Meetings

Click on the theme item for the meeting plan for that theme.Click on the meeting item for references, exercises, and additional reading related to it.

**Theme 1 : Describing Sets : Logic and Proofs**- 32 meetings- Meeting 01 : Mon, Aug 01, 11:00 am-11:50 am
- Meeting 02 : Tue, Aug 02, 10:00 am-10:50 am
- Meeting 03 : Wed, Aug 03, 09:00 am-09:50 am
- Meeting 04 : Thu, Aug 04, 12:00 pm-12:50 pm
- Meeting 05 : Mon, Aug 08, 11:00 am-11:50 am
- Meeting 06 : Tue, Aug 09, 10:00 am-10:50 am
- Meeting 07 : Wed, Aug 10, 09:00 am-09:50 am
- Meeting 08 : Thu, Aug 11, 12:00 pm-12:50 pm
- Meeting 09 : Tue, Aug 16, 10:00 am-10:50 am
- Meeting 10 : Wed, Aug 17, 09:00 am-09:50 am
- Meeting 11 : Thu, Aug 18, 12:00 pm-12:50 pm
- Meeting 12 : Mon, Aug 22, 11:00 am-11:50 am
- Meeting 13 : Tue, Aug 23, 10:00 am-10:50 am
- Meeting 14 : Wed, Aug 24, 09:00 am-09:50 am
- Meeting 15 : Thu, Aug 25, 12:00 pm-12:50 pm
- Meeting 16 : Tue, Aug 30, 10:00 am-10:50 am
- Meeting 17 : Wed, Aug 31, 09:00 am-09:50 am
- Meeting 18 : Thu, Sep 01, 12:00 pm-12:50 pm
- Meeting 19 : Tue, Sep 06, 10:00 am-10:50 am
- Meeting 20 : Wed, Sep 07, 09:00 am-09:50 am
- Meeting 21 : Tue, Sep 13, 10:00 am-10:50 am
- Meeting 22 : Thu, Sep 15, 12:00 pm-12:50 pm
- Meeting 23 : Fri, Sep 16, 11:00 am-11:50 am
- Meeting 24 : Mon, Sep 19, 11:00 am-11:50 am
- Meeting 25 : Tue, Sep 20, 10:00 am-10:50 am
- Meeting 26 : Wed, Sep 21, 09:00 am-09:50 am
- Meeting 27 : Thu, Sep 22, 12:00 pm-12:50 pm
- Meeting 28 : Mon, Sep 26, 11:00 am-11:50 am
- Meeting 29 : Mon, Sep 26, 04:15 pm-05:15 pm
- Meeting 30 : Tue, Sep 27, 10:00 am-10:50 am
- Meeting 31 : Wed, Sep 28, 09:00 am-09:50 am
- Meeting 32 : Thu, Sep 29, 12:00 pm-12:50 pm

Acad : Why this course? Choice between LCCS and LARP courses. The story and the thread the course will follow. Reference textbooks.<br> Admin : Evaluation plans, TA and Instructor contact hours, course homepage, moodle, mailing list. References Chapter 1 in Rosen's Textbook. Exercises Reading Acad : Why this course? Choice between LCCS and LARP courses. The story and the thread the course will follow. Reference textbooks.

