Suppose that f(n) = 3ª + 2 and g(n) = 2″ +n² +3 ○ f(n) = O(g(n)) and f(n) = N(g(n)) ○ f(n) = O(g(n)) and f(n) ‡ N(g(n)) ○ f(n) ‡ O(g(n)) and f(n) = N(g(n)) ○ f(n) ‡ O(g(n)) and f(n) ‡ N(g(n))
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Select the correct answer for each given pair of functions f(n) and g(n)
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- Let A = {c, n, b}, B = {x, y} and C = {0, 1}. Find A)A X B X C B)C X B X A C)B X C X CProve that the following shows LHS = RHS : (+ 2 1 ) = 3 Given: 1=λf.λy.f y 2=λf.λy.f(f y) 3=λf.λy.f(f(f y)) M+N=λx.λy.(M x)((N x)y)Let l be a line in the x-yplane. If l is a vertical line, its equation is x = a for some real number a. Suppose l is not a vertical line and its slope is m. Then the equation of l is y = mx + b, where b is the y-intercept. If l passes through the point (x₀, y₀), the equation of l can be written as y - y₀ = m(x - x₀). If (x₁, y₁) and (x₂, y₂) are two points in the x-y plane and x₁ ≠ x₂, the slope of line passing through these points is m = (y₂ - y₁)/(x₂ - x₁). Instructions Write a program that prompts the user for two points in the x-y plane. Input should be entered in the following order: Input x₁ Input y₁ Input x₂
- Let l be a line in the x-y plane. If l is a vertical line, its equation is x 5a for some real number a. Suppose l is not a vertical line and its slope is m. Then the equation of l is y 5mx 1b, where b is the y-intercept. If l passes through the point (x0, y0,), the equation of l can be written as y 2y0 5m(x 2x0 ). If (x1, y1) and (x2, y2) are two points in the x-y plane and x1 ≠ x2, the slope of line passing through these points is m 5(y2 2y1 )/(x2 2x1 ). Write a program that prompts the user two points in the x-y plane. The program outputs the equation of the line and uses if statements to determine and output whether the line is vertical, horizontal, increasing, or decreasing. If l is a non-vertical line, output its equation in the form y 5mx 1b.Let n be the product of two large primes. Alice wants to send a message m to Bob, where gcd(m, n) = 1. Alice and Bob choose integers a and b that are relatively prime to Φ(n). Alice computes c ≡ ma (mod n) and sends c to Bob. Bob computes d ≡ cb (mod n) and sends d back to Alice. Since Alice knows a, she finds a1 such that aa1 ≡ 1 (mod Φ(n)). Then she computes e ≡ da1 (mod n) and sends e to Bob. Explain what Bob must do to obtain m and show that this works.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.
- Show that for f(n) = 2n2 and g(n) = 20n + 3n2 , f(n) is θ(g(n)). How many ways can it be shown? Also Show that for g(n) = 10n2and f(n) = n! + 3 , f(n) is Ω(g(n)). How many ways can it be shown? Discuss with the instructor.Given f(x)=x2+6x and g(x)=1−x2, find f+g, f−g, fg, and fg. Enclose numerators and denominators in parentheses. For example, (a−b)/(1+n).Give an example of a function 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 )
- Algorithm to An iterative solution to Towers of Hanoi.in: triplet S = s0, s1, s2 representing the current game stateout: triplet R = r0, r1, r2 representing the new game statelocal: pole indices a, b, z ∈ {0, 1, 2}; disc numbers g, h ∈ [2, n]; last(Q) = Q|Q|−1, if1 ≤ |Q|, otherwise, last(Q) = +∞Show that x log x is O(x2) but that x2 is not O(x log x).Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you fill in the values is the correct one. Notice how it is a lot easier to analyze the running time of…