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The area of mathematics known as discrete mathematics deals with objects like integers that can only take distinct, separated values. Because of how dependent the world has become on computers and information technology, this mathematics subject has recently become more well-known. Graph theory, algorithms, and optimization are only a few topics covered by discrete mathematics. Its significance lies in its many real-world applications. Discrete mathematics finds application in developing effective internet routing protocols, cryptography, electronic banking, security, telecommunications, and web search engines. Many students study this subject but find it challenging to ace it without assistance.

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**Q1:**Prove or disprove: a) For all integers a, b, c, if a | b and b|c then a|c. b) For all integers a, b, c, d, if a | b and c | d, then (ac)| (b+d). c) For all integers a, b, c, if a | (b+c), then a | b and a | c. d) For all integers a, b, c, if a | bc, then a | b or a | c. e) For all integers a, b, c, if a | c and b | c, then ab |c².See Answer**Q2:**A logistics task: rectangular boxes containing mechanical components need to be transported to a site. Every box has the same width and height (60 x 10 cm). Lengths of boxes vary from 10 cm to 60 cm. All boxes need to be packed into a large container of dimensions 60 × 60 × 60 cm. In a certain shipment the boxes have the following lengths How can the boxes be packed in the container? Any empty space within the container could be filled with packing material. Design an algorithm for handling the task for n boxes of variable lengths. What is the complexity of this algorithm (see Figure 2.9)?Show your work. See Answer**Q3:**Suppose that a and b are integers, a = 4 (mod 13), and b = 9 (mod 13). Find the integer c with 0 <=c<=12 such that a) c = 9a (mod 13). b) c = 11b (mod 13). c) c= a+b (mod 13). d) c= a^2 + b^2 (mod 13). e) c = a^3 – b^3 (mod 13).See Answer**Q4:**To establish a big-O relationship, find witnesses C and k such that |f(x)| < C|g(x)|whenever x> k. Determine whether each of the following functions is 0(x). a) f(x) = 10 b) f(x) = 3x+ 7 c) f(x) =x² +x+1 d) f(x) = 5 log x e) f(x) = [x/2]See Answer**Q5:**Arrange the following functions in a list so each is big-O of the next one in the list: \text { a) } n^{3}+88 n^{2}+3, \log n^{4}, 3^{n}, n^{2} \log n, n \cdot 2^{n}, 10900 \text { b) } \log n^{2}, \log \log n, n \log n, \log \left(n^{2}+1\right), \log 2^{n}See Answer**Q6:**Distance Calculator If you know a vehicle's speed and the amount of time it has traveled, you can calculate the distance it has traveled as follows: Distance = Speed * Time For example, if a train travels 40 miles per hour for 3 hours, the distance traveled is 120 miles. Create an application with a form similar to the one shown in Figure 5-47.When the user clicks the Calculate button, the application should display an input box asking the user for the speed of the vehicle in miles-per-hour, followed by an-other input box asking for the amount of time, in hours, that the vehicle has traveled. Then it should use a loop to display in a list box the distance the vehicle has traveled for each hour of that time period. Figure 5-48 shows an example of what the application's form should look like. See Answer**Q7:**Describe an algorithm that takes a list of integers a1, a2, ..., an (n > 2) and finds the second-largest integer in the sequence by going through the list and keeping track of the largest and second-largest integer encountered.See Answer**Q8:**Create an application that allows a teacher to enter three test scores each for threestudents. The application should calculate each student's average test score and as-sign a letter grade based on the following grading scale: The application should prompt the user for each student's name and three test scores. Figure 5-54 shows an example of how the application's form might appear after all the data has been entered. See Answer**Q9:**Sort the following list showing the lists obtained at each step. a) Use insertion sort to sort list: 3, 1, 5, 7, 4. b) Use selection sort to sort list: 3, 5, 4, 1, 2.See Answer**Q10:**Describe an algorithm that takes as input a list of n integers in non-decreasing order and produces the list of all values that occur more than once. (List of integers is non-decreasing if each integer in the list is at least as large as the previous integer in list.)See Answer**Q11:**A software company sells three