Admin : Evaluation plans, TA and Instructor contact hours, course homepage, moodle, mailing list.References : Chapter 1 in Rosen's Textbook. Propositional Logic. Truth tables, Logical Equivalences. References Chapter 1 in Rosen's Textbook. Exercises Reading Propositional Logic. Truth tables, Logical Equivalences.References : Chapter 1 in Rosen's Textbook. Equivalences, Argument forms, Rules of inferences. Resolution principle. References Chapter 1 in Rosen's Textbook. Exercises Reading Equivalences, Argument forms, Rules of inferences. Resolution principle.References : Chapter 1 in Rosen's Textbook. More examples of resolution-refutation. References Chapter 1 in Rosen's Textbook. Exercises Reading More examples of resolution-refutation.References : Chapter 1 in Rosen's Textbook. Axiomatizations for propositional logic. Notion of soundness and completeness. Expressibility. References <a href="http://www.logicinaction.org/docs/ch1.pdf">Chapter 1</a>, <a href="http://www.logicinaction.org/docs/ch2.pdf">Chapter 2</a>, <a href="http://www.logicinaction.org/docs/ch3.pdf">Chapter 3</a>. Exercises Reading Axiomatizations for propositional logic. Notion of soundness and completeness. Expressibility.Remarks about expressibility. Monadic predicate logic (Syllogisitc). Propositional Functions, Predicates. Syntax for predicate logic.Predicate Logic - Syntax. References <a href="http://www.logicinaction.org/docs/ch4.pdf">Chapter 4</a> Exercises Reading Remarks about expressibility. Monadic predicate logic (Syllogisitc). Propositional Functions, Predicates. Syntax for predicate logic.Predicate Logic - Syntax.References : Chapter 4 Semantics and Examples. Need for a formal notion of semantics. References <a href="http://www.logicinaction.org/docs/ch4.pdf">Chapter 4</a> Exercises Reading Semantics and Examples. Need for a formal notion of semantics.References : Chapter 4 Formal description of semantics. Models and variable assignments. Tarski's theory of Truth in first order logic. References <a href="http://www.logicinaction.org/docs/ch4.pdf">Chapter 4</a> Exercises Reading Formal description of semantics. Models and variable assignments. Tarski's theory of Truth in first order logic.References : Chapter 4 Validity, Satisfiability, Semantic implication. Examples. Undecidability of validity. Decidability in MFO (Syllogistic) - inly statements. <br> Axiomatizations of First order Logic. References <a href="http://www.logicinaction.org/docs/ch4.pdf">Chapter 4</a> Exercises Reading <a href="http://www.cs.cmu.edu/~cdm/pdf/AutomLogic1-6up.pdf">slides</a> Validity, Satisfiability, Semantic implication. Examples. Undecidability of validity. Decidability in MFO (Syllogistic) - inly statements.

Axiomatizations of First order Logic.Axiomatization for First Order Logic. Theorems and Proofs in the axiomatization. Comments about true vs provable. Soundness and Completeness, Godel's completeness theorem for FO (statement). References <a href="http://www.logicinaction.org/docs/ch4.pdf">Chapter 4</a> Exercises Reading Axiomatization for First Order Logic. Theorems and Proofs in the axiomatization. Comments about true vs provable. Soundness and Completeness, Godel's completeness theorem for FO (statement).References : Chapter 4 Statements about numbers. Natural numbers as a model for an axiomatization. Peano's Axioms. Provable vs Truth. Godel's incompleteness theorem. Mathematical Proofs. Proof techniques: Direct proof. The structure of a proof. References <a href="http://www.xamuel.com/the-axioms-of-peano-arithmetic-modern-version/">Pean's Axioms</a> (We appended the axioms with equality axioms too). <br>Section 1.5-1.8 of Rosen's Book. Exercises Reading Statements about numbers. Natural numbers as a model for an axiomatization. Peano's Axioms. Provable vs Truth. Godel's incompleteness theorem. Mathematical Proofs. Proof techniques: Direct proof. The structure of a proof.References : Pean's Axioms (We appended the axioms with equality axioms too).

