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

M131: Discrete Mathematics
Tutor Marked Assignment

Cut-Off Date: December --, 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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M131 TMA Feedback Form


[A] Student Component

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[B] Tutor Component


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


General Comments:







Tutor name:


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: [3+2 marks] Determine whether the statement is a tautology,
a) using the truth table method,
b) using logic laws.











Q−2: [5×1 marks] Given a predicate P(x) where the domain is the set of real numbers, determine the truth value of each of the following quantifier, Justify your answer:
a) .
b) .
c) .
d) .
e) .

 
Q¬−3: [5×1 marks] Let A×B = {(1,1), (2,2), (3,1), (3,2), (1,2), (1,4), (2,1), (2,4), (3,4)}. Find
a) P(B),
b) Number of binary relations from A to B,
c) ,
d) ,
e) .











Q¬−4: [5×1 marks] For all integers a, b, c, d, determine whether each of the following is TRUE or FALSE, Justify your answer:
a) If a | (b + c), then a | b and a | c.
b) If a | c and b | c, then (ab) | c2.
c) If a | b and c | d, then (ac) | (b + d).
d) If a | b, b | c and c | d, then a | d.
e) If a | b and c | d, then (a + c) | (b + d).

 
Q−5: [1+2+2 marks] Perform the following arithmetic operations for the numbers in the bases indicated and write out answers in base 5 notation:
a) 244 + 132 (base 5).
b) 11101 × 111 (base 2).
c) F7 – B6 (base 16).












Q¬−6: [5 marks] Find the set of all binary relations on {0, 1}.


 
Q¬−7: [3+2 marks] Let A be the set of all ordered pairs of positive integers and R be the relation defined on A where (a, b) R (c, d) means that b - a = d - c.
a) Show that R is an equivalence relation.
b) Find [(3, 5)] and [(7, 1)].












Q¬−8: [5×1 marks] Let A be the set of all integers greater than or equal to 2 and R be the relation on A where (x, y)R if and only if x is a divisor of y. Determine whether each of the following is TRUE or FALSE, Justify your answer:
a) 7 is a minimal element of A.
b) 6 is not a minimal element of A.
c) A does not have any maximal elements.
d) 2 is the least element of A.
e) R is a total order relation.