packages, Package A, Package B, and Package C,which retail for $99, $199, and $299, respectively. Quantity discounts are given ac-cording to the following table: Create an application that allows the user to enter the number of units sold for each software package. The application's form should resemble Figure 4-37. The application should calculate and display the order amounts and the grand total in a Label control. The Clear button must clear all text boxes and calculated labels.The Exit button must close the window. Input validation: Make sure the number of units for each package is numeric, and is not negative.Use the following test data to determine if the application is calculating properly: See Answer**Q12:**Determine the numerical values of the ß coefficients(weighting factors) in such a way that the solution of the water resources allocation problem complies with the assumed hierarchy of water users:Ul > U2 > U3 >U4>U5 > U6 > U7 See Answer**Q13:**List all the steps to search for 25 in the following list: 5, 6, 8, 12, 15, 21, 25, 31. Using: a) Binary search algorithm b) Linear search algorithmSee Answer**Q14:**Show that: a) If a and d: positive integers, then (-a) div d=-a div d if and only if d divides a. b) If n and k are positive integers, then [n/k] = [(n – 1)/k] + 1.See Answer**Q15:**Show that: a) If a |b and b|a, where a and b are integers, then a = b or a=-b. b) If a, b, and c are integers, where a + 0 and c + 0, such that ac | bc, then a | b.See Answer**Q16:**Determine the numerical values of the B coefficients(weighting factors) in such a way that the solution of the water resources allocation problem complies with the assumed hierarchy of water users:U1 > U4 > U3 > U2 See Answer**Q17:**Determine the numerical values of the ß coefficients(weighting factors) in such a way that the solution of the water resources allocation problem complies with the assumed hierarchy of water users:U5 > U1 > U2 > U3 > U4 See Answer**Q18:**Exercise 4A Write functions to perform the following tasks efficiently. Do not change the function names in the template file. (1) Take x and return x³ x² (2) Take two inputs x and y and return their sum and product, in that order. E.g. inputs 3, 4 should return 7, 12. (3) Take a list of numbers and returns their arithmetic mean, geometric mean and harmonic mean, in that order. For example, the input [4,16] should return 10.0, 8.0, 6.4. (4) Take an integer n and returns a list of the first n of the Narayana's cows sequence Cn. (This is a variant of the Fibonacci numbers, defined by C₁ = C₂ = C3 = 1, and Cn = Cn-1 + Сn-3 for n ≥ 4.) (5) Take a pair r≥ 0 and representing a point's polar coordinates and return a pair x, y representing its Cartesian coordinates. (6) Take a pair x, y representing a point's Cartesian coordinates and return a pair r, representing its polar 0 coordinates, where 0 = [0, 2π). Check that your function works for (x, y) in all quadrants.See Answer**Q19:**Exercise 4B (1) Write a function which takes a list of the coefficients of a polynomial P(x) = ª₁ + ª₁x + a² + ... + a₂x² of arbitrary degree n, and a value of x, and returns P(xo). You can use the function given in lectures, ensuring you understand how it works. (2) Use the function to evaluate (a) P₁(x) = x³ + x² + 5x + 1 at x = 2. (b) P₂(x) = 5 x² at x = √3. Are these answers exact? Explain why or why not. (Use a print statements to show the evaluation of your function, and answer the question in a comment.) (3) The Maclaurin series for the natural logarithm ln(1 + x) is given by =(−1)n+1 xn n In(1+x) : = n=1 =x- x² x³ + 2 3 for all x. Use the first five terms in this series in the Horner evaluation function at a suitable value of x to give an approximation of In 3/2. (4) (a) Use your Horner's method function to evaluate the polynomial (x - 2)4 at the point x = 2.0001. (b) Is this answer correct? (c) Give brief reasoning for this answer. (5) In week 3 we wrote a function to convert from binary to decimal. The efficiency of this function can be improved using the same principle as Horner's method. Write such a function (horner_ternary_to_dec) using the ideas of Horner's method which takes a list containing Os, 1s and 2s (representing a base-3 number) and returns the corresponding decimal integer (so the input [1,2,0] returns the integer 15).See Answer**Q20:**#2 (2+1=3 marks) Explain how using base 100, you can develop a test for divisibility by 51, and use it to determine whether 4,012,623,492 is divisible by 51 or not. Show your work.See Answer

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