Announcement/Reminder

General Information

Time and Venue


Course material

Notes and plates

Lecture Handout Title Sequel to something? Remark
1 1 Mathematical statements and predicates Notes Plates
12 Solving equations and inequalities Notes Self-study material: Review on school maths.
21 Simple inequalities justified using `direct proofs' Notes Plates
22 Absolute value and Triangle Inequality for the reals Notes Plates
23 Examples of proofs of statements with conclusion `... iff ...' Notes Plates
24 Basic results on divisibility Notes Plates
25 Summation and Product Notes Self-study material: Review on school maths.
26 Arithmetic progression and geometric progression Notes Self-study material: Review on school maths.
27 Formalization of the Real Number System as understood in School Maths Notes Self-study material: Preview on MATH2050, MATH2070.
31 Examples of proofs-by-contradiction Notes Plates
32 Basic results on complex numbers `beyond school mathematics' Notes Plates
33 Quadratic polynomials Notes Self-study material: Review on school maths.
41 Sets (Revised 10/09.) Notes Plates
51 Mathematical induction Notes Plates
52 De Moivre's Theorem and roots of unity Notes Plates Lecture 3 Handout 2.
53 Binomial coefficients and binomial expansions Notes Self-study material: Review on school maths.
61 Basics of logic in mathematics Notes Plates
62 Applications of logic in mathematics Notes Lecture 6 Handout 1. Self-study material.
71 Examples of proofs concerned with `subset relations' Notes Plates Lecture 4 Handout 1, Lecture 6 Handout 1.
72 Examples of proofs for properties of basic set operations Notes Plates Lecture 4 Handout 1, Lecture 6 Handout 1.
81 Universal quantifier and existential quantifier Notes Plates Lecture 6 Handout 1.
82 Statements with several quantifiers Notes Plates Lecture 8 Handout 1.
83 Existence, uniqueness, and existence-and-uniqueness Notes Lecture 8 Handout 2.
91 Division Algorithm Notes Plates
92 Euclidean Algorithm Notes Plates Lecture 9 Handout 1.
93 What is the system of all natural numbers? Notes Self-study material: Preview on axiomatic set theory.
101 Power set (Revised 10/09.) Notes Plates Lecture 7 Handout 2.
102 Greatest/least element, upper/lower bound Notes Plates
103 Monotonicity and boundedness for infinite sequences of real numbers Notes Plates Lecture 10 Handout 2. Self-study material: Preview on MATH2050.
111 Dis-proofs Notes Plates Lecture 8 Handout 2.
112 Cauchy-Schwarz Inequality and Triangle Inequality Notes Plates Lecture 3 Handout 3.
113 Cauchy-Schwarz Inequality and Triangle Inequality for square-summable sequences Notes Lecture 10 Handout 3, Lecture 11 Handout 2. Self-study material: Preview on MATH2060.
121 Arithmetico-geometric Inequality Notes Plates Lecture 5 Handout 1.
122 The number e Notes Plates
123 Archimedean Principle for the reals Notes Plates Lecture 10 Handout 2. Self-study material: Preview on MATH2050.
131 Notion of functions and its pictorial visualizations Notes Plates
132 Ordered pairs, ordered triples and cartesian products Notes Plates
133 Families Notes Plates
134 Abelian groups, integral domains and fields Notes Plates Preview on MATH2070.
135 Linear algebra beyond systems of linear equations and manipulation of matrices Notes Lecture 13 Handout 4. Self-study material: Preview on MATH2040.
136 Spanning sets, linearly independent sets, and bases Notes Lecture 13 Handout 5. Self-study material: Preview on MATH2040.
137 Basic results on polynomials `beyond school mathematics' Notes Lecture 13 Handout 4. Self-study material: Preview on MATH2070.
138 Roots of polynomials with complex coefficients Notes Lecture 3 Handout 2, Lecture 13 Handout 6. Self-study material: Preview on MATH2070.
141 Surjectivity and injectivity Notes Plates
142 Surjectivity and injectivity for `nice' real-valued functions of one real variable Notes Plates Lecture 14 Handout 1.
143 Intermediate Value Theorem, and the surjectivity and injectivity for continuous real-valued functions of one real-variable Notes Plates Lecture 14 Handout 1. Preview on MATH2050.
144 Surjectivity and injectivity for `simple' complex-valued functions of one complex variable Notes Plates Lecture 14 Handout 1. Preview on MATH2070.
151 Compositions, surjectivity and injectivity Notes Plates Lecture 14 Handout 1.
152 Image sets and pre-image sets Notes Plates
153 Image sets and pre-image sets under `nice' real-valued functions of one real variable Notes Plates Lecture 15 Handout 2.
154 Image sets, pre-image sets of intervals for continuous real-valued functions of one real-variable Notes Lecture 14 Handout 3. Self-study material: Preview on MATH2050.
155 Parametrizations for curves and surfaces Notes Lecture 15 Handout 2. Self-study material: Preview on MATH2010, MATH2020.
156 Curves and surfaces as level sets Notes Lecture 15 Handout 2. Self-study material: Preview on MATH2010, MATH2020.
161 Theoretical results involving image sets and pre-image sets Notes Plates Lecture 15 Handout 2.
162 Characterization of surjectivity with image sets, pre-image sets Notes Plates Lecture 16 Handout 1.
171 Notion of inverse functions Notes Plates Lecture 14 Handout 1.
172 Examples on finding inverse functions for `simple' bijective functions Notes Plates Lecture 17 Handout 1.
173 Relations, functions and `well-defined-ness' for functions Notes Plates Lecture 13 Handout 1.
174 Existence and uniqueness of inverse functions Notes Plates Lecture 17 Handout 1, Lecture 17 Handout 3.
175 Anthology on definitions for the notion of `function' Notes Self-study material.
176 Groups Notes Plates Lecture 13 Handout 4. Self-study material: Preview on MATH2070.
181 Sets of equal cardinality Notes Plates Lecture 17 Handout 1, Lecture 17 Handout 4.
191 Equivalence relations Notes Plates
192 Examples of Equivalence Relations Notes Plates Lecture 19 Handout 1.
201 Integers modulo n Notes Plates Lecture 19 Handout 1.
202 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. Self-study material: Preview on MATH2040.
201 Partial orderings, total orderings, and well-order relations Notes Plates Lecture 10 Handout 2.
212 Partial orderings defined by the subset relation Notes Lecture 21 Handout 1.
213 Axiom of Choice Notes Lecture 13 Handout 3, Lecture 21 Handout 2. Self-study material: Preview on axiomatic set theory.
221 Cantor's diagonal argument Notes Plates Lecture 18 Handout 1.
222 Sets of not necessarily the same size Notes Plates Lecture 18 Handout 1.
231 Schroeder-Bernstein Theorem Notes Plates Lecture 22 Handout 2.
232 Cantor's Theorem and its consequences Notes Plates Lecture 22 Handout 1, Lecture 23 Handout 1.
233 Zermelo-Fraenkel Axioms with the Axiom of Choice Notes Lecture 21 Handout 3, Lecture 22 Handout 2, Lecture 23 Handout 2. Self-study material: Preview on axiomatic set theory.
241 Finite sets versus infinite sets Notes Plates Lecture 18 Handout 1, Lecture 22 Handout 2, Lecture 23 Handout 1.
242 Countable sets and uncountable sets Notes Plates Lecture 18 Handout 1, Lecture 22 Handout 2, Lecture 23 Handout 1.
251 Comparisons amongst the number systems Notes Self-study material.
252 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. Self-study material.
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Exercises

