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Dit 204 Discrete Mathematics Question Paper

Dit 204 Discrete Mathematics

Course:Diploma In Information Technology

Institution: Kca University question papers

Exam Year:2014

UNIVERSITY EXAMINATIONS: 2013/2014
ORDINARY EXAMINATION FOR THE DIPLOMA IN
INFORMATION TECHNOLOGY
DIT 204 DISCRETE MATHEMATICS
DATE: AUGUST, 2014
TIME: 11/2HOURS
INSTRUCTIONS: Answer Any Three Questions
QUESTION ONE
a) Define the following terms
i. logic (2 Marks)
ii. proposition (2 Marks)
b) Let p and q be propositions:
p : I bought a lottery ticket this week
q : I won the million dollar jackpot on Friday
Express each of the following propositions as an English sentence
i. p
p
ii. q p (1 Mark)
iii. p q (1 Mark)
iv. p
c) Show that p
q
q
(1 Mark)
(1 Mark)
q
p
r and p
q
r are logically equivalent.
(6 Marks)
d) A mixed hockey team containing 5 men and 6 women is to be chosen from 7 men
and 9 women. In how many ways can this be done?
e) Let U
1,2,3,4,5,6,7,8,9 , A
elements of the set A
B
1,2,3,4 , B
C.
3,4,5,6 and C
(4 Marks)
1,2,7 .List the
(2 Marks)
1
QUESTION TWO
a) Briefly describe a Hasse diagram.
(3 Marks)
b) Draw a Hasse diagram of the partial ordering, R on
A
1,2,3,6,12,24,36,72
given by R
a, b : a / b
(7 Marks)
i. Find the maximal and minimal elements (2 Marks)
ii. Find the greatest and least elements if they exist (2 Marks)
iii. Find all upper bounds and lower bounds of the set B
3, 6, 12
(2 Marks)
iv. Find least upper bound and greatest lower bound of B
exist.
3, 6, 12 if they
(2 Marks)
v. Is the poset a lattice?
Give reason for your answer.
(2 Marks)
QUESTION THREE
a)
Define the following terms
i. (1 Mark)
ii.
b)
Set Subset (1 Mark)
Construct the truth table for the Boolean expression
f ( x, y , z )
c)
xyz
x yz
(5 Marks)
xy
500 people were asked about their morning vitamin intake. It was found 150 take
vitamin B , 200 take vitamin C , 165 take vitamin E , 57 take both B and
C , 125 take both B and E ,
82 take both C and E , and 52 take all three
vitamins. Use a Venn diagram to find:
i.
How many take both B and E but do not take C ?
ii. How many take none of these vitamins?
(8 Marks)
d)
Prove that
p
q
q
p is a tautology.
2
(5 Marks)
QUESTION FOUR
a)
V , E, g where V
Draw the graph, G
,E
v1 , v 2 , v3 , v 4 , v5
e1 , e2 , e3 , e4 , e5 , e6 , e7 and g is defined by
g e1
g e2
g e3 g e5
g e6 v2 , v4
g e7
i.
v 4 , v4
g e4
v1 , v 2
v 4 , v5
v1 , v3 (3 Marks)
Find the degree of each vertex (5 Marks)
ii. Find all odd degree vertices (2 Marks)
iii. Find the adjacency matrix AG of the graph above (3 Marks)
iv. (3 Marks)
v.
b)
Find the incidence matrix I G of the graph above Identify a pendant vertex of the graph above (2 Marks)
How many vertices are there in a graph with 15 edges if each vertex is of degree 3?
(2 Marks)
QUESTION FIVE
a)
Find a SOP expansion of the following Boolean function using Boolean identities.
f ( x, y , z )
b)
Given A
R
xy
z
(3 Marks)
2,3,4 .Consider the following relation in A
(1, 1), 2,2 , (2, 3), (3, 2), 4,2 , (4, 4) .
i. Draw its directed graph
(3 Marks)
ii. Is R (i) reflexive (ii) symmetric (iii) anti-symmetric
?
iii.
c)
or (iv) transitive
(4 Marks)
Find R 2
x2
Given that f x
(3 Marks)
RoR
2 and g x
x 1 .Find
fog x
i.
1
ii. f
iii. gof
(2 Marks)
x
(2 Marks)
1
(3 Marks)
x
3
4

Dit 204 Discrete Mathematics Question Paper

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