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 © University of London 2015UL15/0323 Page 1 of 9 D1

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This paper is not to be removed from the Examination Halls

UNIVERSITY OF LONDON MT2116 ZA

BSc degrees and Diplomas for Graduates in Economics, Management, Financeand the Social Sciences, the Diplomas in Economics and Social Sciences andAccess Route

Abstract Mathematics 

Tuesday, 26 May 2015 : 14:30 to 17:30

Candidates should answer SIX of the following EIGHT questions: THREE from SectionA and THREE from Section B. All questions carry equal marks.

PLEASE TURN OVER

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SECTION A

Answer any   three  questions from this section.

1.(a) Let  S  be the following statement about integers  x:

S: If  x  is an odd integer, then there is an integer  y  such that x2 = 8y + 1.

(i) Prove the statement  S .

(ii) Write down the converse of  S .

(iii) Write down the contrapositive of the converse of  S .

(iv) Prove that the converse of  S   is true.

(b) (i) Prove that for any three sets  A, B  and  C ,A ∩ (B \ C ) = (A ∩ B) \ (A ∩ C ).

(ii) Is it always the case that  A ∪ (B \ C ) = (A ∪B) \ (A ∪ C )?Justify your answer.

(c) For statements p, q  and  r, prove that the three statements

¬( p ∨ q ) =⇒ r, ¬(q ∨ r) =⇒ p  and  p ∨ q ∨ r  are equivalent.

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2.(a) Prove by induction that, for all natural numbers n,  n(n2 + 5) is amultiple of 6.

(b) For a positive real number a, let (an) be the sequence given by  a1 =  a

and, for  n ≥  1,  an+1 =  anan + 1

. Prove by induction that, for all  n ∈ N,

an =  a

(n− 1)a + 1.

(c) Let  f   : (−1, 1) → R  be given by

f (x) =  x

1 − x2.

Prove that f  is a bijection and determine its inverse.

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3.(a) For  X  = (0,∞), let  R  be the relation on  X  with

x R y   ⇐⇒  x

y ∈ Q.

Prove that R is an equivalence relation on  X . Explain why  Q  is anequivalence class of  R.

(b) Find the real number a such that  z  = −i  is a solution of  P (z ) = 0,where P (z ) is the polynomial

P (z ) = z 3 − z 2 + z  + 1 + a.

For this value of  a, find all solutions to the equation  P (z ) = 0.

(c) Let  a, b and  c  be positive integers, and let  S  be the set of all  n  ∈ N suchthat  n |a, n |b  and  n |c. (So,  S   is the set of positive common divisors of the three numbers  a, b and  c.) Explain briefly why  S  must have agreatest element. Suppose we denote this greatest element of  S  byD(a,b,c). Prove that

D(a,b,c) = gcd (gcd(a, b), c) .

[Recall that gcd(x, y) denotes the greatest common divisor of  x and  y.]

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4.   (a) Find the greatest common divisor, d, of 261 and 66 and determineintegers  x  and  y  such that  d = 261x + 66y.

(b) Prove that, if  a  and  b are integers, and 2a ≡  2b (mod 22), thena ≡  b  (mod 11).

Find all solutions (if there are any) of the following system of simultaneous equations in  Z22, justifying all your steps carefully.

9x + 2y   = 11

4x− 2y   = 14.

(c) For all  θ ∈ R, prove that

1 + ie−iθ

1 − ieiθ  = ie−iθ.

Use this result to show that

1 + sin

π7

 + i cos

π7

7− i

1 + sin

π7

− i cos

π7

7= 0.

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SECTION B

Answer any   three  questions from this section.

5.(a) What does it mean to say that  s is the  supremum  of a non-empty set of real numbers ?

Let  S  be a non-empty set of positive real numbers such that, wheneverx ∈  S , then also  x +   1

x ∈ S .

Show that  S   is   not b ounded above.

Hint:  show that, if  s−   1s

 < x ≤  s, then  x +   1x

 > s.

(b) State the Extreme Value Theorem and the Intermediate ValueTheorem.

(c) Let  f   : [0, 1] → R  be a continuous function such that  f (0) < f (1).

(i) Let  S  = {f (x)  |  0 ≤  x  ≤  1}.Show that  S   is an interval of the form [a, b].

(ii) Show that there exists  c ∈  [0, 1] such that

f (c)− f (1) =

f (0) − f (1)

c.

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6.(a) Define what it means for a sequence (an) to converge to  L ∈ R.

(b) Define a sequence (an) of real numbers by

a1 = 1 and   an+1 = a2n

4 + 1 for every  n ≥  1.

Show that 0 < an  C . Let (dn) be the sequencedefined for every n ∈ N  by

dn = max{bn, cn},

where max{x, y} =  x  for  x ≥  y, and max{x, y} =  y   for  x < y.

Show that (dn) is convergent, with limit  B.

(d) Suppose that f   : R→ R  and  g  : R→ R  are functions that are bothcontinuous at 1, with  f (1) > g(1). Let  h : R→ R  be the functiondefined by

h(x) = max{f (x), g(x)},   for all x ∈ R.

Show that  h is continuous at 1.You may use any results from the course, provided they are clearly 

stated.

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7.(a) Let (G, ) be a group with identity element  e, and let (H, ) be a groupwith identity element  e.

What does it mean to say that a function  φ :  G  →  H   is

a homomorphism ?What are the  kernel  and the  image  of a homomorphism ?

Show that, for any homomorphism  φ  :  G  →  H , we have  φ(e) = e. Alsoshow that, for any element  g ∈  G,  φ(g−1) = (φ(g))−1.

(b) Let (G, ), (H, ), and (K, ) be groups, and suppose that  φ :  G  →  H and  ψ :  H  → K  are homomorphisms.

Show that the composition  ψ ◦ φ :  G  →  K  is a homomorphism.

(c) Let  C∗ denote the set of non-zero complex numbers, and  R∗ the set of non-zero real numbers.

Explain briefly why (C∗,×), where  × is the usual multiplication, is agroup, and why (R∗,×) is a subgroup of (C∗,×).

Show that the function  φ : C∗ → R∗ defined by  φ(a + bi) = a2 + b2

(where  a and  b are real numbers) is a homomorphism. Find the kerneland the image of  φ.

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8.(a) Define what is meant by the order of an element of a group.

(b) Let (F, ◦) be a group.

(i) Show that if  x ∈  F  has infinite order then, for every positive integerm,  xm has also infinite order.

(ii) Show that if  x ∈  F  has a finite order  n  then, for every positiveinteger  m  dividing  n,  xm has order  n/m.

(c) What does it mean to say that a group is abelian  ?

Let (G, ) be an abelian group. Prove that, for every positive integer  n

and all  a, b ∈  G, we have  an

bn

= (a b)n

.

(d) Let (G, ) be an abelian group with identity  e, and let  n  be a fixedpositive integer. Show that the set  {a |  an = e}  is a subgroup of  G.

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