Date |
Topic |
Reference |
Jan 13 |
Introduction to course, Simple examples of Pigeon Hole principle + Administrative announcements |
None |
Jan 14 |
Example of Pigeon Hole principle continued |
None |
Jan 16 |
Introduction to Logic, Propositions, basic operators, negation, conjunction, disjunction, exclusive OR |
Sec. 1.1 [KR] |
Jan 17 |
Implication operator, translating English sentences into formulae |
Sec. 1.2 [KR] |
Jan 18 (extra class) |
Double-Implication, Inverse, Converse, Contrapositive, Precedence Rules for opertors, Logical Equivalences, Logic Puzzles (Knights and Knaves) |
Sec. 1.3 [KR] |
Jan 20 |
Predicates, Examples, Quantifiers, Domain of variables, What happens when domain is empty? Quantifiers with restricted domains |
Sec. 1.4 [KR] |
Jan 21 |
No class. |
None. |
Jan 23 |
Predicates and Quantifiers: examples, Translating English sentences to logical formulas and vice versa. There exists exactly one quantifier. |
Sec. 1.4 [KR] |
Jan 24 |
Tutorial 1 |
None |
Jan 27 |
Logical Equivalences, definition, examples, non-examples. Proving a logical equivalence, disproving a logical equivalence. |
Sec. 1.4 [KR], see Ex. 19. |
Jan 28 |
Free variable and bound variables, scope of quantifiers, examples. Equivalences again. Translating English sentences to logical formulas, identifying correct / incorrect conclusions. |
Sec. 1.4 [KR] |
Jan 30 |
Nested quantifiers, examples. Order of quantifiers. Argument and Argument forms. Example argument forms. Why are some forms valid versus some invalid? |
Sec. 1.5, 1.6 [KR] |
Jan 31 |
Rules of inference and examples. |
Sec. 1.6 [KR] |
Feb 3 |
Resolution, Universal / Existential Generalization / Instantiation and examples. |
Sec. 1.6 [KR] |
Feb 4 |
Introduction to Proofs, direct proof, proof by contrapositive, proof by contradiction. |
Sec. 1.7 [KR] |
Feb 6 |
Examples of proof by contradiction, Euclids proof for infinitely many primes, sqrt(2) is irrational. |
Sec. 1.7 [KR], Proofs from the Book |
Feb 7 |
Tutorial 2 |
None |
Feb 10 |
Existential proofs, Proof by cases examples, Two player games, winning strategies |
Sec. 1.8 [KR] |
Feb 11 |
Chomp : two player game, special cases, n*n, 2*n |
Sec. 1.8 [KR] |
Feb 13 |
Chomp continued : player 2 does not have a winning strategy |
Sec. 1.8 [KR] |
Feb 14 |
Tutorial 2 -- discussion |
None |
Feb 17 |
Quiz 1 |
None |
Feb 18 |
Quiz 1 -- discussion; Introduction to Sets. |
None |
Feb 20 |
Sets, empty set, universal set, can a universal set exist? Russels paradox. |
Section 2.1 [KR] |
Feb 21 |
Responses to Russels paradox, basic set operations -- union, intersection, power set, examples. |
Section 2.2 [KR] |
Feb 24 |
Cartesian product, relations, reflexive, symmetric, antisymmetric and transitive relations. Can a relation be both
symmetric and antisymmetric? Examples, non-examples |
Section 9.1 [KR] |
Feb 25 |
Relations continued, basic counting, number of reflexive, symmetric and antisymmetric relations |
|
Feb 27 |
Relations and functions, examples, non-examples |
Section 9.1 [KR] |
Feb 28 |
Tutorial 3 |
None |
March 2 |
Cardinality of Infinite Sets using functions, cardinality of positive odd integers and positive integers |
Section 2.5 [KR] |
March 3 |
Cardinality of Infinite Sets continued, countably infinite sets, cross product of positive integers, introduction to diagonalization, set of reals is uncountable |
Section 2.5 [KR] |
March 5 |
Alternate definition of infinite sets. A bijection from A \cup {b} to A. Hilberts Grand Hotel Example. Towards uncomputable functions. |
None |
March 6 |
The set of all (C) programs, its cardinality. Example of an uncomputable program. Halting Problem. |
None |
March 9 |
No class |
None |
March 10 |
Non Instructional Day (Holi) |
None |
March 12 |
The technique of proof by Induction, Examples, Induction Trap "All horses are of the same color". |
Section 5.1 [KR] |
March 13 |
Proof by Induction, Examples: Odd Pie fight, tiling a square board with triominoes, Proving a stronger induction hypothesis. |
Section 5.1 [KR] |
March 16 |
Strong Induction and Principle of Well-ordering. Examples on Tournaments. Difference between proof using Induction and Strong Induction |
Section 5.2 [KR] |
March 17-19 |
No Class |
None |
March 20 |
Recursion, examples, proving bounds using induction, some important recursive functions. |
Section 5.3 [KR] Slides |
March 23 |
Recursion continued, recursive sets, lists and binary trees, recursive sequences. |
Section 5.3 [KR] Slides |
March 24 |
Counting, Basic Counting techniques, sum, product, subtraction and division rules. Examples. |
Section 6.1 [KR] Slides |
March 26 |
Pigeonhole principle and applications. |
Section 6.2 [KR] Slides |
March 30 |
Permutations and Combinations. |
Section 6.3 [KR] Slides |
March 31 |
Combinatorial Proofs |
Section 6.4[KR] Slides |
April 2 |
Permutations and Combinations with repetitions |
Section 6.5[KR] Slides Handout |
April 3 |
Principle of Inclusion Exclusion |
Section 8.5[KR] Slides Handout |
April 6 |
Principle of Inclusion Exclusion: Applications |
Section 8.5[KR] Slides Handout |
April 7 |
Application of recurrences: Catalan numbers |
Section 8.1[KR] Slides Handout |
April 9 |
Solving recurrences: Substitution Method |
Section 8.2[KR] Slides Handout |
April 10 |
Tutorial 5 |
Tut5 |
April 13 |
Solving recurrences: Linear Homogeneous Recurrences of degree two with constant coefficients. |
Section 8.2[KR] Slides Handout |
April 14 |
Solving recurrences: Linear Homogeneous Recurrences with constant coefficients. |
Section 8.2[KR] Slides Handout |
April 16 |
Solving recurrences: Linear Non-Homogeneous Recurrences with constant coefficients. Comparing Functions |
Section 8.2[KR] Slides Handout
Comparing-Functions.pdf |
April 17 |
Structured Sets: Closures of properties. |
Section 9.4[KR] Slides Handout |
April 20 |
Structured Sets: Equivalence Relations and Partial Orders. |
Section 9.5,9.6[KR] Slides Handout |
April 21 |
Structured Sets: Posets and Lattices. |
Section 9.6[KR] Slides Handout |
April 23 |
Structured Sets: Semi-groups, Monoids and Groups. |
Section 11.1, 11.2 of C. L . Liu Slides Handout |
April 24 |
Structured Sets: Groups, Subgroups and Genrators. |
Section 11.3, 11.4 of C. L . Liu Slides Handout |
April 27 |
Structured Sets: Lagrange's Theorem and proof. |
Section 11.3 of C. L . Liu Slides Handout |