Foundation of Modern Mathematics (MATH1050A)

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Schedule.

• The schedule of lectures below is tentative.
• Week 1. Monday: 2.01, 2.02. Wednesday: 3.01, 3.02.
• Week 2. Wednesday: 2.03, 2.04, 4.01.
• Week 3. Monday: 4.02, 4.03, 4.04. Wednesday: 5.03, 5.04, 5.05.
• Week 4. Monday: 2.05, 3.05, 3.06. Wednesday: 3.03, 3.04.
• Week 5. Monday: 2.06, 5.08. Wednesday: 2.07, 5.14.
• Week 6. Monday: 2.08, 5.09. Wednesday: 5.12, 6.01.
• Week 7. Monday: 2.10, 5.13, 6.12. Wednesday: 6.02, 6.03, 6.05.
• Week 8. Monday: 6.06, 6.07, 6.08. Wednesday: 6.09, 6.10, 6.11.
• Week 9. Monday: 8.01.
• Week 10. Monday: 6.17, 6.18.
• Week 11. Monday: 7.01, 7.02. Wednesday: 8.02, 8.03.
• Week 12. Monday: 7.03, 7.04. Wednesday: 8.04, 8.05.
• Week 13. Monday: 8.06, 8.07. Wednesday: to be announced.
• This is the schedule of guided study topics that students are supposed to abide by.
• Topic 1: 5.06. Expected to be completed by Week 5 Monday.
• Topic 2: 5.09 and 5.10. Expected to be completed by Week 6 Wednesday.
• Topic 3: 6.13. Expected to be completed by Week 10 Monday.
Video material is available in `Blackboard'. Associated exercises (containing work to be submitted) are available here.

Notes and plates.

• Theme 1. Miscellaneous background topics.

Handout	Title			Schedule	Assumed background
1.01	Quadratic polynomials.	Notes		Self-study review material.
1.02	Summation and product.	Notes		Self-study review material.
1.03	Binomial coefficients and binomial expansions.	Notes		Self-study review material.
1.04	Arithmetic progression and geometric progression.	Notes		Self-study review material.

Example set	Title			Supplement to which handouts?
1.A	Quadratic polynomials and quadratic functions.	Questions	Answers	1.01.
1.B	Arithmetic progressions and geometric progressions.	Questions	Answers	1.04.
1.C	Solving equations and inequalities with algebraic methods.	Questions	Answers
1.D	Equations involving trigonometric functions.	Questions	Answers
1.E	Binomial coefficients and Binomial Theorem.	Questions	Answers	1.03.

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• Theme 2. Inequalities.

Handout	Title			Schedule	Assumed background
2.01	From simple inequalities to basic properties of the reals.	Notes	Plates	Week 1 Monday.
2.02	Absolute Value and Triangle Inequality for the Reals.	Notes	Plates	Week 1 Monday.
2.03	Bernoulli's Inequality.	Notes	Plates	Week 2 Wednesday.	1.04.
2.04	Cauchy-Schwarz Inequality and Triangle Inequality.	Notes	Plates	Week 2 Wednesday.	2.02.
2.05	Weierstrass' Inequality	Notes	Plates	Week 4 Monday.	3.05, 3.06.
2.06	Arithmetico-geometrical Inequality.	Notes	Plates	Week 5 Monday.	3.05, 3.06.
2.07	Greatest/least element, upper/lower bound.	Notes	Plates	Week 5 Wednesday.	5.03, 5.10, 5.11.
2.08	Bounded-Monotone Theorem for infinite sequences.	Notes	Plates	Week 6 Monday.	2.07, 5.10, 5.11.
2.09	Cauchy-Schwarz Inequality and Triangle Inequality for square-summable sequences.	Notes		Self-study preview material on MATH2050, MATH2060.	2.04, 2.08.
2.10	The number e.	Notes		Week 7 Monday.	2.03, 2.05, 2.08.
2.11	Archimedean Principle for the reals.	Notes	Plates	Self-study preview material on MATH2050, MATH2060.	2.07, 2.08.

