University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
2014/15
Solutions to Exercise Set 2
This document contains answers to the exercises set in Week 12. There may well be some
errors; please do let me know of any you spot.
17 This is one of those examples which is
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
2014/15
Solutions to Exercise Set 11
This document contains answers to selected exercises from the Exercises on p42, including
all the exercises set for class 11, as well as a few others. There may well be some
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
MA103 Slide set 10
Rational and Real Numbers
1
Topics
Rational numbers
Arithmetic with rational numbers
Real numbers as infinite decimal fractions
Rationals as periodic decimal fractions
Irrational numbers
Countability of Q
Uncountability of R
2
Arithmeti
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
MA103 Slide set 8
Divisibility and Prime Numbers
1
Topics
Integers (including zero and negative integers)
Divisibility
Division with remainder
Greatest common divisor and the Euclidean algorithm
Prime numbers
Unique factorization into prime numbers
2
The
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Momma
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Meta: we oft ma, aw aw 55 ( w , A in? (67%) L M Id? C' L. facts.
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University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
MA103 Slide set 9
Modular Arithmetic
1
Topics
Clock and weekday arithmetic
Congruence modulo m
Congruence classes, the set Zm
Solving equations in Zm
Inverses
The Chinese Remainder Theorem
2
Clock arithmetic
You go to sleep at 11 oclock and you sleep for
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
2014/15
Solutions to Exercises
This document contains answers to selected exercises from the second half of MA103, including all the exercises set for classes. There may well be some errors; please do let me
kno
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
Exercises 5
Relevant parts of the Lecture notes: Sections 4.1 4.5 and 4.8.
Relevant parts of the text books: Biggs: Chapter 5 and Sections 6.1 6.3;
Eccles: Sections 8.1 8.2, 8.4 8.5, 9.1 9.2 and 10.1.
Always
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
MA 103 Introduction to Abstract Mathematics
Extra Examples Session 1
Department of Mathematics
London School of Economics and Political Science
Logical statements / truth tables
Example 1
Let p , q and r be statements. Prove, using a truth table, that
(p
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
MA 103 Slides 5 a
Introduction to Abstract Mathematics
Functions
1
Functions
Definition
Let X and Y be sets.
A function f from X to Y (also known as a mapping) is a rule
that associates to each element x of X a unique member of Y .
This definition of a fu
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
MA 103
Introduction to Abstract Mathematics
2016/17
Week 1 Introduction
1
Lecturers
Section A (Michaelmas Term):
Bernhard von Stengel
Section B (Lent term):
Graham Brightwell
Class teachers
Sally Barton
Nicholas Cron
Sam Fendrich
Philip Johnson
Barnaby Ro
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
MA 103 Slides 5 b
Introduction to Abstract Mathematics
Counting
1
Functions Important Concepts and Properties
Let X and Y be sets, and f : X Y a function.
f is surjective if: y Y , x X : f (x) = y.
f is injective if: x, x 0 X : x 6= x 0 f (x) 6= f (x 0
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
MA103 Slide set 11
Complex Numbers
1
Topics
Complex numbers a + ib
The complex plane
Geometric interpretation of addition and multiplication
The complex conjugate
Fundamental Theorem of Algebra
Roots of polynomials with real coefficients
Roots of unity
Po
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 7
1
(a) The statementab means : there is a q Z so that b = q a.
The key point for the rest of this question is that you have to engage with this definition,
rather than write down what yo
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 6
Before we start .
One of the most useful mathematical symbols is the equals sign: =. It means equals. It
means that whatever is to the left of it is equal to whatever is to the right of
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
2014/15
Solutions to Exercise Set 5
This document contains answers to Exercises, including all those set in Week 15.
Q56. (a) Fix any c R. Given e > 0, we take = e/3. Now, if x satisfies  x c < , we
have
 f (
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
2014/15
Solutions to Exercise Set 4
This document contains answers to the exercises set in Week 14. There may well be some
errors; please do let me know of any you spot.
Q42. Given any e > 0, choose N N so that
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 10
1
(a) We have (2 + 3 i ) + (5 + 4 i ) = (2 + (5) + (3 + 4) i = 3 + 7 i.
(b) We have (2 + 3 i ) (5 + 4 i ) = (2 (5) 3 4) + (2 4 + 3 (5) i = 22 7 i.
(c) For this expression we find
(2 + 3
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
2014/15
Solutions to Exercise Set 6
This document contains answers to selected Exercises, including all the exercises set in Week
16.
Q73. Following the hint, we consider the function g( x ) = f ( x ) x. Note th
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
2014/15
Solutions to Exercise Set 9
This document contains answers to some exercises, including all those set in Week 19. There
may well be some errors; please do let me know of any you spot.
Q1.
(a) Is the set
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
2014/15
Solutions to Exercise Set 13
This document contains answers to selected exercises from Exercises 2439, including all
those set in Week 13.
Q24. Suppose that ( an )nN is a sequence of nonnegative number
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
2014/15
Solutions to Exercise Set 8
This document contains answers to the exercises set in Week 18. There may well be some
errors; please do let me know of any you spot.
Q29. We claim that, if ( a b)n = e, then
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 8
1
(a) All these are done by repeatedly finding quotient and remainder using the base number
as divisor. So for base 2 we do
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University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 4
1
For every cat born at time n that are two cats born at time n + 1. So if Cn is the number of
cats born at time n, then Cn+1 = 2Cn . With C1 = 1, this number doubles in each time period
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 9
1
(a) The key point is that, if we allow pairs ( a, b) with b = 0, then the relation R is no longer
transitive.
If necessary, you should have a look back at the proof to see where where
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
2014/15
Solutions to Exercise Set 7
This document contains answers to selected Exercises, including all exercises set in Week 17.
There may well be some errors; please do let me know of any you spot.
Q3.
(a) Sho
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 5
1
2
(a) For all n N we get ( )(n) = ( (n) = (n2 ) = 2n and ( )(n) = (n) =
(2n ) = (2n )2 = 22n (= 4n ).
(b) Since we have ( )(1) = 21 = 2 and ( )(1) = 22 = 4, the functions are different
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
MA 103 Slides 1
Introduction to Abstract Mathematics
Statements, Definitions, Logic, Proofs
1
Abstract Mathematics1
Abstract Mathematics or Pure Mathematics
is mathematics that studies entirely abstract concepts;
studies abstract entities with respect t