Back to the general page.

The page is updated from time to time, with new material included. Do click `re-fresh' to ensure you are seeing the most updated page.

Material arranged according to schedule.

The schedule is tentative.

Week 1.

Week 2.

Week 3.

Week 4.

Week 5.

Week 6.

Week 7.

Week 8.

Week 9.

Week 10.

Week 11.

Week 12.

Week 13.

Back to Top

Material arranged according to topics.

Topic 1. Un-formal introduction to logic.

Sub-topic Title Assumed background. Where does it go?
1.1 Statements, predicates, logical connectives and quantifiers. 1.2
1.2 Negation, conjunction, disjunction, and logical equivalence. 1.1 1.3
1.3 Conditional, Biconditional and rules of inference. 1.2
1.4 Universal quantifier and existential quantifier. 1.2, 1.3 1.5
1.5 Statements and predicates with mixed quantifiers. 1.4

Topic 2. Arithmetic in the real number system.

Sub-topic Title Assumed background. Where does it go?
2.1 Arithmetic in the Real Number System. 2.2, 3.1
2.2 Usual ordering for the reals. 2.1 2.3, 4.1
2.3 Examples on proofs of simple conditional inequalities. 2.2 2.4, 2.5, 2.6
2.4 Squares, quadratics, and inequalities. 2.3 2.5
2.5 Absolute value and Triangle Inequality for the Reals. 2.4 3.6
2.6 Bernoulli's Inequality and Weierstrass' Inequality. 2.5, 3.3

Topic 3: Complex numbers.

Sub-topic Title Assumed background. Where does it go?
3.1 Field of complex numbers. 2.1 3.2, 3.3
3.2 Argand plane. 3.1 3.4
3.3 Binomial Theorem, arithmetic progressions and geometric progressions. 3.1 3.4
3.4 Polar form and De Moivre's Theorem. 3.2, 3.3 3.5
3.5 Roots for complex numbers. 3.4
3.6 Modulus and inequalities for complex numbers. 2.5, 3.2, 3.4

Topic 4: Integers and rational numbers.

Sub-topic Title Assumed background. Where does it go?
4.1 Integers and rational numbers. 2.1, 2.2 4.2
4.2 Divisibility for integers. 4.1 4.3
4.3 Division Algorithm, Well-ordering Principle for Integers, and Principle of Mathematical Induction. 4.2 4.4
4.4 Prime numbers, composite numbers, and Euclid's Lemma. 4.3 4.5
4.5 Rationals and irrationals, and surds of non-negative real numbers. 4.4 4.6
4.6 Greatest common divisor. 4.5

Topic 5: Set language.

Sub-topic Title Assumed background. Where does it go?
5.1 Set equality, subset relation, method of specification. 1.3 5.2, 5.3
5.2 Examples on application of the method of specification in set construction. 5.1
5.3 Set operations. 1.3, 5.1 5.4, 5.6
5.4 Examples of proofs and `dis-proofs' concerned with `subset relations'. 1.4, 5.1, 5.2, 5.3 5.6
5.5 Ordered pairs, ordered triples and cartesian products. 5.3
5.6 Power set. 1.4, 5.3, 5.4

Topic 6. Functions and relations.

Sub-topic Title Assumed background. Where does it go?
6.1 Notion of functions and its pictorial visualizations. 5.5 6.2, 6.3, 6.4
6.2 Surjectivity and Injectivity. 1.4, 1.5, 6.1 6.5, 6.6, 6.7
6.3 Image sets and pre-image sets. 1.4, 5.1, 5.2, 6.1 6.8, 6.13
6.4 Compositions, Surjectivity and Injectivity. 1.4, 1.5, 6.2
6.5 Notion of inverse functions. 6.1, 6.2 6.9, 6.10
6.6 Examples on surjectivity and injectivity for `nice' real-valued functions of one real variable. 6.2
6.7 Examples on surjectivity and injectivity for `simple' complex-valued functions of one complex variable. 3.5, 6.2
6.8 Image Sets and pre-image sets under `nice' real-valued functions of one real variable. 6.3
6.9 Examples on finding inverse functions for `simple' bijective functions. 6.5, 6.6, 6.7 6.12
6.10 Relations and the formal definition for the notion of functions. 1.4, 1.5, 6.1 6.11, 6.12
6.11 `Well-defined-ness' for functions. 6.11
6.12 Existence and uniqueness of inverse functions. 6.5, 6.10
6.13 Theoretical results involving image sets and pre-image sets. 5.3, 6.2, 6.3
6.14 Equivalence relations. 6.10 6.15
6.15 Basic examples on equivalence relations. 6.14 6.16
6.16 Integers modulo n. 4.2, 6.14, 6.15 6.17
6.17 Arithmetic in Integers modulo n. 6.10, 6.11, 6.14, 6.15, 6.16 6.18, 6.19
6.18 Equivalence relations defined by level sets of functions. 6.3, 6.14 6.19
6.19 Quotients and equivalence classes. 6.10, 6.14, 6.18

Topic 7: Infinite sets.

Sub-topic Title Assumed background. Where does it go?
7.1 Equipotence. 6.10, 6.12 6.13
7.2 Further examples on equipotence. 7.1 7.3
7.3 Cantor's diagonal argument. 7.2 7.4
7.4 Subpotence and strict subpotence. 7.3 7.5, 7.6
7.5 Schroeder-Bernstein Theorem. 7.4 7.6
7.6 Cantor's Theorem and its consequences. 7.3, 7.4, 7.5 7.8
7.7 Finite sets versus infinite sets. 7.2, 7.4, 7.6 7.8
7.8 Countable sets and uncountable sets. 7.4, 7.5, 7.6, 7.7
Back to Top

Assignments.

Mandatory submission, for assessment purpose? Optional proof-writing exercise (to be submitted separately from mandatory submission)? Due date? Any closely related notes, in terms of mathematical content?
Assignment 1 Question sheet Answers and selected solution Questions (1)-(7), (9a), (9b), (10a), (10b), (11a). Questions (8a), (8b). 29/01 2359hrs. 3.1-3.5.
Assignment 2 Question sheet Answers and selected solution Questions (1), (2), (3a), (3d), (9a), (9b), (9c.i), (9c.iii), (11), (12), (13a), (13b). Question (4a). 19/02 2359hrs. 2.2-2.6, 3.6.
Assignment 3 Question sheet Answers and selected solution Questions (1), (3), (6), (7a), (7b.i), (8a), (8b.ii), (8c.i), (8c.ii), (9), (10), (11), (16), (17a), (17b.i). Questions (8c.iii), (19c). 28/02 2359hrs. 1.2,1.3, 4.1-4.6.
Assignment 4 Question sheet Answers and selected solution Questions (1), (2), (3c), (3d), (4), (5a), (6), (9), (12). Questions (7a), (10a), (13a), (16b). 21/03 2359hrs. 1.4,1.5, 5.1-5.6.
Assignment 5 Question sheet (Q8 amended 27/03.) Answers and selected solution Questions (1), (3), (5), (6), (7a), (7b), (8), (9), (10), (12a), (19), (20), (21). Questions (16a), (17). 05/04 2359hrs. 6.1,6.2, 6.4-6.7, 6.9-6.12.
Assignment 6 Question sheet Answers and selected solution Questions (1),(2),(3),(6). 19/04 2359hrs. 6.13-6.15, 7.1-7.8.

Back to the general page.
Last modified: 1230hrs, 20-04-2024 (HKT)