2015 - The University of Manchester

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Three hours
The number of marks available on this paper is 75.
THE UNIVERSITY OF MANCHESTER
SETS, NUMBERS AND FUNCTIONS
?? January 2015
??.?? – ??.??
Answer ALL FIVE questions in Section A (30 marks in total). Answer THREE of the FIVE
questions in Section B (45 marks in all). If more than THREE questions from Section B are
attempted, then credit will be given for the best THREE answers.
Electronic calculators may be used provided they cannot store text.
1
SECTION A
2
Answer ALL FIVE questions
A1. Construct truth tables for the statements:
(i)
(ii)
(iii)
(iv)
P ⇒Q
P and (not Q)
(not P ) or (not Q)
(P or Q) ⇐ R.
[6 marks]
A2. Prove or disprove each of the following statements:
(i)
(ii)
(iii)
(iv)
(v)
∃r ∈ R≥ ,
∀r ∈ R≥ ,
∀r ∈ R≥ ,
∀s ∈ R≥ ,
∃s ∈ R≥ ,
∃s ∈ R≥ ,
∀s ∈ R≥ ,
∃s ∈ R≥ ,
∃r ∈ R≥ ,
∀r ∈ R≥ ,
r − s = 1/2
r − s = 1/2
r − s = 1/2
r − s = 1/2
r > s ⇒ r2 − 1 > 5r.
[6 marks]
A3.
[6 marks]
A4.
[6 marks]
A5.
[6 marks]
SECTION B
MATH10101
Answer THREE of the five questions
B6.
(i) State the induction principle for statements P (n), where n ∈ Z+ .
(ii) Prove by induction on n that
P
1
(a) n+1
j=1 j = 2 (n + 1)(n + 2)
(b) 3 divides n3 − n
for all positive integers n.
[15 marks]
B7.
(i) State necessary and sufficient conditions on integers a, b, c, d for the fractions a/b and c/d to
represent the same rational number. Describe how to express a/b in lowest terms.
√
(ii) State√the equation satisfied by 3 2 which confirms that it is an algebraic number, and prove
that 3 2 is not rational.
(iii) Calculate, as a fraction in lowest terms, the rational number represented by the recurring
infinite decimal 3.141515.
[15 marks]
B8.
[15 marks]
B9.
[15 marks]
B10.
[15 marks]
END OF EXAMINATION PAPER
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