CS6015 - Logic and Combinatorics for Computer Science

Meetings  

• Theme 1 : Basics of Mathematical Proofs - 11 meetings

  • Meeting 01 : Mon, Aug 03, 11:00 am-11:50 am - Raghavendra Rao

  Administrative announcements.

  References
  Exercises
  Reading

  Administrative announcements.
  

  References:

  None

  • Meeting 02 : Tue, Aug 04, 10:00 am-10:50 am - Raghavendra Rao

  Propositional logic (quick review). Predicate logic. Quantifiers.

  References	[Rosen] Section 1.1, 1.2 and 1.3.
  Exercises
  Reading

  Propositional logic (quick review). Predicate logic. Quantifiers.
  

  References:

  [Rosen] Section 1.1, 1.2 and 1.3.

  • Meeting 03 : Wed, Aug 05, 09:00 am-09:50 am - Raghavendra Rao

  Rules of inference. Arguments and argument forms.

  References	[Rosen] Sections 1.4 and 1.5.
  Exercises
  Reading

  Rules of inference. Arguments and argument forms.
  

  References:

  [Rosen] Sections 1.4 and 1.5.

  • Meeting 04 : Thu, Aug 06, 12:00 pm-12:50 pm -

  Validity. Notion of a proof.

  References	[Rosen]
  Exercises
  Reading

  Validity. Notion of a proof.
  

  References:

  [Rosen]

  • Meeting 05 : Mon, Aug 10, 11:00 am-11:50 am -

  Proof techniques: Direct proof. Proof by contradiction. Contrapositive.

  References	[Rosen]
  Exercises
  Reading

  Proof techniques: Direct proof. Proof by contradiction. Contrapositive.
  

  References:

  [Rosen]

  • Meeting 06 : Tue, Aug 11, 10:00 am-10:50 am -

  More proof techniques. Existential proofs. Uniqueness proofs. Axioms for set theory. Russell's paradox.

  References	[Rosen]
  Exercises
  Reading

  More proof techniques. Existential proofs. Uniqueness proofs. Axioms for set theory. Russell's paradox.
  

  References:

  [Rosen]

  • Meeting 07 : Wed, Aug 12, 09:00 am-09:50 am -

  Notion of cardinality for infinite sets. Countable and uncountable sets. Examples of countable sets: Set of integers, rational numbers.

  References	[Rosen]. A notes on <a href="http://www.math.ucla.edu/~mwilliams/pdf/cardinality.pdf"> Cardinality</a> by Michael Williams.
  Exercises
  Reading

  Notion of cardinality for infinite sets. Countable and uncountable sets. Examples of countable sets: Set of integers, rational numbers.
  

  References:

  [Rosen]. A notes on Cardinality by Michael Williams.

  • Meeting 08 : Thu, Aug 13, 12:00 pm-12:50 pm -

  Uncountability. Cantor's diagonalization.

  References	[Rosen]. A notes on <a href="http://www.math.ucla.edu/~mwilliams/pdf/cardinality.pdf"> Cardinality</a> by Michael Williams.
  Exercises
  Reading

  Uncountability. Cantor's diagonalization.
  

  References:

  [Rosen]. A notes on Cardinality by Michael Williams.

  • Meeting 09 : Mon, Aug 17, 11:00 am-11:50 am -

  More examples of uncountable sets. Principle of mathematical induction. Example.

  References	[Rosen]. A notes on <a href="http://www.math.ucla.edu/~mwilliams/pdf/cardinality.pdf"> Cardinality</a> by Michael Williams.
  Exercises
  Reading

  More examples of uncountable sets. Principle of mathematical induction. Example.
  

  References:

  [Rosen]. A notes on Cardinality by Michael Williams.

  • Meeting 10 : Tue, Aug 18, 10:00 am-10:50 am -

  Mathematical Induction: Examples.

  References	[Rosen]
  Exercises
  Reading

  Mathematical Induction: Examples.
  

