UNIVERSITY EXAMINATION 2018/2019
YEAR I SEMESTER I EXAMINATION FOR THE DEGREE OF BACHELOR OF
SCIENCE IN MATHEMATICS AND COMPUTER SCIENCE, ACTURIAL
SCIENCE, STATISTICS AND COMPUTER SCIENCE
SPM 2100: DISCRETE MATHEMATICS
Date: Thursday, 13th December 2018
Time: 11.00am – 1.00pm
INSTRUCTIONS
Answer question one (compulsory) and any other two questions
Question One (30 Marks)
a) Define the following term
(i) Tautology
(2 marks)
(ii) Contingency
(1 mark)
(iii)
(2 marks)
A Set
b) Let A = {4, 2, 6, 1} and B = {a,
,5,4}. Evaluate:
(i) B-A
(2 marks)
(ii) 𝐴 ∩ 𝐵
(1 marks)
(iii) |𝐴 ∪ 𝐵|
(2 marks)
c) Show that if x is odd then x 2 is odd
(3 marks)
d) Determine the truth value of the following sentences
(i) If 1 + 1 = 2, then some birds can fly
(ii) Nairobi is the only city in kenya and 2 + 3 = 6
(iii)
(2 marks)
(2 marks)
1 + 1 = 31 if and only if monkeys can fly
(2 marks)
e) Determine the validity of the following argument.
(4 marks)
(𝑝 ∧ 𝑞) → 𝑟
𝑝
𝑟
f)
Prove √3 is irrational using contradiction
(5 marks)
g) Let f: ℤ → ℤ with f (x) = 𝑥 2 . Find the domain and the corresponding range of f
(2 marks)
Question Two (20 Marks)
a) Proof by mathematics induction that
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1 + 2+ 3 + ⋯+ 𝑛 =
𝑛(𝑛+1)
(7 marks)
2
b) Use truth tables to test if the following is a tautology, a contradiction or a
contingency.
(7marks)
c) Prove using contrapositive "if 3n + 2 is odd, then n is odd"
(4 marks)
d) Negate the statement ∀𝑥𝜖ℝ, ∃𝑦𝜖ℝ|𝑥 + 𝑦 = 0
(2marks)
Question Three (20 Marks)
a) Let 𝑓, ℎ: ℝ → ℝ such that f (x) = 3x + 2 and h(x) = x 2 + 2x + 1. Find
(i)
( ℎ𝑜𝑓)(𝑥)
(3 marks)
(ii)
(𝑓𝑜ℎ)(2)
(3 marks)
(iii)
𝑓 −1 (𝑥)
(2 marks)
(iv)
Is 𝑓 bijective mapping
(3 marks)
b) State the necessary and sufficient condition of the following statement “if someone
is a mother, she must be a lady”.
(3 marks)
c) Write the inverse, converse and contrapositive of the following statement "if 2 + 3 =
5, then mathematics is easy"
(6 marks)
Question Four (20 Marks)
a) Let A = {1,2,3,4} and B = {a, b, c, d). Find
(i)
P(B) and |P(B)|,where P(B) is the power set of B
(5 marks)
(ii)
AxB
(3 marks)
b) In a survey including 60 people, 25 take milk, 26 tea and 26 coffee, 9 like both milk
and tea, 11 like milk and coffee, 8 like coffee and tea and 8 like none of the three
drinks
(i) Draw a vehn diagram representing then information
(ii) Find the number of people who like all the three drinks
(iii) Find the number of people who like exactly one of the
(4 marks)
(2 marks)
(6 marks)
three drinks.
Question Five (20 Marks)
a) Consider the following propositions p: mathematicians are generous
q: Spiders hate algebra
write the compound propositions symbolized by
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(i)
𝑃𝑉¬𝑞
(2 marks)
(ii)
¬(𝑝⋀𝑞)
(2 marks)
(iii)
¬𝑝 ⇒ ¬𝑞
(2 marks)
(iv)
¬𝑝 ⟺ ¬𝑞
(2 marks)
b) Prove that (𝐴𝑈𝐵)𝑐 = 𝐴𝑐 𝑛𝐵 𝑐
(4 marks)
c) (i) Define the following functions
1. One-to one functions
(2 marks)
2. Onto functions
(2 marks)
(ii) Determine whether the following functions are injective or surjective
functions
1. 𝑓: ℝ → ℝ when 𝑓(𝑥) = 3𝑥 + 7 for all 𝑥𝜖ℝ
(2 marks)
2. 𝑔: ℝ → ℝ when 𝑔(𝑥) = 𝑥 3 for all 𝑥𝜖ℝ
(2 marks)
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