© Didasko 2021. All rights reserved. 1MAT2DMX – Discrete Mathematics forComputer ScienceAssessment 1 – Sets, relations, functions, number systems, and combinatorics (30 + 2 marks)Task weighting: 15% Assessment Objectives▪ Perform calculations related to computer science using sets, functions and relations▪ Understand applications of these topics into computer scienceThis is an INDIVIDUAL assignment. Students are not permitted to work in a group whenwriting this assignment.Copying, PlagiarismThis is an individual assignment. Students are not permitted to work in a group when writingthis assignment. Plagiarism is the submission of another person’s work in a manner that givesthe impression that the work is their own. La Trobe University treats plagiarism seriously.When detected, penalties are strictly imposed.Further information can be found on http://www.latrobe.edu.au/students/academicintegrity/explanation/plagiarismSubmission Guidelines• Your assignment submission should be typed, not written/drawn by hand (except forgraphics).• Submit the electronic copy of your assignment through the subject LMS.• Late submission: Submission after the deadline will incur a penalty of 5% of theavailable marks for that task per day capped at 5 days. No assignment will beaccepted after 5 days. If you have encountered difficulties that lead to latesubmission or no submission, you should apply for special consideration.Submitting your assignmentWhen you have completed your answers, submit the assessment on the Learning Portal.You should submit the following:Submit your answers in a Word or pdf document called xxx_MAT2DMX_Assessment1(where xxx is your student ID number). 2 © Didasko 2021. All rights reserved.SymbolsThe following list of symbols are provided for students to use. They are not compulsory and may notbe sufficient for students’ solutions. Similar symbols are acceptable but must be readable.Sets: ,∪,∩, , , , , , , , ⫅, ⫋, ℕ ℤ ℚ ℝOthers: ≠≥≤÷× ∞Note: Answers for some questions need to be explained/justified. Graphs for Question 3 and 4need to be done with Excel.Question 1: 2 marksLet 𝐴 = {5, 6, 9, 10, 11, 12} and 𝐵 = {𝑛 ∈ 𝑁 | 𝑛 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 5; 2 ≤ 𝑛 ≤ 19}.(a) List all elements of the set 𝐵 in the set notation 𝐵 = {… }.(b) List all elements of the sets 𝐴 ∩ B and 𝐴 ∪ 𝐵 (in set notation {… }).(c) List all elements of the power set 𝑃(𝐵) of 𝐵 (in set notation {… }).(d) What is the cardinality of Cartesian product 𝐴 × 𝐵?Explanation is not required for this question.Question 2: 2 marksLet 𝐴 = {Banana, Potato, Pumpkin} and 𝐵 = {Fruid, Vegetable}. Decide whether each of thefollowing is a relation from 𝐴 to 𝐵. Explain (Set up the answer as in Question 9 of Week 1 Practicalquestions. There is no mark for an answer “Yes”/“No” without explanation).(a) X = {(Banana, Fruid), (Potato, Vegetable), (Banana, Vegetable), (Pumpkin, Fruid)}(b) Y = {(Banana, Fruid), (Potato, Meat), (Potato, Vegetable)}Question 3: 3 marks(a) Let 𝑓(𝑥) = -𝑥3 + 2𝑥2 – 1 and 𝑔(𝑥) = 𝑥2 – 2𝑥 + 3. Simplify (i)(ii)𝑓(𝑥) + 𝑔(𝑥)𝑓(𝑥) × 𝑔(𝑥) In both questions, show all steps in details. See Question 13 of Week 1 Practical questionsfor an example (By simplifying, your final answers should not have parenthesis and thereshould NOT be more than one term for each index of 𝑥. For example, -𝑥4 + 2𝑥4 shouldbe written as 𝑥4.)(b) Let 𝑓(𝑥) = -𝑥3 + 2𝑥2 – 1. Calculate 𝑓(𝑥) for each value of 𝑥 in the following table (seealso the next page). 𝑥𝑓(𝑥)-2-1.8-1.6-1.4-1.2-1-0.8-0.6-0.4 © Didasko 2021. All rights reserved. 