b. Suppose h(n) = 4n + 10 and f(n) = 2n³ + 1. Is ƒ(n)€0(h(n))? Use the definition of big O notation.
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- 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.Let X = {l ∈ Z | l = 5a + 2 for some integer a}, Y = {m ∈ Z | m = 4b + 3 for some integer b}, and Z = {n ∈ Z | n = 4c − 1 for some integer c}. Is X ⊆ Y ? Is Y ⊆ Z?2. Prove from the definition that 2n2 + 100n log n + 1000 = O(n2) 3. For what values of n is 50 n lg n greater than 0.5 n2? Why do we say that 0.5 n2 is asymptotically larger, if 50 n lg n is larger for many values? (Hint: you may need to graph the functions or play around with your calculator.)
- Give big-O estimates for the factorial function and the logarithm of the factorial function, wherethe factorial function f(n) = n! is defined byn! = 1 · 2·3· ... ·nwhenever n is a positive integer, and O! = 1 . For example,1 ! = 1, 2! = 1 · 2=2, 3 ! =1· 2·3 = 6, 4! = 1 · 2·3· 4=24.Note that the function n! grows rapidly. For instance,20! = 2,432,902,008,176,640,000.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.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).
- 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.Transform the following sentences to CNF. 1. ꓯ x (P(x) ꓦ Q(x)) → R(x) 2. ꓯ x ꓯ y ꓯ z (A(x,y) ꓥ A(y,z)) → A(x,z) 3. ꓯ x ꓯ y ꓱ z P(x,z) ꓥ Q(y,z) 4. ꓯ x ⌐ [(P(x) ꓥ Q(x)) ꓦ (R(x) ꓥ S(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…
- 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 ll in the values is the correct one.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. Prove that the coin changing problem exhibits optimal substructure. Design a recursive backtracking (brute-force) algorithm that returns the minimum number of coins needed to make change for n cents for any set of k different coin denominations. Write down the pseudocode and prove that your algorithm is correct.You are given a third-degree polynomial function f(x) as follows. f(x)=x3-43x+42 x0=2.4, x1=1, x2=3, n=90 Note that the answers are given with exactly 3 digits after the decimal point. Could you write the codes by using Pyhton 3 according to the questions in the picture which I uploded?