Example set	Title			Supplement to which handouts?
2.A	Absolute values, Triangle Inequality and beyond.	Questions		2.01, 2.02.
2.B	Miscellanies on inequalities.	Questions		2.01, 2.02, 2.03.
2.C	Inequalities and mathematical induction.	Questions		2.01, 2.02, 2.03, 2.05, 2.06, 3.05, 3.06, 4.01.
2.D	Greatest/least element and upper/lower bound.	Questions		2.07.
2.E	Cauchy-Schwarz Inequality.	Questions		2.04.

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• Theme 3. Integers, rationals and irrationals.

Handout	Title			Schedule	Assumed background
3.01	Basic results on divisibility.	Notes	Plates	Week 1 Wednesday.
3.02	Rationals and irrationals.	Notes	Plates	Week 1 Wednesday.	3.01.
3.03	Division Algorithm.	Notes	Plates	Week 4 Wednesday.	3.01.
3.04	Euclidean Algorithm.	Notes	Plates (Attachment)	Week 4 Wednesday.	3.03.
3.05	Argument by mathematical induction.	Notes	Plates	Week 4 Monday.
3.06	Examples on argument by mathematical induction.	Notes	Plates	Week 4 Monday.	3.04.
3.07	Principle of Mathematical Induction and Well-ordering Principle for Integers.	Notes		Self-study advanced material.	3.05, 5.04.

Example set	Title			Supplement to which handouts?
3.A	Mathematical induction.	Questions		3.05, 3.06.
3.B	Divisibility, Division Algorithm, rationals and irrationals.	Questions	Answers	3.01, 3.02, 3.03, 3.04, 3.05, 3.06.
3.C	Generalizations of divisibility and rationality.	Questions		3.01, 3.02, 3.03, 4.01, 4.02.

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• Theme 4. Complex numbers.

Handout	Title			Schedule	Assumed background
4.01	Basic algebraic results on complex numbers `beyond school mathematics'.	Notes	Plates	Week 2 Wednesday.
4.02	Argand plane.	Notes	Plates	Week 3 Monday.	4.01.
4.03	Polar form.	Notes	Plates	Week 3 Monday.	4.02.
4.04	De Moivre's Theorem and roots of unity.	Notes	Plates	Week 3 Monday.	4.03.
4.05	Roots of polynomials with complex coefficients.	Notes		Self-study preview material on MATH2070.	4.04.

Example set	Title			Supplement to which handouts?
4.A	Complex numbers.	Questions	Answers	4.01, 4.02, 4.03.
4.B	De Moivre's Theorem and Roots of unity.	Questions	Answers	4.01, 4.02, 4.03, 4.04.

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• Theme 5. Logic and sets.

Handout	Title			Schedule	Assumed background
5.01	Statements, proofs, and mathematical language.	Notes		Self-study review material.
5.02	Basics in set language.	Notes	Plates	Self-study review material.
5.03	Method of specification.	Notes	Plates	Week 3 Wednesday.	5.02.
5.04	Set operations.	Notes	Plates	Week 3 Wednesday.	5.03.
5.05	Examples of proofs concerned with `subset relations'.	Notes	Plates	Week 3 Wednesday.	5.03, 5.04.
5.06	Basics of logic in mathematics.	Notes	Plates	Guided Study Topic 1; expected to be completed by Week 5 Monday.
5.07	Applications of logic in mathematics.	Notes		Self-study material	5.06
5.08	Examples of proofs for properties of basic set operations.	Notes	Plates	Week 5 Monday.	5.04, 5.06.
5.09	Power set.	Notes	Plates	Week 6 Monday.	5.04, 5.08.
5.10	Universal quantifier and existential quantifier.	Notes	Plates	Guided Study Topic 2; expected to be completed by Week 6 Wednesday.	5.06.
5.11	Statements with several quantifiers.	Notes	Plates	Guided Study Topic 2; expected to be completed by Week 6 Wednesday.	5.10.
5.12	Dis-proofs by counter-example.	Notes	Plates	Week 6 Wednesday.	5.10, 5.11.
5.13	Dis-proofs by wholesale refutation.	Notes	Plates	Week 7 Monday.	5.10, 5.11.
5.14	Ordered pairs, ordered triples and cartesian products.	Notes	Plates	Week 5 Wednesday.	5.02, 5.03, 5.04, 5.08.
5.15	Families.	Notes		Self-study advanced material.	5.10, 6.01.
5.16	Axiom of Choice.	Notes		Self-study advanced material.	5.15, 7.06, 7.07, 8.03, 8.05.