  References:

  [Rosen]

  • Meeting 11 : Wed, Aug 19, 09:00 am-09:50 am -

  Disproving theorems: counter examples. Fallacies in proofs.

  References
  Exercises
  Reading	The book "Mathematical Fallacies Flaws and Flimflam" by Ed Barbeau has a good collection of well known mathematical fallacies.

  Disproving theorems: counter examples. Fallacies in proofs.
  

  References:	None
  Reading:	The book "Mathematical Fallacies Flaws and Flimflam" by Ed Barbeau has a good collection of well known mathematical fallacies.

• Theme 2 : Describing Sets : Logic & Computation - 15 meetings

  • Meeting 12 : Thu, Aug 20, 12:00 pm-12:50 pm -

  Introduction to first order logic.

  References	The book "Logic in Computer Science" by Michael Huth and Mark Ryan is a good book on Logic from a computer science perspective. Lectures will be in part based on this book.
  Exercises
  Reading

  Introduction to first order logic.
  

  References:

  The book "Logic in Computer Science" by Michael Huth and Mark Ryan is a good book on Logic from a computer science perspective. Lectures will be in part based on this book.

  • Meeting 13 : Mon, Aug 24, 11:00 am-11:50 am -

  Decidability. Undecidability of a First order formula. Expressivenes of first order formula.

  References	Logic in Computer Science by Huth and Ryan.
  Exercises
  Reading

  Decidability. Undecidability of a First order formula. Expressivenes of first order formula.
  

  References:

  Logic in Computer Science by Huth and Ryan.

  • Meeting 14 : Mon, Aug 24, 04:45 pm-05:35 pm -

  [Make up for THU 27-08-2015]. Undecidability of checking validity of FO w.f.f. Expressiveness of First order logic: Path(u,v) cannot be expressed in FO. Compactness theorem (statement only).

  References	FOr undecidability, the reduction is based on the one given in the book "Logic for Computer Scientists" by Uwe Schoening. For expressiveness, please See "Logic in Computer Science" by Michael Huth and Mark Ryan.
  Exercises
  Reading

  [Make up for THU 27-08-2015]. Undecidability of checking validity of FO w.f.f. Expressiveness of First order logic: Path(u,v) cannot be expressed in FO. Compactness theorem (statement only).
  

  References:

  FOr undecidability, the reduction is based on the one given in the book "Logic for Computer Scientists" by Uwe Schoening. For expressiveness, please See "Logic in Computer Science" by Michael Huth and Mark Ryan.

  • Meeting 15 : Tue, Aug 25, 10:00 am-10:50 am -

  Notion of a proof in first order logic. Soundness and Completeness of FO. (Statement only).

  References	[Enderton]
  Exercises
  Reading

  Notion of a proof in first order logic. Soundness and Completeness of FO. (Statement only).
  

  References:

  [Enderton]

  • Meeting 16 : Wed, Aug 26, 09:00 am-09:50 am -

  Formal model for number theory. Peano arithmetic. Goedel's incompleteness theorem(Statement only).

  References	Class notes + [Enderton]
  Exercises
  Reading

  Formal model for number theory. Peano arithmetic. Goedel's incompleteness theorem(Statement only).
  

  References:

  Class notes + [Enderton]

  • Meeting 17 : Mon, Aug 31, 11:00 am-11:50 am -

  Verification of systems using logic. Model Checking. Linear Time Temporal Logic.

  References	<a href="http://www.cs.bham.ac.uk/research/projects/lics/"> Book</a> by Michael Huth and Mark Ryan "Logic in Computer Science"
  Exercises
  Reading

  Verification of systems using logic. Model Checking. Linear Time Temporal Logic.
  

  References:

  Book by Michael Huth and Mark Ryan "Logic in Computer Science"

  • Meeting 18 : Tue, Sep 01, 10:00 am-10:50 am -

  Linear Time Temporal logic semantics and examples.

