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Sma 2100: Discrete Mathematics Question Paper

Sma 2100: Discrete Mathematics

Course:Bachelor Of Science In Actuarial Science

Institution: Dedan Kimathi University Of Technology question papers

Exam Year:2012

SMA 2100: Discrete Mathematics Page 1 of 3
KIMATHI UNIVERSITY COLLEGE OF TECHNOLOGY
UNIVERSITY EXAMINATIONS 2012/2013 ACADEMIC YEAR
FIRST YEAR FIRST SEMESTER EXAMINATION FOR THE DEGREE OF
BACHELOR OF SCIENCE IN ACTUARIAL SCIENCE
SMA 2100: DISCRETE MATHEMATICS
15th AUGUST, 2012 DURATION: 2 HOURS
Instructions To Candidates
1. Answer Question One and any other Two Questions.
2. Show clearly the number of questions attempted.
Question One (30 Marks)
a) Define the following terms:
i) universal set
ii) function
iii) real number (3 Marks)
b) Let P and Q be the sets:
P ??x : x ? 4k,k??,0 ? k ? 5?and Q ??1,4,9,16,??
Determine
i) P?Q
ii) P ?Q
iii) Q in set–builder form (6 Marks)
c) (i) List the element of each of the following sets:
? ? 2 A ? x : x ?16and 2x ? 6
B ??x : x ? x?
B ??x : x ?5 ? 5? (3 Marks)
(ii) Determine which of the sets ?, ?0? , ??? are equal. (1 Mark)
d) Given that A and B are sets such that n?U? ? 22 , ? ? 12 c n A ? , n?B? ?14,
? ? 5 c c n A ?B ? . Find ? ? c n A?B (3 Marks)
SMA 2100: Discrete Mathematics Page 2 of 3
e) (i) Define a rational number
(ii) Prove that 2 ? a is an irrational number where a??. (4 Marks)
f) (i) Let f :??? and g :???
where ? ?
1
2 1
f x
x
?
?
and ? ?
3
2
g x
x
?
?
Find which of the relations are functions. (2 Marks)
(ii) The functions ? ? 2 f x ? x ? 2 and g ? x? ? x ? 2 are defined from ? to ? .
Find g ? f , ?g ? f ???2? , and ? ? 1 f x ? (3 Marks)
g) (i) Define the following terms:
Proposition
Predicate (2 Marks)
(ii) Let p = “ misers like money” and
q = “my cousins are not greedy”.
Write the following in ordinary language as simply as possible:
p? q , p? ~ q , p ? ~q
(3 Marks)
Question Two (20 Marks)
a) Use set algebra to prove that
A ? ( A n B) = A – B (6 Marks)
b) Use the set notation to prove that
(A U B)c = A c n B c (7 Marks)
c) A college soccer team is expected to: have clean boots , practice, and arrive on
time. During a certain week 7 players had clean boots, 9 practiced and 9 arrived on
time. Six had clean boots and practiced, while 6 had clean boots and arrived on time.
Seven practiced and arrived on time. Two players did not obey any of the rules. Use a
Venn diagram to determine how many obeyed all the three rules. (7 Marks)
Question Three (20 Marks)
a) Let A = B = { 1 ,2, 3, 4 }and f1, f2, . . . f4 be relations from A to B
If f1 = { (1, 2), ( 2, 3), (3, 4) }
f2 = { ( 1, 4), (2, 1), ( 3, 2), (4, 4) }
f3 = {( 1, 2), (2, 3), (3, 4), (4, 1)}
f4 = { (1 ,4), (2, 3), (3, 2), (4, 1) }
Find which of the relations are:
(i) functions (ii) injections
(iii) surjections and (iv) bijections ( 8 Marks)
b) Let the functions f :??? and g :??? be defined by
f(x) = v (x-2) and g (x) = x2 + 4 .
Find (i) fog (ii) gof (4 Marks)
SMA 2100: Discrete Mathematics Page 3 of 3
c) Given f :??? and f(x) =
2
x
+ 3 , find g(x) and h(x) such that
(hog)(x) = f(x). Use these results to determine the inverse of f(x). (4 Marks)
d) Define f :??? , g :??? by f(x) = 2x – 4 and g (x) = x2
(i) Show that f is injective but not surjective.
(ii) Show that g is not injective.
(4 Marks)
Question Four (20 Marks)
a) State the Axiom of mathematical induction (3 Marks)
b) Proof by mathematical induction that:
(i) 3n + 2n -1 is divided by 4
(ii) (1 x 2 ) + (2 x 3) + (3 x 4) + …+ n(n + 1) = n (n+1)(n + 2)
3 (14 Marks)
c) Prove that;
If a, b, m and n are integers and if c¦a and c¦b then c¦(ma + n b). (3 Marks)
Question Five (20 Marks)
a) Consider the statement “ If this car is made in Kenya then it is a good car”
Write down a linguistic statement which is
(i) the negation
(ii) the converse
(iii) the contra positive of the statement ( 3 Marks)
b) Show that the propositions
~ (p ? q ) and (~ p ? ~ q) are logically equivalent.
Write down the equivalent statement in the laws of sets
( 6 Marks)
c) Construct the truth table for the proposition
(p? q) OR ( p? ~q)
Hence determine if the proposition is a contradiction, a tautology or logically
indeterminate
(6Marks)
d) Prove that A ? B ? A n B = A ( 5 Marks)

Sma 2100: Discrete Mathematics Question Paper

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