3 -0.200.20.40.60.811.21.41.61.82 Table 1Then use Excel (Scatter) to plot graph of 𝑓(𝑥) = -𝑥3 + 2𝑥2 – 1. Label axes with “𝑥-axis”and “𝑦-axis”. Copy and paste graph into your answer document.Question 4: 3 marks(a) Simplify the expression 𝑥-2𝑦3 × (𝑥2𝑦3)2. Explain what rules you use in each step. Answerwithout explanation will not get mark.(b) Let 𝑓(𝑥) = 𝑙𝑜𝑔3(3𝑥 – 2). Calculate 𝑓(𝑥) (up to 2 decimal places) for each value of 𝑥 in thefollowing table. 𝑥𝑓(𝑥)11.5234567891011121314151617181920 Table 24 © Didasko 2021. All rights reserved.Then use Excel (Scatter) to plot graph of 𝑓(𝑥) = 𝑙𝑜𝑔2(2𝑥 + 3). Label axes with “𝑥-axis” and“𝑦-axis”. Copy and paste graph into your answer document.Question 5: 4 marks(a) Convert 101012 to decimal.(b) Convert 4𝐵𝐹16 to decimal.(c) Convert 193 to binary.(d) Convert 1213 to octal.Set up your answer (See Week 2 Practical questions). Answer without detailed work will not getmark.Question 6: 2 + 1 + 1 marks:Consider the following addition in the base 8: 72358 + 7648. We can perform the addition as follow,using table for alignment.Table 3Thus, 72358 + 7648 = 102218.In this table, carrying values are placed in a separated row instead of being written as subscriptionfor your typing convenience (Also it is how machine is modelized). Writing carrying values assubscription is only used by human.Using similar format, perform following addition in corresponding systems.(a) 1101012 + 111012.(b) 7358 + 2618(c) 8𝐶𝐹16 + 𝐵2916Your answers must include table with all fours row as in the example above.Question 7: 1 + 1 + 1 marks(a) A company introduces a type of barcode containing 2 capital letters followed by 5 digits.(i) How many different barcodes can that company generate? Explain.(ii) How many would there be if letters could not be repeated but digits could be?Explain.(b) The company now introduces new type of barcode in which each barcode contains either2 or 3 capital letters following by either 5 or 6 digits. How many different barcodes can that companygenerate? Explain.(Explain: Set up the answer as in Examples 3 and 4 of Chapter 3 Handout).Question 8: 2 + 1 marksYou are asked to use Inclusive Exclusive Principle for this question. To explain, set up youranswer as in Questions 1-3 of Week 3 Practical questions. Clearly shown 4 separated step.(a) In MAT2DMX class, 11 students know Java, 17 know Python. There are 23 students whoknows at least one language Java or Python. How many students in this class know both Java andPython? Explain.(b) In a class of 27 students, 8 students can speak Japanese, 9 can speak French, and 4 canspeak both French and Japanese. How many students in this class cannot speak French orJapanese (neither French nor Japanese)? Explain. summand 17235summand 2764carrying1110result10221 © Didasko 2021. All rights reserved. 5Question 9: 1 + 1 marks(a) How many different strings of 5 characters can be made from 7 characters A, B, C, D, E,F, G, H if each character appears at most one time in each string? Such a string is “ABCEF”. Explainand evaluate your answer.(b) A student needs to enrol 4 subjects from 9 available subjects for her last semester. Howmany different choices of 4 subjects that she can make? Explain and evaluate your answer.Question 10: 2 + 2 marksAn IT consultant has 7 clients to visit a day. Two of clients are banks named A and B. The consultantcan visit clients one by one in any order.(a) How many arrangements such that she can visits exactly 3 clients between 2 bank clientsA and B?(b) How many arrangements such that she can visits at least 3 clients between 2 bank clientsA and B?Explain your answer.6 © Didasko 2021. All rights reserved.Marking rubric: QuestionsPartsTotalBreak down12a0.5b0.5c0.5d0.522a1b133a11b0.50.543a11b0.50.554a1b1c1d164a2b1c173a11b183a2b192a1b1104a2b2Total30

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