Lectures
| Lectures | Title | Recommended reading |
|---|---|---|
| Lect 1 Th 9/24 | Introduction. Mathematical symbols you should know. Type of problems we will solve: counting, proofs, ... | Table of mathematical symbols |
| Lect 2 T 9/29 | Logic. Propositions. Compound propositions: conjunction, disjunction, exclusive or. Logical equivalences. De Morgan's laws. | Important logical equivalences |
Lect 3 Th 10/1 | Logic: Conditional. Contrapositive of a conditional. Biconditional Methods of proof: direct proofs, indirect proofs | |
| Lect 4 T 10/6 | Methods of proof Rules of inference. Proof by contradiction. Proof by case. Mistakes in proofs | Examples of mistakes in proofs |
Lect 5 Th 10/8 | Logic: quantifiers. Constructive and non constructive proofs Set theory Definition of sets. Terminology: elements, inclusion, union, intersection | See wikipedia articles on Russell's paradox and the barber paradox |
Lect 6 T 10/13 | Set theory: Important properties; Membership tables cardinality; inclusion exclusion principle; computer representation of sets | |
Lect 7 Th 10/15 | Quiz 1: Logic and methods of proof Functions. Definition. Injection, Surjection, Bijection. | |
Lect 8 T 10/20 | Functions Bijection; Special functions: floor and ceiling Growth of Functions. The concept of O ("big O"). | Some useful sites on: |
Lect 9 Th 10/22 | Growth of functions Big-O, Big-Omega, Big-Theta notations Algorithms. Definition. Pseudo code. Time complexity. Example: Linear search | See wikipedia article on Algorithms |
Lect 10 Th 10/27 | Algorithms Binary search. Complexity: logarithmic, linear, polynomial, exponential. Number theory. Divisibility. Prime numbers. Fundamental theorem of arithmetics | Prime numbers |
Lect 11 Th 10/29 | Number theory Sieve of Eratosthene. Definition of gcd and lcm. Bezout's identity. Division. Modular arithmetics. Congruence | Sieve of Eratosthenes |
Lect 12 T 11/3 | Number theory Modular arithmetics. Fermat's little theorem. | Summary Notes on Number Theory |
Lect 13 Th 11/5 | Number theory Computing GCD: Euclid's algorithm Sequences Arithmetic and geometric progressions. Summations | |
Lect 14 T 11/10 | Midterm Logic. Proofs. | |
Lect 15 Th 11/12 | Proof by induction Induction. Strong induction. Fibonacci numbers. | |
Lect 16 T 11/17 | Proof by induction Strong induction. Recursion. Tower of Hanoi. | |
Lect 17 T 11/19 | Counting Product rule. Sum rule. Pigenhole principle |