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Bit 1206: Discrete Mathematics Day Question Paper

Bit 1206: Discrete Mathematics Day

Course:Bachelor Of Science In Information Technology

Institution: Kca University question papers

Exam Year:2014

1
UNIVERSITY EXAMINATIONS: 2013/2014
ORDINARY EXAMINATION FOR THE BACHELOR OF SCIENCE
IN INFORMATION TECHNOLOGY
BIT 1206 DISCRETE MATHEMATICS DAY
DATE: APRIL, 2014 TIME: 2 HOURS
INSTRUCTIONS: Answer Question ONE and any other TWO
QUESTION ONE
(a) Define the following terms as used in linear algebra
i. A set (1 Mark)
ii. Power set (1 Mark)
(b) Let ( ) 16 2 f x x and g(x) x 8 , find
i. ( ) 1 f x (3 Marks)
ii. g f (x) (3 Marks)
(c) Given that A 5,6 and B c,d . List the elements of A B (1 Mark)
i. List the elements of (A B) (2 Marks)
ii. How many relations are from A to B (2 Marks)
(d) Let A 1,2,3,4,5 and B 0,1,4,5,9,16,20 .Let f : A B defined by
( ) 2 1 2 f x x x , state the domain, co-domain and range of f . (4 Marks)
(e) Let 1,2,3,4,5,6,7,8,9 , A 1,2,3,4 , B 2,4,6,8 and C 3,4,5,6 .List the
elements of the following sets.
i. A B C (2 Marks)
ii. (A B) A B (3 Marks)
(f) Construct a truth table for p q r p q r .Is this tautology?
Justify your answer. (5 Marks)
(g) Clearly define a Boolean algebra (3 Marks)
2
QUESTION TWO
(a) Let A 1,2,3 , B a,b,c andC x, y, z .Consider the following relation R
and S. R 1,a , 2,b , 2,c , 3,b and S a, y , b, z , c, x , c, y
i. Find the composition relation RoS (4 Marks)
ii. Find the matrices R M , S M and RoS M . (4 Marks)
iii. Find the elements of c R , c S and c (RoS) (6 Marks)
(b) Let A 2,4,5,10,12,20,25 and R a,b : a divides b
i. List the elements of R (2 Marks)
ii. Is (A, R) a poset? Justify your answer (4 Marks)
QUESTION THREE
(a) Construct a relation on the set a,b,c,d that is
i. Reflexive, symmetric and transitive. (2 Marks)
ii. Neither reflexive, reflexive ,symmetric, antsymmetric, nor transitive.
(2 Marks)
(b) State the converse, inverse and contra positive of the following conditional
statement
“If it snows today, then I will ski tomorrow” (3 Marks)
(c) Show that p q and p q are logically equivalent. (5 Marks)
(d) Consider the following propositions:
p: I bought a lottery ticket this week
q: I won the million dollar jackpot on Friday
Write the compound propositions symbolized by:
i. ( p) (2 Marks)
ii. q p (2 Marks)
iii. p q (2 Marks)
iv. p q (2 Marks)
3
QUESTION FOUR
(a) i. Draw a directed graph with the adjacency matrix
0 1 1 0
1 0 0 1
1 0 1 0
0 1 0 1
A
ii. With vertices ordering 1 2 3 4 v ,v ,v ,v (3 Marks)
iii. Use the graph in (i) to find the in-degree, out-degree and degree of each
vertex (5 Marks)
(b) Let B 1,2,3,6,10,15,30 .Define x y lcm x, y , x.y gcd x, y ,
x
x
30
.Is B
together with its operations a Boolean algebra? (6 Marks)
(c) Construct a truth table of the Boolean function f (x, y, z) x z y z
(6 Marks)
QUESTION FIVE
(a) State the domain of the following functions
i.
16
2
( ) 2 x
x
f x (2 Marks)
ii. ( ) 7 12 2 f x x x (5 Marks)
(b) If A 1,2,3,4 and B a,b,c,d , determine type of the following functions
i. f 1,a , 2,a , 3,b , 4,d (2 Marks)
ii. g 1,d , 2,b , 3,a , 4,a (2 Marks)
iii. h 1,d , 2,b , 3,a , 4,c (2 Marks)
(c) In survey of 60 people it is found that 25 like to drink milk, 26 coffee, and 26 tea.
9 like milk and tea, 11 like milk and coffee, 8 like coffee and tea and 8 like none
of the three.
i. Find the number of people who like all the three drinks (3 Marks)
ii. Use the value obtained in (i) to represent information in a Venn-diagram.
(3 Marks)
4
iii. Find the number of people who drink milk at all (1 Mark)

Bit 1206: Discrete Mathematics Day Question Paper

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