  References	<a href="http://www.cs.bham.ac.uk/research/projects/lics/"> Book</a> by Michael Huth and Mark Ryan "Logic in Computer Science"
  Exercises
  Reading

  Linear Time Temporal logic semantics and examples.
  

  References:

  Book by Michael Huth and Mark Ryan "Logic in Computer Science"

  • Meeting 19 : Wed, Sep 02, 09:00 am-09:50 am -

  Modelling systems using LTL. Mutual exlcusion: an example.

  References	<a href="http://www.cs.bham.ac.uk/research/projects/lics/"> Book</a> by Michael Huth and Mark Ryan "Logic in Computer Science" See Chapter 4.
  Exercises
  Reading

  Modelling systems using LTL. Mutual exlcusion: an example.
  

  References:

  Book by Michael Huth and Mark Ryan "Logic in Computer Science" See Chapter 4.

  • Meeting 20 : Thu, Sep 03, 12:00 pm-12:50 pm -

  Modal Logic: syntax and semantics

  References	<a href="http://www.cs.bham.ac.uk/research/projects/lics/"> Book</a> by Michael Huth and Mark Ryan "Logic in Computer Science", Chapter 5.
  Exercises
  Reading

  Modal Logic: syntax and semantics
  

  References:

  Book by Michael Huth and Mark Ryan "Logic in Computer Science", Chapter 5.

  • Meeting 21 : Mon, Sep 07, 11:00 am-11:50 am -

  Effective Computability: Models of computation

  References	[HMU]. Also see Dexter Kozen: "Automata and Computability"
  Exercises
  Reading

  Effective Computability: Models of computation
  

  References:

  [HMU]. Also see Dexter Kozen: "Automata and Computability"

  • Meeting 22 : Wed, Sep 09, 09:00 am-09:50 am -

  Recursive and Recursively enumerable languages. Decidability and semi-decidability. The halting problem. Undecidability.

  References	[HMU]. Also see Dexter Kozen: "Automata and Computability"
  Exercises
  Reading

  Recursive and Recursively enumerable languages. Decidability and semi-decidability. The halting problem. Undecidability.
  

  References:

  [HMU]. Also see Dexter Kozen: "Automata and Computability"

  • Meeting 23 : Thu, Sep 10, 12:00 pm-12:50 pm -

  Halting problem is undecidable.

  References
  Exercises
  Reading

  Halting problem is undecidable.
  

  References:

  None

  • Meeting 24 : Mon, Sep 14, 11:00 am-11:50 am -

  PCP is undecidable

  References
  Exercises
  Reading

  PCP is undecidable
  

  References:

  None

  • Meeting 25 : Tue, Sep 15, 10:00 am-10:50 am -

  More on undecidability.

  References
  Exercises
  Reading

  More on undecidability.
  

  References:

  None

  • Meeting 26 : Wed, Sep 16, 09:00 am-09:50 am -

  To Be Announced

  References
  Exercises
  Reading

  To Be Announced
  

  References:

  None

• Theme 3 : Size of Sets : Counting & Combinatorics - 17 meetings

  • Meeting 27 : Mon, Sep 21, 11:00 am-11:50 am - Jayalal Sarma

  Review of Basics. Permutations, Combinations. Double Counting as a combinatorial proof technique.

  References	[KR] Page 335 - 340 and Pages 354 - 369.<br>
  Exercises	Exercises from [KR] : Exercises for section 5.1, 5.3, 5.4.
  Reading	Sections 5.1, 5.2, 5.3 in the Benjamin-Quinn [BQ] Book.

  Review of Basics. Permutations, Combinations. Double Counting as a combinatorial proof technique.
  

  References:	[KR] Page 335 - 340 and Pages 354 - 369.
  Exercises:	Exercises from [KR] : Exercises for section 5.1, 5.3, 5.4.
  Reading:	Sections 5.1, 5.2, 5.3 in the Benjamin-Quinn [BQ] Book.

