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M131: Discrete Mathematics
Tutor Marked Assignment
Cut-Off Date: December 7th, 2012
Total Marks: 40
Contents
Feedback form ……….……………..…………..…………………….…...…..
Question 1 ……………………..………………………………………..………
Question 2 ……………………………..………………..………………………
Question 3 ………………………………..………………..……………………
Question 4 ………………..……………………………………..………………
Question 5 ……………………..………………………………………..………
Question 6 ……………………………..………………..………………………
Question 7 ………………………………..………………..……………………
Question 8 ………………………………..………………..……………………
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M131 TMA Feedback Form
[A] Student Component
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[B] Tutor Component
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QUESTION
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MARK
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SCORE
TOTAL
Tutor’s Comments:
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This TMA covers only chapters 1, 2, 3 and 8. The TMA 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: [2+3 marks]
a) Let p and q be the propositions
p: Your car is out of gas,
q: You can't drive your car,
r: There is a traffic jam.
Write the following propositions using p, q, r and logical connectives:
i. You can't drive your car if it is out of gas or there is a traffic jam.
ii. You can drive your car if and only if it isn’t out of gas and there is no traffic
jam.
b) Prove that q   p  q   q is a tautology using:
i.
Truth table.
ii. Laws of logic.
Q−2: [2+3 marks]
a) Suppose that the universe of discourse of the atomic formula P(x, y) is {1, 2, 3,
4, 5}. Write out the following propositions using disjunctions and conjunctions:
i. x P x ,2 .
ii. x P3, y .
b) Let L(x, y) be the predicate "x likes y" and let the universe of discourse be the
set of all people. Use quantifiers to express each of the following statements:
i. Everyone likes everyone.
ii. Everyone likes someone.
iii. Someone does not like anyone.
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Q−3: [3+2 marks]
a) Consider the universal set U = {x: x is integer, 1 ≤ x ≤ 10}, A = {3, 6, 9}, B =
{x: x > 4} and C = {x: x is an odd integer}.
i. List the elements of the set B  A  C .
ii. Write the bit string of the set  A  B   C .
iii. List the elements of the set represented by the bit string 11 1101 0110.
b) By showing that each side is a subset of the other side, prove that
A  B  C    A  B    A  C  .


Q−4: [2+3 marks]
a) Consider the two integers 1170 and 5940.
i. Write the prime factorizations of the above two numbers.
ii. Find their LCM and GCD.
b) Convert
i. 11101101 from binary to decimal,
ii. 11101101 from binary to hexadecimal,
iii. 11101101 from binary to a number of base 5.
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Q−5: [2+3 marks]
a) Solve the system of equations:
(28 div 5) x + (–24 mod 7) y = 121 div 8
(28 mod 6) x + (–8 div 5) y = 111 mod 3
b) Using the encryption function f (p) = (10 – p) mod 26, 0 ≤ p ≤ 25, to:
i. Encrypt the message “HAVE A NICE DAY”.
ii. Decrypt the message “EWWH SRQHGXR”.
Q−6: [5×1 marks] List the ordered pairs in the relation R from A = {0, 1, 2, 3} to B =
{0, 1, 2, 3, 4} where (a, b)  R if and only if
a) a + b = 3.
b) –1< b – a < 2.
c) a | b.
d) |a – b| = 0.
e) b = 2a.
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Q−7: [5×1 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) Represent the given relation as a digraph.
b) Find the matrix representation of R.
c) Is R reflexive, irreflexive, symmetric, antisymmetric, transitive? Explain.
d) Calculate R2.
e) Calculate R  R-1.
Q−8: [2+3 marks]
a) Which of the following relations on {1, 2, 3, 4} are equivalence relations?
Explain.
i. {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}.
ii. {(1, 1), (1, 2), (1, 4), (2, 2), (2, 4), (3, 3), (4, 1), (4, 2), (4, 4)}.
b) Let A = {2, 3, 4, 6, 8, 9, 12, 16} and (x, y)  R if and only if x, y  A and x
divides y.
i. Draw the Hasse diagram of R.
ii. What are the maximal and minimal elements?
iii. Find lub and glb of {4, 8}.
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