MA103 Introduction to Abstract Mathematics
Lecture 34
Graham Brightwell
22 February 2018
Graham Brightwell
MA103 Introduction to Abstract Mathematics Lecture 34
Exercises 17, for classes in Week 8
Que
MA103 Introduction to Abstract Mathematics
Lecture 31
Graham Brightwell
12 February 2018
Graham Brightwell
MA103 Introduction to Abstract Mathematics Lecture 31
Exercises 16, for classes in Week 7
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MA103 Introduction to Abstract Mathematics
Lecture 33
Graham Brightwell
19 February 2018
Graham Brightwell
MA103 Introduction to Abstract Mathematics Lecture 33
Exercises 17, for classes in Week 8
Que
MA103 Introduction to Abstract Mathematics
Lecture 32
Graham Brightwell
15 February 2018
Graham Brightwell
MA103 Introduction to Abstract Mathematics Lecture 32
Exercises 16, for classes in Week 7
Que
MA103 Introduction to Abstract Mathematics
Lecture 22
Graham Brightwell
11 January 2018
Graham Brightwell
MA103 Introduction to Abstract Mathematics Lecture 22
Exercises for classes in Week 2
For the
MA103 Introduction to Abstract Mathematics
Lecture 23
Graham Brightwell
15 January 2018
Graham Brightwell
MA103 Introduction to Abstract Mathematics Lecture 23
Exercises 12, for classes in Week 3
For
MA103 Introduction to Abstract Mathematics
Lecture 25
Graham Brightwell
22 January 2018
Graham Brightwell
MA103 Introduction to Abstract Mathematics Lecture 25
Exercises 13, for classes in Week 4
Plea
MA103 Introduction to Abstract Mathematics
Lecture 26
Graham Brightwell
25 January 2018
Graham Brightwell
MA103 Introduction to Abstract Mathematics Lecture 26
Exercises 13, for classes in Week 4
Plea
MA103 Introduction to Abstract Mathematics
Lecture 24
Graham Brightwell
19 January 2017
Graham Brightwell
MA103 Introduction to Abstract Mathematics Lecture 24
Exercises 12, for classes in Week 3
For
MA103 Introduction to Abstract Mathematics
Lecture 27
Graham Brightwell
29 January 2018
Graham Brightwell
MA103 Introduction to Abstract Mathematics Lecture 27
Exercises 14, for classes in Week 5
For
2016/17
MA103
Introduction to Abstract Mathematics
Second part, Analysis and Algebra
Amol Sasane
Revised by Jozef Skokan, Konrad Swanepoel, and Graham Brightwell
c London School of Economics 2016
Copy
MA 103 Slides 3
Introduction to Abstract Mathematics
Proof by Induction
1
The principle of induction
The Induction Principle
Suppose P(n) is a statement involving natural numbers n.
Suppose the follow
Introduction to Abstract Mathematics
MA 103
Exercises 3
Relevant parts of the Lecture notes: Sections 3.1 3.8 and 3.10 3.12.
Relevant parts of the text books: Biggs: Chapter 5;
Eccles: Chapter 5.
A
Introduction to Abstract Mathematics
MA 103
Exercises 6
Relevant parts of the Lecture notes: Sections 4.6 4.12. and 5.1 5.4.
Relevant parts of the text books: Biggs: Section 6.3 6.4, 7.2 7.3 and 7.6
Introduction to Abstract Mathematics
MA 103
Exercises 1
Relevant parts of the Lecture notes: Chapter 1 and Sections 2.1 2.8, 2.12 and 2.13.
Relevant parts of the text books: Biggs: Chapter 1 and Sec
Introduction to Abstract Mathematics
MA 103
Exercises 7
Relevant parts of the Lecture notes: Chapter 6.
Relevant parts of the text books: Biggs: Chapter 8;
Eccles: Chapters 15 17 and Section 23.1.
Logical statements / truth tables
Example 1
Let p , q and r be statements. Prove, using a truth table, that
MA 103 Introduction to Abstract Mathematics
Extra Examples Session 1
(p r ) (q r ) (p r ) (q
The Fibonacci numbers
The Fibonacci numbers go back to Leonardo of Pisa
(c. 11701250), who was better known as Fibonacci.
MA 103 Slides 4 a
Introduction to Abstract Mathematics
Fibonacci Numbers
Fibon
Abstract Mathematics1
Abstract Mathematics or Pure Mathematics
MA 103 Slides 1
is mathematics that studies entirely abstract concepts;
studies abstract entities with respect to their intrinsic natur
Using strong induction
When using strong induction on recurrence relations such as the
Fibonacci numbers, we usually only have to go a few steps back:
to prove P(n), we only use P(n 1) and P(n 2).
MA
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
an equivalence relation, since
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 3
1
(a) To prove : for all n N, n3 + 5n is a multiple of 3.
Proof : Let P(n) be the statement that n3 + 5n is a multiple of 3.
Base c
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
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 7
1
(a) The statement a|b means: there is a q Z so that b = qa.
The key point for the rest of this question is that you have to engag
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 2
1
(a) (i)
We can write this set as A = cfw_ x N | x = 5y for some y N .
But there are many other ways; for instance:
A = cfw_ m N |
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
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 4
1
(a) For every cat born at time n, there are two cats born at time n + 1. So if Cn is the number
of cats born at time n, then Cn+1
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 6
1
First we note that if the person picks 3 socks, he might end up with one blue, one red and
one grey one. So at least 4 socks will
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 1
1
The statement if n is a multiple of 16, then n is not a multiple of 6 , is a statement of the
form if P, then Q. A counterexample