  • Meeting 28 : Tue, Sep 22, 10:00 am-10:50 am - Jayalal Sarma

  Counting by Bijections. Examples. Approximate Bijection.

  References	Sections 6.1 in the Benjamin-Quinn [BQ] Book.
  Exercises
  Reading

  Counting by Bijections. Examples. Approximate Bijection.
  

  References:

  Sections 6.1 in the Benjamin-Quinn [BQ] Book.

  • Meeting 29 : Wed, Sep 23, 09:00 am-09:50 am - Jayalal Sarma

  Catalan numbers. Diagonal Avoiding paths. Counting by bijections.

  References	Section 2.4 in <a href="http://mathcircle.berkeley.edu/BMC6/pdf0607/catalan.pdf">Notes by Tom Davis</a>.
  Exercises
  Reading

  Catalan numbers. Diagonal Avoiding paths. Counting by bijections.
  

  References:

  Section 2.4 in Notes by Tom Davis.

  • Meeting 30 : Tue, Sep 29, 10:00 am-10:50 am - Jayalal Sarma

  Permutation and Combinations with repetitions. Multichoosing. Interpretations. Identities related to multichoosing.

  References
  Exercises
  Reading

  Permutation and Combinations with repetitions. Multichoosing. Interpretations. Identities related to multichoosing.
  

  References:

  None

  • Meeting 31 : Wed, Sep 30, 09:00 am-09:50 am - Jayalal Sarma

  Inclusion-Exclusion Principle. Proof. Applications

  References	<a href="http://math.mit.edu/~fox/MAT307-lecture04.pdf">Lecutre notes</a>. The proof of PI was not done as per this (please use class notes). But examples are from here.
  Exercises	Use inclusion-exclusion to derive a formula for the number of numbers less than n, but co-prime to n.
  Reading

  Inclusion-Exclusion Principle. Proof. Applications
  

  References:	Lecutre notes. The proof of PI was not done as per this (please use class notes). But examples are from here.
  Exercises:	Use inclusion-exclusion to derive a formula for the number of numbers less than n, but co-prime to n.

  • Meeting 32 : Thu, Oct 01, 12:00 pm-12:50 pm - Jayalal Sarma

  Derangements, Chromatic Polynomial of a graph.

  References	Section 6.6 in [KR]<br> Section 9.1 in <a href="http://www.ltcc.ac.uk/courses/enumerative_combinatorics/l9.pdf">Notes on Enumerative Combinatorics by Cameron</a>.
  Exercises
  Reading

  Derangements, Chromatic Polynomial of a graph.
  

  References:

  Section 6.6 in [KR] Section 9.1 in Notes on Enumerative Combinatorics by Cameron.

  • Meeting 33 : Mon, Oct 05, 11:00 am-11:50 am - Jayalal Sarma

  Pigeon Hole Principle and two applications.

  References	Kennth Rosen's Book Section 5.2. I used some examples from the old notes as well.
  Exercises	If I have at least two friends on Facebook, there will be two friends in my friends list such that I have the same number of mutual friends with them.
  Reading

  Pigeon Hole Principle and two applications.
  

  References:	Kennth Rosen's Book Section 5.2. I used some examples from the old notes as well.
  Exercises:	If I have at least two friends on Facebook, there will be two friends in my friends list such that I have the same number of mutual friends with them.

  • Meeting 34 : Tue, Oct 06, 10:00 am-10:50 am - Jayalal Sarma

  More Applications of PHP. Dirichlet's Approximation principle, Monotone subsequence lemma.

  References	<ul> <li><a href="https://www.expii.com/topic/2468">Dirichlet's Principle</a></li> <li><a href="http://www2.math.ethz.ch/education/bachelor/lectures/hs2013/math/tpdm/lecture-notes.pdf">Lecture 2 of the notes by Benny Sudakov</a></li> </ul>
  Exercises
  Reading	<a href="http://stat.wharton.upenn.edu/~steele/Publications/PDF/VOTMSTOEAS.pdf">Variations of Erdoz-Szekeres Monnotone Subsequence Theorem</a>

  More Applications of PHP. Dirichlet's Approximation principle, Monotone subsequence lemma.
  

