CS6030 - Logic and Combinatorics for Computer Science

Meetings  

• Theme 1 : Describing Sets : Logic and Proofs - 32 meetings

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

  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.

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

  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.

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

  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.

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

  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.

  • Meeting 05 : Mon, Aug 08, 11:00 am-11:50 am

  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.
  

  References

  :

  Chapter 1, Chapter 2, Chapter 3.

  • Meeting 06 : Tue, Aug 09, 10:00 am-10:50 am

  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

  • Meeting 07 : Wed, Aug 10, 09:00 am-09:50 am

  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

  • Meeting 08 : Thu, Aug 11, 12:00 pm-12:50 pm

  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

  • Meeting 09 : Tue, Aug 16, 10:00 am-10:50 am

  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.
  

  References	:	Chapter 4
  Reading	:	slides

  • Meeting 10 : Wed, Aug 17, 09:00 am-09:50 am

  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

  • Meeting 11 : Thu, Aug 18, 12:00 pm-12:50 pm

  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.

  • Meeting 12 : Mon, Aug 22, 11:00 am-11:50 am

  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.

  • Meeting 13 : Tue, Aug 23, 10:00 am-10:50 am

  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.

  • Meeting 14 : Wed, Aug 24, 09:00 am-09:50 am

  Principle of mathematical induction. Three examples.

  References	Rosen's Book.
  Exercises
  Reading

  Principle of mathematical induction. Three examples.
  

  References

  :

  Rosen's Book.

  • Meeting 15 : Thu, Aug 25, 12:00 pm-12:50 pm

  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.

  • Meeting 16 : Tue, Aug 30, 10:00 am-10:50 am

  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.

  • Meeting 17 : Wed, Aug 31, 09:00 am-09:50 am

  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

  • Meeting 18 : Thu, Sep 01, 12:00 pm-12:50 pm

  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

  • Meeting 19 : Tue, Sep 06, 10:00 am-10:50 am

  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

  • Meeting 20 : Wed, Sep 07, 09:00 am-09:50 am

  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

  • Meeting 21 : Tue, Sep 13, 10:00 am-10:50 am

  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

  • Meeting 22 : Thu, Sep 15, 12:00 pm-12:50 pm

  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

  • Meeting 23 : Fri, Sep 16, 11:00 am-11:50 am

  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

  • Meeting 24 : Mon, Sep 19, 11:00 am-11:50 am

  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

  • Meeting 25 : Tue, Sep 20, 10:00 am-10:50 am

  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

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  None

  • Meeting 26 : Wed, Sep 21, 09:00 am-09:50 am

  Deduction Theorem. Completing the proof of completeness theorem

  References
  Exercises
  Reading

  Deduction Theorem. Completing the proof of completeness theorem
  

  References

  :

  None

  • Meeting 27 : Thu, Sep 22, 12:00 pm-12:50 pm

  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

  • Meeting 28 : Mon, Sep 26, 11:00 am-11:50 am

  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

  • Meeting 29 : Mon, Sep 26, 04:15 pm-05:15 pm

  Connectedness of Graphs is not even infinitely axiomatizable.

  References
  Exercises
  Reading

  Connectedness of Graphs is not even infinitely axiomatizable.
  

  References

  :

  None

  • Meeting 30 : Tue, Sep 27, 10:00 am-10:50 am

  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

  • Meeting 31 : Wed, Sep 28, 09:00 am-09:50 am

  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

  • Meeting 32 : Thu, Sep 29, 12:00 pm-12:50 pm

  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

  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.

  • Meeting 34 : Tue, Oct 04, 10:00 am-10:50 am

  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.

  • Meeting 35 : Wed, Oct 05, 09:00 am-09:50 am

  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.

  • Meeting 36 : Thu, Oct 06, 12:00 pm-12:50 pm

  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

  • Meeting 37 : Thu, Oct 13, 12:00 pm-12:50 pm

  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.

  • Meeting 38 : Fri, Oct 14, 09:00 am-09:50 am

  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.

  • Meeting 39 : Mon, Oct 17, 11:00 am-11:50 am

  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.

  • Meeting 40 : Tue, Oct 18, 10:00 am-10:50 am

  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 Principle
  Reading	:	Variations of Erdoz-Szekeres Monnotone Subsequence Theorem

  • Meeting 41 : Wed, Oct 19, 09:00 am-09:50 am

  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

  • Meeting 42 : Thu, Oct 20, 12:00 pm-12:50 pm

  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

  • Meeting 43 : Mon, Oct 24, 11:00 am-11:50 am

  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.

  • Meeting 44 : Wed, Oct 26, 09:00 am-09:50 am

  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.

  • Meeting 45 : Thu, Oct 27, 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	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.

  • Meeting 46 : Mon, Oct 31, 11:00 am-11:50 am

  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.

  • Meeting 47 : Tue, Nov 01, 10:00 am-10:50 am

  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.

  • Meeting 48 : Wed, Nov 02, 09:00 am-09:50 am

  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

  • Meeting 49 : Thu, Nov 03, 12:00 pm-12:50 pm

  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

  :

  Section 7.1-7.5, 7.6 in [KR] Book

  • Meeting 50 : Tue, Nov 08, 10:00 am-10:50 am

  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

  :

  Section 7.7-7.8 in [KR] Book

  • Meeting 51 : Wed, Nov 09, 09:00 am-09:50 am

  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 Source

  • Meeting 52 : Thu, Nov 10, 12:00 pm-12:50 pm

  Proof. 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

  Short Exam 1

  References
  Exercises
  Reading

  Short Exam 1
  

  References

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  None

  • Meeting 54 : Thu, Sep 08, 08:00 am-08:50 am

  Quiz - I

  References
  Exercises
  Reading

  Quiz - I
  

  References

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  None

  • Meeting 55 : Tue, Oct 25, 08:00 am-08:50 am

  Quiz - II

  References
  Exercises
  Reading

  Quiz - II
  

  References

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  None

  • Meeting 56 : Mon, Nov 07, 11:00 am-11:50 am

  Short Exam 2

  References
  Exercises
  Reading

  Short Exam 2
  

  References

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  None

  • Meeting 57 : Fri, Nov 18, 01:00 pm-04:00 pm

  Endsemester Examination

  References
  Exercises
  Reading

  Endsemester Examination
  

  References

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  None