MT131 : Discrete Mathematics Tutor Marked Assignment Cut-Off Date: November --, 2022 Total Marks: 40 Contents Feedback form ……….……………..…………..…………………….…...….. 2 Question 1 ……………………..………………………………………..……… 3 Question 2 ……………………………..………………..……………………… 4 Questio

MT131 : Discrete Mathematics
Tutor Marked Assignment
Cut-Off Date: November --, 2022 Total Marks: 40
Contents
Feedback form ……….……………..…………..…………………….…...….. 2
Question 1 ……………………..………………………………………..……… 3
Question 2 ……………………………..………………..……………………… 4
Question 3 ………………………………..………………..…………………… 5
Question 4 ………………..……………………………………..……………… 6
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MT131 TMA Feedback Form
[A] Student Component
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[B] Tutor Component
Comments
Weight
Mark
Q_1
6
Q_2
8
Q_3
8
Q_4
8
30
General Comments:
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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: [3+3 marks]
State the converse, inverse and contrapositive of each of the following statements:
I will buy a house if I can save more than $1000 per month.
¬p∧q⟶p∨q.
Determine whether the following proposition is a tautology:
¬p∨¬r⟶q⟷p⟶¬q∧r.
Q−2: [4+4 marks]
Find the set of solutions of each of the linear congruence:
x ≡ 3 (mod 5).
2x≡5 (mod 9).
Suppose f:R⟶Z, where fx=2x-1.
If A={x:1x4}, find f(A).
If B={3, 4, 5, 6, 7}, find f(B).
If C={-9, -8}, find f-1(C).
Is f one-to-one? Explain.
Q−3: [4+4 marks]
Find the smallest positive integer a in the encryption function
f(x)=(ax+7) mod 26, 0≤x≤25,
such that the function encodes the letter “H” by “C”. Use the function to encrypt the message “NEED HELP”.
Prove that: If n is an integer that is not a multiple of 3, then n2≡1 mod 3.
Q−4: [4+4 marks]
Let R be the relation on the set A={1, 2, 3, 4} defined by aRb if and only if 2a>b+1. Find the matrix representing R∘R.
Suppose that the relation R is defined on the set Z where aRb means a= ±b. Show that R is an equivalence relation.