  References:	Dirichlet's Principle Lecture 2 of the notes by Benny Sudakov
  Reading:	Variations of Erdoz-Szekeres Monnotone Subsequence Theorem

  • Meeting 35 : Wed, Oct 07, 09:00 am-09:50 am -

  Erdos-Szekeres Subsequence Theorem. Happy Ending Problem. Ramsey Theory. Ramsey Numbers. Erdos-Szekeres upper bound.

  References	<a href="http://people.maths.ox.ac.uk/gouldm/ramsey.pdf">Section 6 of the Notes</a>
  Exercises
  Reading	<a href="http://www.math.ucsd.edu/~ronspubs/90_06_ramsey_theory.pdf">Essay on Ramsey Numbers</a>

  Erdos-Szekeres Subsequence Theorem. Happy Ending Problem. Ramsey Theory. Ramsey Numbers. Erdos-Szekeres upper bound.
  

  References:	Section 6 of the Notes
  Reading:	Essay on Ramsey Numbers

  • Meeting 36 : Thu, Oct 08, 12:00 pm-12:50 pm -

  Lower Bounds for Diagonal Ramsey Numbers. The probabilistic method.

  References	<a href="http://people.maths.ox.ac.uk/gouldm/ramsey.pdf">Section 6 of the Notes</a>
  Exercises	<a href="">Infinite version of Ramsey Numbers. Applications to Program Termination Arrguments.</a>
  Reading

  Lower Bounds for Diagonal Ramsey Numbers. The probabilistic method.
  

  References:	Section 6 of the Notes
  Exercises:	Infinite version of Ramsey Numbers. Applications to Program Termination Arrguments.

  • Meeting 37 : Mon, Oct 12, 11:00 am-11:50 am -

  Solving Recurrences. Linear Homogeneous Recurrences with constant coefficients. Characterestic equation. Case when the characterestic equation has distinct roots.

  References
  Exercises
  Reading

  Solving Recurrences. Linear Homogeneous Recurrences with constant coefficients. Characterestic equation. Case when the characterestic equation has distinct roots.
  

  References:

  None

  • Meeting 38 : Tue, Oct 13, 10:00 am-10:50 am -

  More examples of recurrence relations and their solutions.

  References
  Exercises
  Reading

  More examples of recurrence relations and their solutions.
  

  References:

  None

  • Meeting 39 : Wed, Oct 14, 09:00 am-09:50 am -

  Linear Non-homogeoneous Recurrence relations with constant coefficients. Particular solution. A characterization of the general solution.

  References
  Exercises
  Reading

  Linear Non-homogeoneous Recurrence relations with constant coefficients. Particular solution. A characterization of the general solution.
  

  References:

  None

  • Meeting 40 : Thu, Oct 15, 12:00 pm-12:50 pm -

  Example of Non-homogeneous recurrence relations. Forms of particular solution from the forms of the non-homogeneous part of the linear recurrence relations. Solution examples.

  References
  Exercises
  Reading

  Example of Non-homogeneous recurrence relations. Forms of particular solution from the forms of the non-homogeneous part of the linear recurrence relations. Solution examples.
  

  References:

  None

  • Meeting 41 : Tue, Oct 20, 10:00 am-10:50 am -

  Solving recurrences by substitution. Index substitution and Functional Substitution. Examples, Counting. Derangements using Recurrence relations. Generating Function for infinite sequences. Extended binomial theorem.

  References	Rosen's Textbook - Section 6.3 and 6.4<br> <a href="https://mikespivey.wordpress.com/2011/11/22/derangements/">Counting Derangements</a>
  Exercises
  Reading

  Solving recurrences by substitution. Index substitution and Functional Substitution. Examples, Counting. Derangements using Recurrence relations. Generating Function for infinite sequences. Extended binomial theorem.
  

