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Q−1: [4+4 marks]Determine whether the proposition is a tautology.Let be a universal set and let and be sets. Findi.The sets and .ii.The power set .

Q−2: [2+3+3 marks] Find integers and such that .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 ,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 whob)Students 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 3 is obtained.

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MT131 (M131): Discrete Mathematics

Tutor Marked Assignment

Cut-Off Date: November –, 2020

Total Marks: 40

Contents

Feedback form ……….……………..…………..…………………….………..

Question 1 ……………………..………………………………………..………

Question 2 ……………………………..………………..………………………

Question 3 ………………………………..………………..……………………

Question 4 ………………..……………………………………..………………

Question 5 ………………..……………………………………..………………

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Plagiarism Warning:

As per AOU rules and regulations, all students are required to submit their own

TMA work and avoid plagiarism. The AOU has implemented sophisticated

techniques for plagiarism detection. You must provide all references in case

you use and quote another person’s work in your TMA. You will be penalized

for any act of plagiarism as per the AOU’s rules and regulations.

Declaration of No Plagiarism by Student (to be signed and submitted by

students with TMA work):

I hereby declare that this submitted TMA work is a result of my own efforts and

I have not plagiarized any other person’s work. I have provided all references

of information that I have used and quoted in my TMA work.

Student Name : _

MT131 TMA Feedback Form

[A] Student Component

MT131 – Discrete Mathematics

2020-2021 / Fall

1

Student Name

:

[B] Tutor Component

Comments

Weight

Q_1

8

Q_2

8

Q_3

8

Q_4

8

Q_4

8

Mark

40

General Comments:

Tutor name:

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]

MT131 – Discrete Mathematics

2020-2021 / Fall

2

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
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attachment
Tags:
programming languages
discrete mathematics
Set Theory
tautology
Functions and Matrices
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