M131: Discrete Mathematics Tutor Marked Assignment Cut-Off Date: May 3rd, 2016 Total Marks: 40 Contents Feedback form ……….……………..…………..…………………….…...….. 2 Question 1 ……………………..………………………………………..……… 3 Question 2 ……………………………..………………..………………………

M131: Discrete Mathematics
Tutor Marked Assignment

Cut-Off Date: May 3rd, 2016 Total Marks: 40
Contents
Feedback form ……….……………..…………..…………………….…...….. 2
Question 1 ……………………..………………………………………..……… 3
Question 2 ……………………………..………………..……………………… 3
Question 3 ………………………………..………………..…………………… 4
Question 4 ………………..……………………………………..……………… 4
Question 5 ……………………..………………………………………..……… 5
Question 6 ……………………………..………………..……………………… 5
Question 7 ………………………………..………………..…………………… 6
Question 8 ………………………………..………………..…………………… 6
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M131 TMA Feedback Form


[A] Student Component

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Comments Weight Mark
Q_1 5
Q_2 5
Q_3 5
Q_4 5
Q_5 5
Q_6 5
Q_7 5
Q_8 5
40


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The TMA covers only chapters 1, 2, 4 and 9 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: [5 marks] Construct the truth table of the proposition .















Q−2: [5×1 marks] Determine the truth values of the following statements where the domain of discourse is the set of real numbers, justify your answers:
a) .
b) .
c) .
d) .
e) .











Q¬−3: [2+3 marks]
a) Find two finite sets A and B such that A ∈ B and A ⊂ B.
b) Give a proof of or a counterexample to the following statement:
A  (B  C) = (A  B)  (A  C).













Q−4: [5×1 marks] Convert the number 10101011 in base 2 to a number in base
a) 4.
b) 8.
c) 16.
d) 5.
e) 7.

















Q−5: [5 marks] Solve the following system of linear equations:
(24 div 7) x + (25 mod 7) y = (-2 mod 13)
(88 mod 6) x + (-10 div 6) y = (3 div 13)













Q¬−6: [5 marks] Let A = {x, y, z}, B = {a, b, c, d} and C = {1, 2, 3}. Write the ordered pairs in the relations R and S defined from A to B and B to C, respectively, represented by the below matrices. Find MS○R. What do you conclude from the result?















Q¬−7: [5 marks] Let R be the binary relation on the set of integer numbers Z defined by R = {(a, b) a + b is an even number}. Show that R is an equivalence relation. What are the sets in the partition of Z arising from R?















Q¬−8: [3+1+1+1 marks] Let A = {2, 3, …, 20} and R be the relation on A where (x, y)  R if and only if x is a divisor of y.
a) Show that R is partial order relation.
b) Find the minimal elements of A.
c) Find the maximal elements of A.
d) What is the upper bound of {3, 6}?