Download MST121 TMA04 FEED BACK FORM

M131: Discrete Mathematics
Tutor-Marked Assignment
Cut-Off Date:
Total Marks: 60
Contents :
Feed back form …………………… ………………………………….. p.2
Question 1 (6marks)……………………………………………………...p.3
Question 2 (11 marks)……………………………………………………p.3
Question 3 (10 marks)……………………………………………………p.4
Question 4 (10 marks)……………………………………………………p.5
Question 4 (16 marks)……………………………………………………p.5
Question 4 (7 marks)……………………………………………………..p.6
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M131 TMA FEED BACK FORM
[A] STUDENT COMPONENT
Student Name:
Student Number:
Section Number:
_____________________________________________________
[B] TUTOR COMPONENT
Tutor Name:
__________________________________________________________________________
QUESTION
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MARK
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Mark
TUTORS COMMENTS:
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TMA [Summer 09/10] COVERING CHAPTERS:1,3
&8
This TMA consists of six questions for a total of 60 marks. Please solve each
question in the space provided .You should show the details of your solutions and
not just write the final result.
_____________________________________________________
Question -1 : (6 marks )
Let p ,q and r be the following propositions :
p : Sandstorms occur in this region .
q : Driving is hazardous on the road .
r : Camels cross the road.
(a) ((3 marks ) Express each of the following propositions as an English sentence :
(i) (p  q)  (q  r )
(ii) ( p  r )  (q  r )
(b) (3 marks ) Write the following propositions using p, q and r and logical
connectives :
(i) Sandstorms do not occur in this region but camels cross the road and driving is
hazardous on the road .
(ii) If camels cross the road then driving is hazardous on the road if and only if
sandstorms occur in this region.
Question-2 (11 marks)
(a) ( 4 marks ) Construct the truth table for each of the propositions in Question -1 (a):
(i) (p  q)  (q  r )
(ii) ( p  r )  (q  r )
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(b) (4 marks ) (i) Show that statement :
(( p  q)  q) is a contradiction :
(ii) Show that the statements ( p  q)  ( p  r ) and p  (q  r ) are logically equivalent.
(c) (3 marks ) Determine the truth value of each of the following statements where the
domain of discourse is the set of rational numbers.
(i) x( x 2  5)
(ii) x  x 2  x 
(iii) x( x 2  x)
Question-3: (10 marks)
Using the cipher : f (p)  (p+5) (mod 26):
(a) (5 marks ) Encrypt the message: YOU ARE FINE.
(b) (5 marks ) Decrypt the message: MTB FWJ DTZ.
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Question-4: (10 marks)
(a) (6 marks ) For the numbers below write their prime factorization and compute their
gcd and lcm :
(i)138600
(ii) 11760
(b) (4 marks) Verify whether each number below is congruent to 13 modulo 19 :
(i) 2814
(ii) 4022
Question-5: (16 marks)
(a) (8 marks ) For each of the following relations on the set {a,b,c,d}, decide whether it
is reflexive , symmetric , antisymmetric and transitive:
(i) {(a,a) , (a,c) , (b,a) , (b,b) , (c,a) , (c,c) , (d,d) }
(ii) {(a,a) , (a,c) , (b.c) , (d,a) , (d ,d)}
(b) (2 marks ) Let A  {a1, a2 , a3} and B  {b1, b2 , b3 , b4} .Write the ordered pairs in the
1 0 0 1
relation R from A to B represented by the matrix 0 1 0 1
0 0 1 0
(c) (6 marks ) Let R, and S be relations on the set {1,2,3} represented by the matrices
below (where the rows and columns correspond to the integers listed in increasing
order)
1 0 1
M R  0 0 1
0 1 0
1 1 0
, M S  0 0 1
1 0 0
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Find the matrices that represent the relations:
(i)
R S
(ii)
R  S (iii)
R S
Question-6: (7 marks) Let R be a relation defined on the set of ordered pairs of
positive integers by ((a,b),(c,d))  R iff ad = bc.
(a) (3 marks ) Show that R is an equivalence relation .
(b) (4 marks ) (i) Write down the equivalence class of the pair (1,2) .
(ii) Give an interpretation of the equivalence classes of the relation R .
[ Hint : Look at the ratio a/b corresponding to (a,b)]
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