1 
1 
Mathematical statements and predicates 
Notes 
Plates 


1  2 
Solving equations and inequalities 
Notes 


Selfstudy material: Review on school maths. 
2  1 
Simple inequalities justified using `direct proofs'

Notes 
Plates 


2  2 
Absolute value and Triangle Inequality for the reals 
Notes 
Plates 


2  3 
Examples of proofs of statements with conclusion `... iff ...'

Notes 
Plates 


2  4 
Basic results on divisibility 
Notes 
Plates 


2  5 
Summation and Product 
Notes 


Selfstudy material: Review on school maths. 
2  6 
Arithmetic progression and geometric progression 
Notes 


Selfstudy material: Review on school maths. 
2  7 
Formalization of the Real Number System as understood in School Maths 
Notes 


Selfstudy material: Preview on MATH2050, MATH2070. 
3  1 
Examples of proofsbycontradiction 
Notes 
Plates 


3  2 
Basic results on complex numbers `beyond school mathematics' 
Notes 
Plates 


3  3 
Quadratic polynomials 
Notes 


Selfstudy material: Review on school maths. 
4  1 
Sets (Revised 10/09.) 
Notes 
Plates 


5  1 
Mathematical induction 
Notes 
Plates 


5  2 
De Moivre's Theorem and roots of unity 
Notes 
Plates 
Lecture 3 Handout 2. 

5  3 
Binomial coefficients and binomial expansions 
Notes 


Selfstudy material: Review on school maths. 
6  1 
Basics of logic in mathematics 
Notes 
Plates 


6  2 
Applications of logic in mathematics 
Notes 

Lecture 6 Handout 1. 
Selfstudy material. 
7  1 
Examples of proofs concerned with `subset relations' 
Notes 
Plates 
Lecture 4 Handout 1, Lecture 6 Handout 1. 

7  2 
Examples of proofs for properties of basic set operations 
Notes 
Plates 
Lecture 4 Handout 1, Lecture 6 Handout 1. 

8  1 
Universal quantifier and existential quantifier 
Notes 
Plates 
Lecture 6 Handout 1. 

8  2 
Statements with several quantifiers 
Notes 
Plates 
Lecture 8 Handout 1. 

8  3 
Existence, uniqueness, and existenceanduniqueness 
Notes 

Lecture 8 Handout 2. 

9  1 
Division Algorithm 
Notes 
Plates 


9  2 
Euclidean Algorithm 
Notes 
Plates 
Lecture 9 Handout 1. 

9  3 
What is the system of all natural numbers? 
Notes 


Selfstudy material: Preview on axiomatic set theory. 
10  1 
Power set (Revised 10/09.) 
Notes 
Plates 
Lecture 7 Handout 2. 

10  2 
Greatest/least element, upper/lower bound 
Notes 
Plates 


10  3 
Monotonicity and boundedness for infinite sequences of real numbers 
Notes 
Plates 
Lecture 10 Handout 2. 
Selfstudy material: Preview on MATH2050. 
11  1 
Disproofs 
Notes 
Plates 
Lecture 8 Handout 2. 

11  2 
CauchySchwarz Inequality and Triangle Inequality 
Notes 
Plates 
Lecture 3 Handout 3. 

11  3 
CauchySchwarz Inequality and Triangle Inequality for squaresummable sequences 
Notes 

Lecture 10 Handout 3, Lecture 11 Handout 2. 
Selfstudy material: Preview on MATH2060. 
12  1 
Arithmeticogeometric Inequality 
Notes 
Plates 
Lecture 5 Handout 1. 

12  2 
The number e 
Notes 
Plates 


12  3 
Archimedean Principle for the reals 
Notes 
Plates 
Lecture 10 Handout 2. 
Selfstudy material: Preview on MATH2050. 
13  1 
Notion of functions and its pictorial visualizations 
Notes 
Plates 


13  2 
Ordered pairs, ordered triples and cartesian products 
Notes 
Plates 


13  3 
Families 
Notes 
Plates 


13  4 
Abelian groups, integral domains and fields 
Notes 
Plates 

Preview on MATH2070. 
13  5 
Linear algebra beyond systems of linear equations and manipulation of matrices 
Notes 

Lecture 13 Handout 4. 
Selfstudy material: Preview on MATH2040. 
13  6 
Spanning sets, linearly independent sets, and bases 
Notes 

Lecture 13 Handout 5. 
Selfstudy material: Preview on MATH2040. 
13  7 
Basic results on polynomials `beyond school mathematics' 
Notes 

Lecture 13 Handout 4. 
Selfstudy material: Preview on MATH2070. 
13  8 
Roots of polynomials with complex coefficients 
Notes 

Lecture 3 Handout 2, Lecture 13 Handout 6. 
Selfstudy material: Preview on MATH2070. 
14  1 
Surjectivity and injectivity 
Notes 
Plates 


14  2 
Surjectivity and injectivity for `nice' realvalued functions of one real variable 
Notes 
Plates 
Lecture 14 Handout 1. 

