University of London The London School of Economics and Political Science
Mathematics 2
MATHS MT105B

Fall 2017
Mathematics 2
M. Anthony
MT105b, 279005b
2011
Undergraduate study in
Economics, Management,
Finance and the Social Sciences
This subject guide is for a 100 course offered as part of the University of
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 abstr
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.
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 Bart
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 elemen
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 an
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
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 class
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 ari
University of London The London School of Economics and Political Science
MATHS ma103

Winter 2011
Momma
[352+ Q 73/75, <32, 8'3 84
p95" Q! cigim, 2 .
60943 863450 W 3 51% (6+(>~d)
M/ an a W (63*), me Wm an!
a, WWW . W, 0, W pm M M L'A/wt
cfw_75 wan0,. Malawi 6 Z352 JnMI/ 95W! 56
Meta: we oft ma
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
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 nu
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 cla
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
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 enga
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
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) =
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 err
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 s
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
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
2008
10
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
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. Sup
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; ple
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 hin
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
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
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 t
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