University of London The London School of Economics and Political Science
Introduction to Abstract Mathematics
MATHEMATIC MA103

Fall 2015
MA 103 Slides 2
Introduction to Abstract Mathematics
Sets and Quantifiers
1
Sets
Loosely speaking, a set is a collection of objects.
A set is usually described by listing or describing its members
ins
University of London The London School of Economics and Political Science
Introduction to Abstract Mathematics
MATHEMATIC MA103

Fall 2015
MA103 extra slides week 3 / 4
Geometric proofs of summation formulas for the
first n odd numbers, natural numbers, and cubes
1
The first n odd numbers sum to n 2 1
Many of you will have noticed that w
University of London The London School of Economics and Political Science
Introduction to Abstract Mathematics
MATHEMATIC MA103

Fall 2015
MA 103 Slides 4b
Introduction to Abstract Mathematics
Using Strong Induction: Egyptian Fractions
1
Using strong induction
When using strong induction on recurrence relations such as the
Fibonacci numb
University of London The London School of Economics and Political Science
Introduction to Abstract Mathematics
MATHEMATIC MA103

Fall 2015
MA 103
Introduction to Abstract Mathematics
2017/18
Week 1 Introduction
1
Lecturers
Section A (Michaelmas Term):
Prof Bernhard von Stengel
Section B (Lent term):
Prof Graham Brightwell
Class teachers
University of London The London School of Economics and Political Science
Introduction to Abstract Mathematics
MATHEMATIC MA103

Fall 2015
MA 103 Introduction to Abstract Mathematics
Extra Examples Session 2
Department of Mathematics
London School of Economics and Political Science
Another proof
Example 1
Prove the following statement:
F
University of London The London School of Economics and Political Science
ABSTRACT MATHEMATICS
MATHEMATIC MA203

Fall 2014
140
Solutions
Solution to Exercise 5.29. (From the calculation done in the given hint, we guess that the derivative
of g at c is the map Rn h 7 2f (c)f (c)h, and we prove our claim below.)
We have for
University of London The London School of Economics and Political Science
ABSTRACT MATHEMATICS
MATHEMATIC MA203

Fall 2014
Solutions to the exercises from Chapter 4
129
Solution to Exercise 4.33. Let X = (0, 1], Y = R with the usual Euclidean metrics. Define f : (0, 1]
R by
1
(x (0, 1]).
f (x) =
x
Then f is continuous. T
University of London The London School of Economics and Political Science
ABSTRACT MATHEMATICS
MATHEMATIC MA203

Fall 2014
Solutions to the exercises from Chapter 4
121
Solutions to the exercises from Chapter 4
Solution to Exercise 4.3. That (1) implies (2 ) is immediate since (1) implies (2), and (2) implies (2 ).
Now su
University of London The London School of Economics and Political Science
ABSTRACT MATHEMATICS
MATHEMATIC MA203

Fall 2014
Solutions
134
Solutions to the exercises from Chapter 5
Solution to Exercise 5.9. We have
f (a)
(a) = det f (a)
f (b)
g(a)
g(a)
g(b)
1
1 =0
1
since the first and second rows of the matrix are linearly
University of London The London School of Economics and Political Science
ABSTRACT MATHEMATICS
MATHEMATIC MA203

Fall 2014
Solutions
Solutions to the exercises from Chapter 1
Solution to Exercise 1.7.
(D1) As d(x, y) is either 0 or 1 for all x, y X, clearly d(x, y) 0.
Also, for all x X, d(x, x) = 0 by definition.
If x, y
University of London The London School of Economics and Political Science
ABSTRACT MATHEMATICS
MATHEMATIC MA203

Fall 2014
Solutions
104
Solution to Exercise 2.34. (If part) Suppose that (an )nN converges to 0. Let > 0. Then there
exists an N N such that whenever n > N , an 0 < . As an 0, we have that an < for all n > N
University of London The London School of Economics and Political Science
ABSTRACT MATHEMATICS
MATHEMATIC MA203

Fall 2014
Solutions
90
Solution to Exercise 1.25. Consider the open ball B(x, r) = cfw_y X : d(x, y) < r in X. If y B(x, r),
then d(x, y) < r. Define r = r d(x, y) > 0. We claim that B(y, r ) B(x, r). Let z B(y
University of London The London School of Economics and Political Science
ABSTRACT MATHEMATICS
MATHEMATIC MA203

Fall 2014
Solutions to the exercises from Chapter 2
97
Solutions to the exercises from Chapter 2
Solution to Exercise 2.5. Let > 0. Since (an )nN is a Cauchy sequence, there exists an N N
such that for all n, m
University of London The London School of Economics and Political Science
ABSTRACT MATHEMATICS
MATHEMATIC MA203

Fall 2014
Solutions
116
Solution to Exercise 3.29.
(1) True.
Since the series
X
n=1
an  converges, we have necessarily that
lim an  = 0.
n
Thus there is an N N such that for all n > N , an  < 1. But now f
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2014
Mathematical Methods, Lecture ThirtySix
36. Dierential and Dierence Equations, 3 of 5
36.1 Dierence equations
Dierence equations are recurrence relations satised by a sequence cfw_yx . The
sequence c
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2014
Mathematical Methods, Lecture ThirtyTwo
32. Multivariate Calculus, 4 of 5
32.1 The second derivative of a function
d2 f (x)
of a function f : Rn R is dened by taking the
dx2
derivative of the (transp
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2014
Mathematical Methods, Lecture TwentySeven
27. Linear Transformations, 5 of 6
27.1 Diagonalisation
In the last lecture we saw examples of linear transformations T : R2 R2 where a
basis B of R2 can be
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2014
Mathematical Methods, Lecture TwentyEight
28. Linear Transformations, 6 of 6
We now focus on symmetric matrices and a special form of diagonalisation applicable
to symmetric matrices, known as orthog
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2014
Mathematical Methods, Lecture Twenty
20. Vector Spaces, 4 of 4
In this section, we develop the concepts of angle and length, so that these concepts
can be applied to a general vector space V where ang
University of London The London School of Economics and Political Science
Further Quantitative Methods
MATHEMATIC MA207

Spring 2016
FQM Solutions to Exercises 7
1. The gradient vector for the function h(x, y, z) = x2 + y 2 + z 2 is given by
2x
1
1 1 1
h = 2y
At
, ,
,
h =
1
2 2
2
2z
2
so this is the normal vector to the surface
University of London The London School of Economics and Political Science
Further Quantitative Methods
MATHEMATIC MA207

Spring 2016
FQM Solutions to Exercises 6
1
a = 2
2
1. We have
1
b= 1
4
2
c = b a = 1
2
The cosines of the three angles are given by
ab
1 + 2 + 8
1
=
= ;
a b
9 18
2
2+24
ac
=
= 0;
a c
9 9
21+8
bc
1
= =
b c
1
University of London The London School of Economics and Political Science
Further Quantitative Methods
MATHEMATIC MA207

Spring 2016
FQM Solutions to Exercises 1
1.
Y R = ( 500
10000
1.05 0.95
1000 ) 1.05 1.05 = ( 12395
1.37 1.42
12395 )
so it is riskless since the guaranteed return is 12395 in each state.
Z = ( 1000
2000
cost(Z) =