Section 1.5-1.8 of Rosen's Book.Indirect proof or Proving the contrapositive. Example(s). Proof by contradiction. Examples. Proof by cases. Examples. References Section 1.5-1.8 of Rosen's Book. Exercises Reading Indirect proof or Proving the contrapositive. Example(s). Proof by contradiction. Examples. Proof by cases. Examples.References : Section 1.5-1.8 of Rosen's Book. A conjecture and confirming the conjecture by a proof. Existence proofs. Non-constructive proofs. Three examples. References Section 1.6 and 1.8 of Rosen's Book. Exercises Reading A conjecture and confirming the conjecture by a proof. Existence proofs. Non-constructive proofs. Three examples.References : Section 1.6 and 1.8 of Rosen's Book. Principle of mathematical induction. Three examples. References Rosen's Book. Exercises Reading Principle of mathematical induction. Three examples.References : Rosen's Book. Strong Induction. Structural Induction. Examples. Well-ordering vs Mathematical Induction. References Rosen's Book. Exercises Reading Strong Induction. Structural Induction. Examples. Well-ordering vs Mathematical Induction.References : Rosen's Book. More example proofs. Infiniteness of prime numbers. How do we define infiniteness? Finiteness? Notion of cardinality of sets. "Equal cardinality" as a relation. Schroeder-Bernstein theorem. Finite and infinite sets. Examples. References Refer to class notes. Exercises Reading More example proofs. Infiniteness of prime numbers. How do we define infiniteness? Finiteness? Notion of cardinality of sets. "Equal cardinality" as a relation. Schroeder-Bernstein theorem. Finite and infinite sets. Examples.References : Refer to class notes. Cardinality. Finite and Infinte subsets.<br> Any infinite subset of N has the same cardinality as N. Countable and countably infinite sets. References <a href="http://www.math.ucla.edu/~ysella/Math115AHCardinality.pdf">Notes</a> Exercises Reading Cardinality. Finite and Infinte subsets.

Any infinite subset of N has the same cardinality as N. Countable and countably infinite sets.References : Notes Uncountability. Cantor's diagonalization. More examples of uncountable sets. Countability of NxN. Union, Cartesian product of countable sets. References Exercises Reading Uncountability. Cantor's diagonalization. More examples of uncountable sets. Countability of NxN. Union, Cartesian product of countable sets.References : None Diagonalization in set form - there cannot be an surjection from a set to its power set. Infinitely many types of infinities. References Exercises Reading Diagonalization in set form - there cannot be an surjection from a set to its power set. Infinitely many types of infinities.References : None An unexpected use of the technique. Are there computational tasks which are not effectively programmable? Computational Tasks vs Languages. Programs vs Strings. A non-constructive proof. References Exercises Reading An unexpected use of the technique. Are there computational tasks which are not effectively programmable? Computational Tasks vs Languages. Programs vs Strings. A non-constructive proof.References : None A constructive proof that there are problems for which no effective programs exist using diagonalization. In particular, the problem of given a program P and an input x, the task of testing if it halts on x, undecidable. References Exercises Reading A constructive proof that there are problems for which no effective programs exist using diagonalization. In particular, the problem of given a program P and an input x, the task of testing if it halts on x, undecidable.References : None Is it because we do not know the x in advance? The variant of the problem where only the program is the input to us and x in which we are interested in checking P is halting or not is fixed apriori to be the string "LCCS". A proof that even this is not effectively programmable. References Exercises Reading Is it because we do not know the x in advance? The variant of the problem where only the program is the input to us and x in which we are interested in checking P is halting or not is fixed apriori to be the string "LCCS". A proof that even this is not effectively programmable.References : None Back to logic : Applying the proof techniques that we learned. Soundness Theorem for Propositional Logic. Proof by induction. References Exercises Reading Back to logic : Applying the proof techniques that we learned. Soundness Theorem for Propositional Logic. Proof by induction.References : None Proof of Completeness Theorem. Eliminating the premises. Completeness theorem with empty premises. Reintroducing the premises. Completing the first and the last step. References Exercises Reading Proof of Completeness Theorem. Eliminating the premises. Completeness theorem with empty premises. Reintroducing the premises. Completing the first and the last step.References : None Proofs for the formula from the lines of the truth table. Combining proofs. Deduction theorem (statement). References Exercises Reading Proofs for the formula from the lines of the truth table. Combining proofs. Deduction theorem (statement).References : None Deduction Theorem. Completing the proof of completeness theorem References Exercises Reading Deduction Theorem. Completing the proof of completeness theoremReferences : None Soundness and Completeness Theorems for First Order Logic. Godel's Incompleteness Theorem. References Exercises Reading Soundness and Completeness Theorems for First Order Logic. Godel's Incompleteness Theorem.References : None Expressibility of First Order Logic. Theory for graphs. 2-colorability cannot be expressed by a formula. 2-colorabilty cannot be axiomatized finitely but can be expressed by an infinite number of formulas. References Exercises Reading Expressibility of First Order Logic. Theory for graphs. 2-colorability cannot be expressed by a formula. 2-colorabilty cannot be axiomatized finitely but can be expressed by an infinite number of formulas.References : None Connectedness of Graphs is not even infinitely axiomatizable. References Exercises Reading Connectedness of Graphs is not even infinitely axiomatizable.References : None Birds eye-view : Second Order Logic, Existential Second Order, Universal Second Order, Monadic Second order Logics. <br> Modes of Truth. Modal Logic: syntax and semantics. Truth in a world. Example formulas. References Exercises Reading Birds eye-view : Second Order Logic, Existential Second Order, Universal Second Order, Monadic Second order Logics.

