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The exam paper for a university-level mathematics course, math 105(r. The exam covers various topics including statements in ordinary english, logic, sets, inequalities, induction, and set theory. The exam includes multiple-choice questions, problem-solving questions, and proof-based questions. Students are required to answer all questions in section a and three questions in section b to obtain full marks.
Typology: Exams
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Examiner: Prof. S.M. Rees, Extension 44063.
Time allowed: Two and a half hours.
Full marks will be obtained by complete answers to all questions in Section A and three questions in Section B. The best 3 answers in Section B will be taken into account.
from the equation), and determine whether each one is true.
a) For x ∈ R, (x^2 + 2x − 3 = 0) ⇔ (x = 1 ∨ x = −3) b) For x ∈ R, (x > 2) ⇒ (x > 1 ∧ x < 3)
[6 marks]
a) ∀x ∈ R, x ∈ Q. b) ∀x ∈ R, x < 3 ⇒ x^2 < 9.
[4 marks]
c) x = 1, X = (1, 5). d) x = 3, X = (−∞, 3]. e) x = 1 − 2 i, X = R. f) x =
3, X = Q. [6 marks]
b) − 1 <
2 − x 1 + x
< 1, x 6 = −1.
[5 marks]
[6 marks]
In each of the following cases, determine whether ∼ is an equivalence relation on X.
a) X = Z and x ∼ y ⇔ 3 |x − y. b) X = Z and x ∼ y ⇔ xy is even. c) X = Q and x ∼ y ⇔ xy ∈ Z d) X = C and x ∼ y ⇔ x − y = m + ni for m, n ∈ Z
[15 marks]
x 0 = 2, xn+1 = xn −
x^2 n − 3 2 xn
xn 2
2 xn
Prove the following.
(i) xn > 0 for n ≥ 0. (Hint: use induction.)
(ii) x^2 n+1 − 3 =
(x^2 n − 3)^2 4 x^2 n
for n ≥ 0. (Hint: use the formula for xn+1.)
(iii) x^2 n − 3 > 0 for n ≥ 0. (Hint: use induction and (ii).)
(iv) 1 ≤ xn+1 < xn for n ≥ 0. (Hint: use the formula for xn+1 and use (iii) for all n ∈ N.)
[15 marks]
[15 marks]
(i) Give the definition of a Dedekind cut (ii) Determine which (if any) of the following sets A are Dedekind cuts. Give brief reasons for your answers.
a) A = {x ∈ Q : x > 1 }
b) A = {x ∈ Q : x ≤ 2 }
(iii) Show that A = {x ∈ Q : x^2 + x − 1 < 0 ∨ x < 0 } is bounded above by 1 and has no maximal element, and hence, or otherwise, show that A is a Dedekind cut. Hint Show that x^2 + x − 1 is strictly increasing on [−^12 , ∞), and show that if a ∈ A with 0 < a < 1, and 0 < ε < 1 then (a + ε)^2 + a + ε − 1 < a^2 + a − 1 + 4ε.
Hence show that if it is also true that ε ∈ Q and ε < −
a^2 + a − 1 4
then a + ε ∈ A.
[15 marks]
Paper Code MATH 105(R) Page 5 of 5 END