14  3 
Intermediate Value Theorem, and the surjectivity and injectivity for continuous realvalued functions of one realvariable 
Notes 
Plates 
Lecture 14 Handout 1. 
Preview on MATH2050. 
14  4 
Surjectivity and injectivity for `simple' complexvalued functions of one complex variable 
Notes 
Plates 
Lecture 14 Handout 1. 
Preview on MATH2070. 
15  1 
Compositions, surjectivity and injectivity 
Notes 
Plates 
Lecture 14 Handout 1. 

15  2 
Image sets and preimage sets 
Notes 
Plates 


15  3 
Image sets and preimage sets under `nice' realvalued functions of one real variable 
Notes 
Plates 
Lecture 15 Handout 2. 

15  4 
Image sets, preimage sets of intervals for continuous realvalued functions of one realvariable 
Notes 

Lecture 14 Handout 3. 
Selfstudy material: Preview on MATH2050. 
15  5 
Parametrizations for curves and surfaces 
Notes 

Lecture 15 Handout 2. 
Selfstudy material: Preview on MATH2010, MATH2020. 
15  6 
Curves and surfaces as level sets 
Notes 

Lecture 15 Handout 2. 
Selfstudy material: Preview on MATH2010, MATH2020. 
16  1 
Theoretical results involving image sets and preimage sets 
Notes 
Plates 
Lecture 15 Handout 2. 

16  2 
Characterization of surjectivity with image sets, preimage sets 
Notes 
Plates 
Lecture 16 Handout 1. 

17  1 
Notion of inverse functions 
Notes 
Plates 
Lecture 14 Handout 1. 

17  2 
Examples on finding inverse functions for `simple' bijective functions 
Notes 
Plates 
Lecture 17 Handout 1. 

17  3 
Relations, functions and `welldefinedness' for functions 
Notes 
Plates 
Lecture 13 Handout 1. 

17  4 
Existence and uniqueness of inverse functions 
Notes 
Plates 
Lecture 17 Handout 1, Lecture 17 Handout 3. 

17  5 
Anthology on definitions for the notion of `function' 
Notes 


Selfstudy material. 
17  6 
Groups 
Notes 

Lecture 13 Handout 4. 
Selfstudy material: Preview on MATH2070. 
18  1 
Sets of equal cardinality 
Notes 
Plates 
Lecture 17 Handout 1, Lecture 17 Handout 4. 

19  1 
Equivalence relations 
Notes 
Plates 


19  2 
Examples of Equivalence Relations 
Notes 
Plates 
Lecture 19 Handout 1. 

20  1 
Integers modulo n 
Notes 
Plates 
Lecture 19 Handout 1. 

20  2 
More on vector spaces and linear transformations 
Notes 
Plates 
Lecture 13 Handout 6, Lecture 15 Handout 2, Lecture 17 Handout 4, Lecture 19 Handout 1. 
Selfstudy material: Preview on MATH2040. 
21  1 
Partial orderings, total orderings, and wellorder relations

Notes 
Plates 
Lecture 10 Handout 2. 

21  2 
Partial orderings defined by the subset relation 
Notes 

Lecture 21 Handout 1. 

21  3 
Axiom of Choice 
Notes 

Lecture 13 Handout 3, Lecture 21 Handout 2. 
Selfstudy material: Preview on axiomatic set theory. 
22  1 
Cantor's diagonal argument 
Notes 
Plates 
Lecture 18 Handout 1. 

22  2 
Sets of not necessarily the same size 
Notes 
Plates 
Lecture 18 Handout 1. 

23  1 
SchroederBernstein Theorem 
Notes 
Plates 
Lecture 22 Handout 2. 

23  2 
Cantor's Theorem and its consequences 
Notes 
Plates 
Lecture 22 Handout 1, Lecture 23 Handout 1. 

23  3 
ZermeloFraenkel Axioms with the Axiom of Choice 
Notes 

Lecture 21 Handout 3, Lecture 22 Handout 2, Lecture 23 Handout 2. 
Selfstudy material: Preview on axiomatic set theory. 
24  1 
Finite sets versus infinite sets 
Notes 
Plates 
Lecture 18 Handout 1, Lecture 22 Handout 2, Lecture 23 Handout 1. 

24  2 
Countable sets and uncountable sets 
Notes 
Plates 
Lecture 18 Handout 1, Lecture 22 Handout 2, Lecture 23 Handout 1. 

25  1 
Comparisons amongst the number systems 
Notes 


Selfstudy material. 
25  2 
Construction of the integer system from the natural number system 
Notes 

Lecture 13 Handout 4, Lecture 17 Handout 4, Lecture 19 Handout 1, Lecture 21 Handout 1, Lecture 25 Handout 1. 
Selfstudy material. 