Modes of Truth. Modal Logic: syntax and semantics. Truth in a world. Example formulas.References : None Modelling mutual exclusion. Safety and Liveness. Truth in an execution path - a sequence of worlds evolving over time. Need of Linear-time Temporal Logic (LTL). References Exercises Reading Modelling mutual exclusion. Safety and Liveness. Truth in an execution path - a sequence of worlds evolving over time. Need of Linear-time Temporal Logic (LTL).References : None Syntax and Semantics. An overview of how it is used in the design and correction of systems. Modelling systems using LTL. Mutual exclusion: an example. References Exercises Reading Syntax and Semantics. An overview of how it is used in the design and correction of systems. Modelling systems using LTL. Mutual exclusion: an example.References : None **Theme 2 : Size & Structure : Combinatorics & Algebraic Structures**- 20 meetings- Meeting 33 : Mon, Oct 03, 11:00 am-11:50 am
- Meeting 34 : Tue, Oct 04, 10:00 am-10:50 am
- Meeting 35 : Wed, Oct 05, 09:00 am-09:50 am
- Meeting 36 : Thu, Oct 06, 12:00 pm-12:50 pm
- Meeting 37 : Thu, Oct 13, 12:00 pm-12:50 pm
- Meeting 38 : Fri, Oct 14, 09:00 am-09:50 am
- Meeting 39 : Mon, Oct 17, 11:00 am-11:50 am
- Meeting 40 : Tue, Oct 18, 10:00 am-10:50 am
- Meeting 41 : Wed, Oct 19, 09:00 am-09:50 am
- Meeting 42 : Thu, Oct 20, 12:00 pm-12:50 pm
- Meeting 43 : Mon, Oct 24, 11:00 am-11:50 am
- Meeting 44 : Wed, Oct 26, 09:00 am-09:50 am
- Meeting 45 : Thu, Oct 27, 12:00 pm-12:50 pm
- Meeting 46 : Mon, Oct 31, 11:00 am-11:50 am
- Meeting 47 : Tue, Nov 01, 10:00 am-10:50 am
- Meeting 48 : Wed, Nov 02, 09:00 am-09:50 am
- Meeting 49 : Thu, Nov 03, 12:00 pm-12:50 pm
- Section 7.1-7.5, 7.6 in [KR] Book
- Meeting 50 : Tue, Nov 08, 10:00 am-10:50 am
- Section 7.7-7.8 in [KR] Book
- Meeting 51 : Wed, Nov 09, 09:00 am-09:50 am
- Meeting 52 : Thu, Nov 10, 12:00 pm-12:50 pm

