MT 131 AOU Discrete Mathematics Integers & Encryption Function Worksheet

Description

Q­−2: [2+3+3 marks] Find integers and such that .Find the GCD and LCM for the numbers 2772 and 37800.Encrypt the message “I NEED HELP” by translating the letters into numbers, applying the encryption function ,and then translating the numbers back into letters.
Q­−3: [4×2] Let and be the functions from to .Find .Is onto? Is g one-to-one? Explain you answer.Check whether exists. If it is so, define . Otherwise, provide a reason for not existence of .Check whether exists. If it is so, define . Otherwise, provide a reason for not existence of .
Q­−4: [4+4 marks] Of 32 people who save paper or bottles (or both) for recycling, 30 save paper and 14 save bottles. Find the number of people whoStudents need to answer 8 out of 10 questions in biology exam.save bothsave only paper, andsave only bottles.Find the number of the ways a student can choose the 8 questions.In how many ways can a student choose 8 questions if the first three questions are mandatory?In how many ways can a student choose 8 questions if at least 4 of the first 5 questions must be answered?
Q­−5: [2+3+3 marks] Consider an experiment of rolling two dice, and assume that each simple event in the sample space is as likely as any other find the probability that:A sum of 7 turns up;A sum of 7 or 11 turns up; A sum is greater than

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MT131 (M131): Discrete Mathematics
Tutor Marked Assignment
Cut-Off Date: November –, 2020
Total Marks: 40
Contents
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Question 1 ……………………..………………………………………..………
Question 2 ……………………………..………………..………………………
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MT131 – Discrete Mathematics
2020-2021 / Fall
1
MT131 TMA Feedback Form
[A] Student Component
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[B] Tutor Component
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Q_1
8
Q_2
8
Q_3
8
Q_4
8
Q_4
8
Mark
40
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MT131 – Discrete Mathematics
2020-2021 / Fall
2
The TMA covers only chapters 1, 2, 4, 6 and 7 and consists of eight questions for
a total of 40 marks. Please solve each question in the space provided. You should
give the details of your solutions and not just the final results.
Q−1: [4+4 marks]
a) Determine whether the proposition (((𝑝 ∨ 𝑞) ∧ (𝑝 ⟶ 𝑟)) ∧ (𝑞 ⟶ 𝑟)) ⟶ 𝑟
is a tautology.
b) Let 𝑈 = {𝑥 ∈ 𝑍| − 2 ≤ 𝑥 ≤ 10} be a universal set and let 𝐴 = {𝑥 ∈ 𝑁|1 ≤
𝑥 < 5} and 𝐵 = {𝑥 ∈ 𝑍|1 ≤ 𝑥 ≤ 6} be sets. Find ̅̅̅̅̅̅̅. i. The sets (𝐴 − 𝐵) ∪ (𝐵 − 𝐴) and 𝐴⨁𝐵 ii. The power set 𝑃(𝐴). MT131 – Discrete Mathematics 2020-2021 / Fall 3 Q−2: [2+3+3 marks] a) Find integers 𝑎 and 𝑏 such that 𝑎 + 𝑏 ≡ 𝑎 − 𝑏 (mod 5). b) Find the GCD and LCM for the numbers 2772 and 37800. c) Encrypt the message “I NEED HELP” by translating the letters into numbers, applying the encryption function 𝑓(𝑥) = (3𝑥 + 7) mod 26, 0 ≤ 𝑥 ≤ 25, and then translating the numbers back into letters. MT131 – Discrete Mathematics 2020-2021 / Fall 4 Q−3: [4×2] Let 𝑔(𝑥) = 5𝑥 2 + 3 and ℎ(𝑥) = 7𝑥 − 4 be the functions from 𝑅 to 𝑅. a) Find 𝑔 ∘ ℎ. b) Is 𝑔 onto? Is g one-to-one? Explain you answer. c) Check whether 𝑔−1 exists. If it is so, define 𝑔−1 . Otherwise, provide a reason for not existence of 𝑔−1 . d) Check whether ℎ−1 exists. If it is so, define ℎ−1 . Otherwise, provide a reason for not existence of ℎ−1 . MT131 – Discrete Mathematics 2020-2021 / Fall 5 Q−4: [4+4 marks] a) Of 32 people who save paper or bottles (or both) for recycling, 30 save paper and 14 save bottles. Find the number of people who i. save both ii. save only paper, and iii. save only bottles. b) Students need to answer 8 out of 10 questions in biology exam. i. Find the number of the ways a student can choose the 8 questions. ii. In how many ways can a student choose 8 questions if the first three questions are mandatory? iii. In how many ways can a student choose 8 questions if at least 4 of the first 5 questions must be answered? MT131 – Discrete Mathematics 2020-2021 / Fall 6 Q−5: [2+3+3 marks] Consider an experiment of rolling two dice, and assume that each simple event in the sample space is as likely as any other find the probability that: a) A sum of 7 turns up; b) A sum of 7 or 11 turns up; c) A sum is greater than 3 is obtained. MT131 – Discrete Mathematics 2020-2021 / Fall 7 Purchase answer to see full attachment Tags: discrete mathematics tautology logical form main connection contingent proportion User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.