Example set	Title			Supplement to which handouts?
5.A	Set notations and method of specification.	Questions	Answers	5.02, 5.03.
5.B	Subset relation.	Questions		5.03, 5.04.
5.C	Logical connectives, quantifiers, negations and dis-proofs.	Questions	Answers	5.06, 5.10, 5.11, 5.12, 5.13.
5.D	Set operations.	Questions		5.08, 5.09, 5.12, 5.13.

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• Theme 6. Functions.

Everything in Themes 1-5 (except `Families', `Axiom of Choice') are assumed as background knowledge.

Handout	Title			Schedule	Assumed background
6.01	Functions.	Notes	Plates	Week 6 Wednesday.
6.02	Surjectivity and Injectivity.	Notes	Plates	Week 7 Wednesday.	6.01.
6.03	Surjectivity and injectivity for `nice' real-valued functions of one real variable.	Notes	Plates	Week 7 Wednesday.	6.02.
6.04	Intermediate Value Theorem, and the surjectivity and injectivity for continuous real-valued functions of one real-variable.	Notes		Self-study preview material on MATH2050.	6.02, 6.03.
6.05	Surjectivity and injectivity for `simple' complex-valued functions of one complex variable.	Notes	Plates	Week 7 Wednesday.	6.02.
6.06	Compositions, Surjectivity and Injectivity.	Notes	Plates	Week 8 Monday.	6.02.
6.07	Notion of inverse functions.	Notes	Plates	Week 8 Monday.	6.02.
6.08	Examples on finding inverse functions for `simple' bijective functions.	Notes	Plates	Week 8 Monday.	6.07.
6.09	Relations and the formal definition for the notion of functions.	Notes	Plates	Week 8 Wednesday.	6.01.
6.10	`Well-defined-ness' for functions.	Notes	Plates	Week 8 Wednesday.	6.01, 6.09.
6.11	Existence and uniqueness of inverse functions.	Notes	Plates	Week 8 Wednesday.	6.07, 6.09.
6.12	Image sets and pre-image sets.	Notes	Plates	Week 7 Monday.	6.01.
6.13	Image Sets and pre-image sets under `nice' real-valued functions of one real variable.	Notes	Plates	Guided Study Topic 3; expected to be completed by Week 10 Monday.	6.12.
6.14	Image sets, pre-image sets of intervals for continuous real-valued functions of one real-variable.	Notes		Self-study preview material on MATH2050.	6.04, 6.12, 6.13.
6.15	Parametrizations for curves and surfaces.	Notes		Self-study preview material on MATH2010, MATH2020.	6.12.
6.16	Curves and surfaces as level sets.	Notes		Self-study preview material on MATH2010, MATH2020.	6.12.
6.17	Theoretical results involving image sets and pre-image sets.	Notes	Plates	Week 10 Monday.	6.12.
6.18	Characterization of surjectivity with image sets, pre-image sets.	Notes	Plates	Week 10 Monday.	6.02, 6.17.
6.19	Abelian groups, integral domains and fields.	Notes	Plates	Self-study preview material on MATH2070.	6.01.
6.20	Groups.	Notes		Self-study preview material on MATH2070.	6.01, 6.19.

