n, n'log n,…
A: We need to arrange the given functions in order of increasing complexity. ***As per the guidelines…
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: According to the rule of sums: Suppose $$ g_{1} \in O(f_{1}), \space{} and \space{} g_{2} \in…
A: Given data: If g1∈O(f1) and g2∈O(f2) then g1+g2 ∈O(max(f1,f2)) h∈O(n log n) and g∈O(n)
Q: Compare the order of growth of 1. 2^n and n! 2. log |n| and √n
A: Here in this question we have given two function and we have to compare their order of growth.so we…
Q: 03: Consider the scores of 12 students in Computer course (X) and Mathematic course (Y) given as…
A: The Manhattan Distance between two points (X1, Y1) and (X2, Y2) is given by |X1 – X2| + |Y1 – Y2|…
Q: What is the result of the following equation (12*m - (m*n % 13 + m/n) * k/9). Assume m = 9, n = 5…
A: Using the associativity and precedence the output of the first question is : 90 Here the for loop…
Q: What are the complexities of the following code segments in terms of n? Give an upper bound. a) int…
A: Let's understand step by step : Answer (a) : int i=1;while (i<= n) { int j = i; while (j…
Q: 12. Give a Divide and Conquer algorithm for calculating xn wheren is a positive integer. [Note: If n…
A: We need to write an algorithm and Java code to evaluate the given function using divide and conquer…
Q: For each of the following expressions below, write the corresponding Big-Oh complexity Expression…
A: 1. O(2n) 2. O(n7)
Q: Let f(n) = n Let g(n) = log(n³) What is the asymptotic relation between f(n) and g(n)? Check all…
A: Given: To choose the correct option.
Q: for (j = n - 1; j> 0; j = j - 1) Factorial ( n ) { if n=1 for (i = 0; i A[ i+ 1]) return n *…
A: Time complexity : Time complexity is the total amount of time required by an algorithm to perform…
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: Given x is an array of values and x[0]=2, x[1]=3, x[2]=5, x[n]=0 , for all other values of n Given…
Q: Let f(n) = 2n and g(n) = n. a) Show that f is O(g) using specific values of C and no. b) Show that g…
A: Big-O Notation O(n): Let f(n) and g(n) be the two functions described and the function f is said to…
Show detailed workout. Give your logical reasoning, where required, in simple and comprehensive
sentences.
Step by step
Solved in 2 steps with 1 images
- Find f(1), f(2), f(3), f(4), and f(5) if f(n) is defined re-cursively by f(0) = 3 and for n = 0, 1, 2, ... a) f(n + 1) = −2f(n).b) f(n + 1) = 3f(n) + 7.Consider the following:[Int]$N1 = 2.7[Int]$N2 = 2.7What is the answer for $N1 + $N2 Group of answer choices 4 5 6 2.72.719. Let f(n) be defined recursively by f(0) = 3, and f(n + 1) = 3f(n)/3, for n = 0, 1, 2, . . . Then, f(10) = _____________.
- Show that the following statements are equivalent, where n is an integer greater than or equal to 2. Feel free to consider any of the following pairs. 1. “n is even” and “n – 1 is odd”2. “n is even” and “n2 is even”3. “n – 1 is odd” and “n2 is even” with Step By step explanation pleaseModeling the spread of a virus like COVID-19 using recursion. Let N = total population (assumed constant, disregarding deaths, births, immigration, and emigration). S n = number who are susceptible to the disease at time n (n is in weeks). I n = number who are infected (and contagious) at time n. R n = number who are recovered (and not contagiuous) at time n. The total population is divided between these three groups: N = S n + I n + R n There are several hidden assumptions here that may or may not apply to COVID-19, such as a recovered person is assumed to not be able to get the disease a second time, at least within the time window being examined. On week 0 (the start), you assume a certain small number of people have the infection (just to get things going). Everyone else is initially susceptible, and no one is recovered. There are two constants of interest: Let period = time period that it takes for an infected person to recover (recover meaning they become not infectious to…Write a program that reads N from the user and compute the following series: (N-1)-((N-2)^2) /8+ ((N-3)^3) /27-....................+1/N^3
- Show that the following argument is valid for any natural number n ∈N. p1 →p2 p2 →p3 ... pn−1 →pn ¬pn ∴¬(p1 ∨···∨pn−1................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; }complete the following pseudo coded algorithm for computing the factorial of n if n equals 0 n! is 1. else n! = ______. a) n*(n-1) b) n*(n+1) c) (n-1)! d) n*(n-1)!
- 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.Generate random matrices of size n × n where n = 100, 200, . . . , 1000. Also generate a random b ∈ R n for each case. Each number must be of the form m.dddd (Example : 4.5444) which means it has 5 Significant digits in totalShow Let f(.) be a computable, strictly monotonic function, that is, f(n+ 1) > f(n) for all n. Show B = {f(n) | n ∈ N} is recursive.