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. 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. More on double counting. Counting by Bijections. Examples. Approximate Bijection. References Sections 6.1 in the Benjamin-Quinn [BQ] Book. Exercises Reading More on double counting. Counting by Bijections. Examples. Approximate Bijection.References : Sections 6.1 in the Benjamin-Quinn [BQ] Book. Multichoosing. Mutiple interpretations and bijections between them. Deriving an expression for the same using double counting (effectively a proof by bijection). References Sections 5.3 in the Benjamin-Quinn [BQ] Book. Exercises Reading Multichoosing. Mutiple interpretations and bijections between them. Deriving an expression for the same using double counting (effectively a proof by bijection).References : Sections 5.3 in the Benjamin-Quinn [BQ] Book. Diagonal Avoiding paths. Counting by bijections. References Section 2.4 in Notes by <a href="http://mathcircle.berkeley.edu/BMC6/pdf0607/catalan.pdf">Tom Davis</a> Exercises Reading Diagonal Avoiding paths. Counting by bijections.References : Section 2.4 in Notes by Tom Davis Inclusion-Exclusion Principle. Proof. Applications. Cases when the intersection estimate depends on the size of the index set (and not structure). Derangements. Number of surjections. 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 Reading Inclusion-Exclusion Principle. Proof. Applications. Cases when the intersection estimate depends on the size of the index set (and not structure). Derangements. Number of surjections.References : Lecutre notes. The proof of PI was not done as per this (please use class notes). But examples are from here. More applications of PIE. A case when the intersection estimate depends on the structure of the index set and not just size. Chromatic Polynomial of a graph. References <a href="http://math.mit.edu/~csikvari/chromatic_polynomial.pdf">Proposition 1.4-1.5</a>. But we did in a simplified way in class. Refer notes for this. Exercises Reading More applications of PIE. A case when the intersection estimate depends on the structure of the index set and not just size. Chromatic Polynomial of a graph.References : Proposition 1.4-1.5. But we did in a simplified way in class. Refer notes for this. Pigeon Hole Principle and Three applications. References Kennth Rosen's Book Section 5.2. Exercises Reading Pigeon Hole Principle and Three applications.References : Kennth Rosen's Book Section 5.2. More Applications of PHP. Erdos-Szekeres Subsequence Theorem. Approximation principles for irrational numbers. References Kennth Rosen's Book Section 5.2.<Br> <a href="https://www.expii.com/topic/2468">Dirichlet's Principle</a> 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. Erdos-Szekeres Subsequence Theorem. Approximation principles for irrational numbers.References : Kennth Rosen's Book Section 5.2.

