Download MT1101 Three hours The total number of marks on the paper is 80

MT1101
Three hours
The total number of marks on the paper is 80. A further 20 marks are available from work during
the semester making a total of 100.
UNIVERSITY OF MANCHESTER
SETS, NUMBERS AND FUNCTIONS: SAMPLE EXAM
Someday
Sometime
Answer ALL SIX questions in Section A (26 marks in all) and THREE of the five questions in
Section B (54 marks in all).
Electronic calculators may be used, provided that they cannot store text.
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P.T.O.
MT1101
SECTION A
Answer ALL SIX questions
A1. Construct truth tables for the statements:
1. P ⇐ Q
2. (not P ) or (not Q)
3. P and (not Q)
4. not (P or Q).
[4 marks]
A2. Prove or disprove each of the following statements:
1. ∃y ∈ R, ∀x ∈ R, x + y = 0
2. ∀x ∈ R, ∃y ∈ R, x + y = 0
3. ∀x ∈ R, ∃y ∈ R, xy = 1
4. ∃y ∈ R, ∀x ∈ R, xy = 1
5. ∃y ∈ R, ∀x ∈ R, xy = 0.
[5 marks]
A3. Explain what is meant by:
1. the contrapositive of a statement P ⇒ Q
2. the image of a function
3. an irrational number
4. an uncountable set.
[4 marks]
A4.
[4 marks]
A5.
[4 marks]
A6.
[4 marks]
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P.T.O.
MT1101
SECTION B
Answer THREE of the five questions
B1. Describe the induction principle for statements P (n), n ∈ Z+ . Explain how the integers 2n and
n! are defined inductively.
Prove by induction on n that
Pn
1
(1)
j=1 j = 2 n(n + 1)
(2) 3 divides 4n + 5
for all positive integers n.
[18 marks]
B2. For any function f : X → Y , define the domain and codomain of f . Describe the properties
required for f to be (i) an injection, (ii) a surjection, (iii) a bijection.
Explain what is meant by an inverse of f , and prove that f is invertible if and only if it is a
bijection.
Let h: R → R be defined by h(x) = x2 , ∀x ∈ R, and show that h is neither an injection nor
a surjection. Explain how the domain and codomain may be restricted so that h becomes (i) an
injection but not a surjection, (ii) a surjection but not an injection, (iii) a bijection. In case (iii),
give an explicit description of h−1 .
[18 marks]
B3. (i) Explain what is meant by a denumerable set, and prove that Z is denumerable. Give two
examples (without proof) of infinite sets which are not denumerable.
Give a description of Z as the union of
(1) a finite set and a denumerable set
(2) three denumerable sets.
[9 marks]
(ii)
[9 marks]
B4.
[18 marks]
B5.
[18 marks]
END OF EXAMINATION PAPER
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