  References:

  Rosen's Textbook - Section 6.3 and 6.4 Counting Derangements

  • Meeting 42 : Wed, Oct 21, 09:00 am-09:50 am -

  Generating Function Method for solving linear recurrence relations. Linear case first.

  References	[Rosen] Section 6.4.
  Exercises
  Reading

  Generating Function Method for solving linear recurrence relations. Linear case first.
  

  References:

  [Rosen] Section 6.4.

  • Meeting 43 : Mon, Oct 26, 11:00 am-11:50 am -

  Non-homogeneous case, Higher Degree Case, Non-linear special cases : Catalan Numbers.

  References	<a href="http://www.math.drexel.edu/~foucart/TeachingFiles/F12/M235Lect4.pdf">Notes</a>
  Exercises
  Reading

  Non-homogeneous case, Higher Degree Case, Non-linear special cases : Catalan Numbers.
  

  References:

  Notes

• Theme 4 : Structured Sets : Operations & Relations - 10 meetings

  • Meeting 44 : Tue, Oct 27, 10:00 am-10:50 am -

  Structured Sets. Relations, Representing relations. Graphs and Hypergraphs. Reflexivity, Symmetry and Transitivity properties. Testing the properties given the graph.

  References	Section 7.1-7.5 in Kenneth Rosen's Textbook
  Exercises
  Reading

  Structured Sets. Relations, Representing relations. Graphs and Hypergraphs. Reflexivity, Symmetry and Transitivity properties. Testing the properties given the graph.
  

  References:

  Section 7.1-7.5 in Kenneth Rosen's Textbook

  • Meeting 45 : Wed, Oct 28, 09:00 am-09:50 am -

  Equivalence Classes. Closure of relations. Reflexive, Symmetric and Transitive closures. Computing the transitive closure.

  References	Section 7.1-7.5 in Kenneth Rosen's Textbook
  Exercises	design an algorithm for finding the equivalence classes of a given equivalence relation (given as a graph).
  Reading

  Equivalence Classes. Closure of relations. Reflexive, Symmetric and Transitive closures. Computing the transitive closure.
  

  References:	Section 7.1-7.5 in Kenneth Rosen's Textbook
  Exercises:	design an algorithm for finding the equivalence classes of a given equivalence relation (given as a graph).

  • Meeting 46 : Thu, Oct 29, 12:00 pm-12:50 pm -

  Posets, Examples, Total order, Hasse diagrams.

  References	Section 7.6 in Kenneth Rosen's Textbook
  Exercises
  Reading

  Posets, Examples, Total order, Hasse diagrams.
  

  References:

  Section 7.6 in Kenneth Rosen's Textbook

  • Meeting 47 : Mon, Nov 02, 11:00 am-11:50 am -

  Total order and Well-order, Well-ordering induction. An example proof using well-ordering.

  References	Section 7.6 in Kenneth Rosen's Textbook
  Exercises
  Reading

  Total order and Well-order, Well-ordering induction. An example proof using well-ordering.
  

  References:

  Section 7.6 in Kenneth Rosen's Textbook

  • Meeting 48 : Tue, Nov 03, 10:00 am-10:50 am -

  Maximal, Minimal, greatest, least elements. GLBs and LUBs. Lattices. Examples and non-examples. Identities on lattices. Distributive lattices.

  References	Section 7.6 in Kenneth Rosen's Textbook
  Exercises
  Reading

  Maximal, Minimal, greatest, least elements. GLBs and LUBs. Lattices. Examples and non-examples. Identities on lattices. Distributive lattices.
  

  References:

  Section 7.6 in Kenneth Rosen's Textbook

  • Meeting 49 : Wed, Nov 04, 09:00 am-09:50 am -

  Complete posets and their structure. Knaster-Tarski Fixed Point theorem. Proof. Cantor-Bernstein-Schroeder Theorem. Applications.