Dirichlet's PrincipleReading : Variations of Erdoz-Szekeres Monnotone Subsequence Theorem Proof of Dirichlet's Approximation principle using PHP. Ramsey Numbers. Estimating R(3,3). References <a href="https://www.expii.com/topic/2468">Dirichlet's Principle</a> Exercises Reading Proof of Dirichlet's Approximation principle using PHP. Ramsey Numbers. Estimating R(3,3).References : Dirichlet's Principle Erdos-Szekeres upper boundfor Ramsey's Number. Lower Bounds for Diagonal Ramsey Numbers. The probabilistic method. References Exercises Reading Erdos-Szekeres upper boundfor Ramsey's Number. Lower Bounds for Diagonal Ramsey Numbers. The probabilistic method.References : None Solving Recurrences. Linear Homogeneous Recurrences with constant coefficients. Characterestic equation. Case when the characterestic equation has distinct roots. A structure theorem for all solutions. More examples of recurrence relations and their solutions. References Section 6.1-6,2 of [KR] book. Exercises Reading Solving Recurrences. Linear Homogeneous Recurrences with constant coefficients. Characterestic equation. Case when the characterestic equation has distinct roots. A structure theorem for all solutions. More examples of recurrence relations and their solutions.References : Section 6.1-6,2 of [KR] book. Linear Non-homogeoneous Recurrence relations with constant coefficients. Particular solution. A characterization of the general solution. References Section 6.1-6,2 of [KR] book. Exercises Reading Linear Non-homogeoneous Recurrence relations with constant coefficients. Particular solution. A characterization of the general solution.References : Section 6.1-6,2 of [KR] book. 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 Section 6.2 of [KR] book. 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 : Section 6.2 of [KR] book. Derangements using Recurrence relations. Solving recurrences by substitution. Index substitution and Functional Substitution. References <a href="https://mikespivey.wordpress.com/2011/11/22/derangements/">Source</a>. Section 6.2 (end) of [KR] book. Exercises Reading Derangements using Recurrence relations. Solving recurrences by substitution. Index substitution and Functional Substitution.References : Source. Section 6.2 (end) of [KR] book. Generating functions for solution of recurrence relations. Generating Function Method for solving linear homogeneous case recurrence relations. Using generating functions to prove combinatorial identities. References [KR] Book. Exercises Reading Generating functions for solution of recurrence relations. Generating Function Method for solving linear homogeneous case recurrence relations. Using generating functions to prove combinatorial identities.References : [KR] Book. Examples for solution of recurrences using generating functions. Using generating functions to solve special cases of non-linear recurrence relations. Catalan Numbers recurrence. References <a href="http://www.math.drexel.edu/~foucart/TeachingFiles/F12/M235Lect4.pdf">notes</a> Exercises Reading Examples for solution of recurrences using generating functions. Using generating functions to solve special cases of non-linear recurrence relations. Catalan Numbers recurrence.References : notes Structured Sets. Relations. Reflexivity, Symmetry, Antisymmetry and Transitivity. Representing relations. From properties of relations to properties of graphs. Equivalence relations, Partial Orders, Total Orders, Well-order. Hasse diagram. Examples and non-examples. Minimal and Maximal elements. References <ul> <li>Section 7.1-7.5, 7.6 in [KR] Book</li> </ul> Exercises Reading Structured Sets. Relations. Reflexivity, Symmetry, Antisymmetry and Transitivity. Representing relations. From properties of relations to properties of graphs. Equivalence relations, Partial Orders, Total Orders, Well-order. Hasse diagram. Examples and non-examples. Minimal and Maximal elements.References : Poset isomorphisms. Subset Poset and Boolean Hypercube. GLBs and LUBs. Lattices. Examples and non-examples. References <ul> <li>Section 7.7-7.8 in [KR] Book</li> </ul> Exercises Reading Poset isomorphisms. Subset Poset and Boolean Hypercube. GLBs and LUBs. Lattices. Examples and non-examples.References : Identities on lattices. Distributive lattices.<br> Complete posets and their structure. Knaster-Tarski Fixed Point theorem. References [KR] Book<br><a href="http://www.drmaciver.com/2012/02/the-power-of-fixed-point-theorems-in-set-theory/">Internet Source</a> Exercises Reading Identities on lattices. Distributive lattices.

Complete posets and their structure. Knaster-Tarski Fixed Point theorem.References : [KR] Book

Internet SourceProof. Cantor-Bernstein-Schroeder Theorem. Applications.<br><br> Using Knaster-Tarski's theorem to prove Schroeder-Bernstein theorem. Applications of fixed points to argue about semantics of while loops. References <a href="http://www.drmaciver.com/2012/02/the-power-of-fixed-point-theorems-in-set-theory/">Internet Source</a> Exercises Reading Proof. Cantor-Bernstein-Schroeder Theorem. Applications.

Using Knaster-Tarski's theorem to prove Schroeder-Bernstein theorem. Applications of fixed points to argue about semantics of while loops.References : Internet Source **Evaluation Meetings**- 5 meetings- Meeting 53 : Mon, Aug 29, 11:00 am-11:50 am
- Meeting 54 : Thu, Sep 08, 08:00 am-08:50 am
- Meeting 55 : Tue, Oct 25, 08:00 am-08:50 am
- Meeting 56 : Mon, Nov 07, 11:00 am-11:50 am
- Meeting 57 : Fri, Nov 18, 01:00 pm-04:00 pm

Short Exam 1 References Exercises Reading Short Exam 1References : None Quiz - I References Exercises Reading Quiz - IReferences : None Quiz - II References Exercises Reading Quiz - IIReferences : None Short Exam 2 References Exercises Reading Short Exam 2References : None Endsemester Examination References Exercises Reading Endsemester ExaminationReferences : None