What to submit as Assignment? Due date of Assignment?
Exercise 1 Question sheet Answers and selected solutions Questions (1a), (1c), (1e), (3a), (3b), (4a), (4b). 13/09 2359hrs
Exercise 2 Question sheet Answers and selected solutions Questions (3f), (3h), (3j), (6), (7), (9). 20/09 2359hrs
Exercise 3 Question sheet Answers and selected solutions Questions (1), (2), (3), (4), (8), (9), (10). 27/09 2359hrs
Exercise 4 Question sheet Answers and selected solutions Questions (3a), (4), (5), (7), (10a), (11a). 04/10 2359hrs
Exercise 5 Question sheet Answers and selected solutions Questions (1a), (2), (4a), (4b), (5). 11/10 2359hrs
Exercise 6 Question sheet Answers and selected solutions Questions (1a), (2), (3), (4), (5b), (5e), (6), (7), (9), (12a). 18/10 2359hrs
Exercise 7 Question sheet Questions (1), (5a), (6a), (7a). 25/10 2359hrs
Exercise 8 Question sheet Questions (3a), (3d), (3i), (6a). 01/11 2359hrs
Exercise 9 Question sheet 08/11 2359hrs
Exercise 10 Question sheet 15/11 2359hrs
Exercise 11 Question sheet 29/11 2359hrs
Exercise 12 Question sheet 06/12 2359hrs
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Supplementary Exercises

Supplement to Exercise 1 Question sheet Answers
Supplement to Exercise 2 Question sheet Answers
Supplement to Exercise 3 Question sheet Answers
Supplement to Exercise 4 Question sheet Answers
Supplement to Exercise 5 Question sheet Answers
Supplement to Exercise 6 Question sheet Answers
Supplement to Exercise 7 Question sheet
Supplement to Exercise 8 Question sheet
Supplement to Exercise 9 and Exercise 10 Question sheet
Supplement to Exercise 11 and Exercise 12 Question sheet
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Proof-writing Exercises (Optional)

What to submit? Due date of submission?
Proof-writing Exercise 1 Question sheet Index of comments Answers and selected solution All questions. 14/09 2359hrs
Proof-writing Exercise 2 Question sheet Index of comments Answers and selected solution Question (1a). 21/09 2359hrs
Proof-writing Exercise 3 Question sheet Index of comments Answers and selected solution Questions (1d), (2d). 28/09 2359hrs
Proof-writing Exercise 4 Question sheet Index of comments Answers and selected solution Questions (1c), (3). 05/10 2359hrs
Proof-writing Exercise 5 Question sheet Question (3b). 12/10 2359hrs
Proof-writing Exercise 6 Question sheet Question (2). 19/10 2359hrs
Proof-writing Exercise 7 Question sheet Question (1). 26/10 2359hrs
Proof-writing Exercise 8 Question sheet Question (1). 28/10 2359hrs
Proof-writing Exercise 9 Question sheet 09/11 2359hrs
Proof-writing Exercise 10 Question sheet 16/11 2359hrs
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Supplementary notes on elementary linear algebra

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Links



Last modified: 1900hrs, 23-10-2021 (HKT)