M131: Discrete Mathematics Tutor Marked Assignment Cut-Off Date: April --, 2015 Total Marks: 40 Contents Feedback form ……….……………..…………..…………………….…...….. 2 Question 1 ……………………..………………………………………..……… 3 Question 2 ……………………………..………………..……………………

M131: Discrete Mathematics
Tutor Marked Assignment

Cut-Off Date: April --, 2015 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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QUESTION 1 2 3 4 5 6 7 8
MARK 5 5 5 5 5 5 5 5
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Tutor’s Comments:


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] Using logic laws determine whether or not is a tautology.












Q−2: [2+3 marks]
a) Write the inverse, converse, contrapositive and negation of the statement “The student will pass the course if g ≥ 50”.
b) Suppose that the universe of discourse of the atomic formula P(x, y) is the set {0, 1, 2, 3, 4, 5}. Write each of the following propositions using disjunctions, conjunctions and only one negation:
i.
ii.
iii.

 
Q¬−3: [5×1 marks] Consider the universal set N, A = {m: m|16} and B = {n: n ≤ 16 and n ≡ 17 (mod 3)}.
a) List the elements of A.
b) List the elements of B.
c) Find .
d) Find .
e) Find .











Q¬−4: [3+2 marks]
a) Find LCM (234, 1188) and GCD (234, 1188).
b) Find the binary representation of (1188)16 – (234)16.

 
Q−5: [1+2+2 marks] Consider the encrypting function f (x) = (3 – x) mod 26, 0 ≤ x ≤ 25.
a) Find f -1(x).
b) Encrypt the message “GOOD GIRL”.
c) Decrypt the message “DPJ LKJAZQKL”.















Q¬−6: [5×1 marks] List the ordered pairs in the relation R on A = {-2, -1, 0, 1, 2} where (a, b)  R if and only if
a) a + b = 1.
b) –1< b – a < 3.
c) a2 = b2.
d) |a + b| = 1.
e) b = 2a.


 
Q¬−7: [3+2 marks] Let A = {a, b, c, d} and R = {(a, a), (a, b), (b, a), (c, c), (c, d), (d, c)} be a relation on A.
a) Find the matrix representation of R2.
b) Is R2 an equivalence relation? Explain.












Q¬−8: [2+3 marks]
a) Find the transitive closure of R if
b) Suppose A = {2, 3, 6, 9, 10, 12, 14, 18, 20} and R is the partial order relation defined on A where xRy means x is a divisor of y. Draw the Hasse diagram for R.