  References	<a href="http://www.drmaciver.com/2012/02/the-power-of-fixed-point-theorems-in-set-theory/">Notes</a>
  Exercises
  Reading

  Complete posets and their structure. Knaster-Tarski Fixed Point theorem. Proof. Cantor-Bernstein-Schroeder Theorem. Applications.
  

  References:

  Notes

  • Meeting 50 : Thu, Nov 05, 12:00 pm-01:20 pm -

  Using Knaster-Tarski's theorem to prove Schroeder-Bernstein theorem. Applications of fixed points to argue about semantics of while loops. <br> <br>Outline : Another use of structure of a lattice. Incidence algebra. Delta, Zeta and Mobious functions of a lattice. Mobious inversion. Deriving Principle of inclusion-exclusion as a Corollary.

  References	<a href="http://www.drmaciver.com/2012/02/the-power-of-fixed-point-theorems-in-set-theory/">Notes</a>
  Exercises
  Reading

  Using Knaster-Tarski's theorem to prove Schroeder-Bernstein theorem. Applications of fixed points to argue about semantics of while loops.
  
  Outline : Another use of structure of a lattice. Incidence algebra. Delta, Zeta and Mobious functions of a lattice. Mobious inversion. Deriving Principle of inclusion-exclusion as a Corollary.
  

  References:

  Notes

  • Meeting 51 : Mon, Nov 09, 11:00 am-11:50 am -

  Structured Sets. Closure, Associativity, Identity, Inverse, Commutativity. Groups, Semigroups, monoids, groupoids.

  References	Kozen's Textbook material
  Exercises
  Reading

  Structured Sets. Closure, Associativity, Identity, Inverse, Commutativity. Groups, Semigroups, monoids, groupoids.
  

  References:

  Kozen's Textbook material

  • Meeting 52 : Tue, Nov 10, 10:00 am-10:50 am -

  Subgroups. Cosets. Lagrange's theorem. Group action on sets. Orbits.

  References	Use notes from class.
  Exercises
  Reading

  Subgroups. Cosets. Lagrange's theorem. Group action on sets. Orbits.
  

  References:

  Use notes from class.

  • Meeting 53 : Mon, Nov 16, 11:00 am-12:00 pm -

  Polya's counting method. Burnside's Lemma

  References	<a href="http://www.saylor.org/site/wp-content/uploads/2011/05/Burnsides-Lemma.pdf">Notes</a>. The example used in class was different.
  Exercises
  Reading

  Polya's counting method. Burnside's Lemma
  

  References:

  Notes. The example used in class was different.

• Theme 5 : Evaluation Meetings - 5 meetings

  • Meeting 54 : Tue, Sep 08, 10:00 am-10:50 am -

  Quiz 1 based on contents of Lectures 1 to 23

  References
  Exercises
  Reading

  Quiz 1 based on contents of Lectures 1 to 23
  

  References:

  None

  • Meeting 55 : Thu, Sep 24, 12:00 pm-12:50 pm -

  Short Exam 1 based on contents of Lectures 1 to 31

  References
  Exercises
  Reading

  Short Exam 1 based on contents of Lectures 1 to 31
  

  References:

  None

  • Meeting 56 : Mon, Oct 19, 11:00 am-11:50 am -

  Quiz 2 based on contents of Lectures 24 to 44

  References
  Exercises
  Reading

  Quiz 2 based on contents of Lectures 24 to 44
  

  References:

  None

  • Meeting 57 : Thu, Nov 12, 12:00 pm-12:50 pm -

  Short Exam 2 based on contents of Lectures 40 to 52

  References
  Exercises
  Reading

  Short Exam 2 based on contents of Lectures 40 to 52
  

  References:

  None

  • Meeting 58 : Mon, Nov 23, 09:30 am-12:30 pm -

  Based on the materials covered in the entire course.

  References
  Exercises
  Reading

  Based on the materials covered in the entire course.
  

  References:

  None