Example set	Title			Supplement to which handouts?
6.A	Surjectivity, injectivity, and inverse functions.	Questions		6.01, 6.02, 6.03, 6.05, 6.06, 6.07, 6.08.
6.B	Image sets and pre-image sets.	Questions		6.12, 6.13, 6.15, 6.16.
6.C	Formal definition for the notion of functions.	Questions		6.02, 6.07, 6.09, 6.10, 6.11.
6.D	Theoretical results on functions.	Questions		6.02, 6.06, 6.07, 6.09, 6.11, 6.12, 6.17, 6.18.

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• Theme 7. Relations.

Everything in Themes 1-6 (except `Axiom of Choice') are assumed as background knowledge.

Handout	Title			Schedule	Assumed background
7.01	Equivalence relations.	Notes	Plates	Week 11 Monday
7.02	Examples on equivalence relations.	Notes	Plates	Week 11 Monday	7.01.
7.03	Integers modulo n.	Notes	Plates	Week 12 Monday	7.01, 7.02.
7.04	Arithmetic in Integers modulo n.	Notes	Plates	Week 12 Monday	6.19, 7.03.
7.05	Partial orderings and total orderings.	Notes	Plates	Self-study advanced material.
7.06	Well-order relations and the Well-ordering Principle.	Notes	Plates	Self-study advanced material.	7.05.
7.07	Partial orderings defined by the subset relation, and Zorn's Lemma.	Notes		Self-study advanced material.	7.06.

Example set	Title			Supplement to which handouts?
7.A	Relations.	Questions	Answers	7.01, 7.02, 7.03, 7.04, 7.05, 7.06.

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• Theme 8. Cardinality.

Everything in Themes 1-6 are assumed as background knowledge.

Handout	Title			Schedule	Assumed background
8.01	Sets of equal cardinality.	Notes	Plates	Week 9 Monday
8.02	Cantor's diagonal argument.	Notes	Plates	Week 11 Wednesday	8.01.
8.03	Sets of not necessarily the same size.	Notes	Plates	Week 11 Wednesday	8.01.
8.04	Schroeder-Bernstein Theorem.	Notes	Plates	Week 12 Wednesday	8.03.
8.05	Cantor's Theorem and its consequences.	Notes	Plates	Week 12 Wednesday	8.01, 8.02, 8.03, 8.04.
8.06	Finite sets versus infinite sets.	Notes	Plates	Week 13 Monday	8.01, 8.03, 8.04.
8.07	Countable sets and uncountable sets.	Notes	Plates	Week 13 Monday	8.01, 8.03, 8.04, 8.05.

Example set	Title			Supplement to which handouts?
8.A	Cardinality.	Questions	Answers	8.01, 8.02, 8.03, 8.04, 8.05, 8.06.

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• Theme 9. Miscellaneous advanced topics.

The material in this theme is of an advanced level, and is meant for self-study. It is however accessible from what has been covered in MATH1010/1018 and MATH1030/1038, and from what has been covered in the other themes in this course.

Handout	Title				Assumed background
9.01	Formalization of the Real Number System as understood in School Maths.	Notes			2.01, 2.07, 2.08, 2.11.
9.02	Surds of positive real numbers.	Notes			2.08, 9.01.
9.03	Ideas behind the construction of the real number system with rational numbers.	Notes			2.08, 2.11, 9.01.
9.04	What is the System of all Natural Numbers?	Notes
9.05	Linear algebra beyond systems of linear equations and manipulation of matrices.	Notes			6.19, 6.20.
9.06	Spanning sets, linearly independent sets, and bases.	Notes			9.05.
9.07	More on vector spaces and linear transformations.	Notes			9.06.
9.08	Basic results on polynomials `beyond school mathematics'.	Notes			6.19
9.09	Zermelo-Fraenkel Axioms with the Axiom of Choice.	Notes			5.15, 7.07, 8.05

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