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

Acad : Introduction to the course. 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 Slides uploaded to moodle. Exercises Reading Acad : Introduction to the course. 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 : Slides uploaded to moodle. Propositional Logic. Syntax. Semantics, Truth tables. Implication. References [KR] Book Section 1.1 Exercises Reading Propositional Logic. Syntax. Semantics, Truth tables. Implication.References : [KR] Book Section 1.1 Tautology, Contradiction, Contingents, Satisfiability. Algorithmics. Implication. Contrapostive, Converse, Inverse, Examples. References [KR] Book Section 1.1 Exercises Reading Tautology, Contradiction, Contingents, Satisfiability. Algorithmics. Implication. Contrapostive, Converse, Inverse, Examples.References : [KR] Book Section 1.1 Logical Equivalences. Checking if a formula is a tautology. Algorithmic question. References [KR] Book Section 1.2 Exercises Reading Logical Equivalences. Checking if a formula is a tautology. Algorithmic question.References : [KR] Book Section 1.2 Arguments and argument forms, validity of argument forms. Rules of inferences. Why are we not adding more rules of inferences? Discussion. Axiomatizations. References [KR] Book Section 1.5. For discussion on axiomatization refer class notes. Exercises Reading Arguments and argument forms, validity of argument forms. Rules of inferences. Why are we not adding more rules of inferences? Discussion. Axiomatizations.References : [KR] Book Section 1.5. For discussion on axiomatization refer class notes. Example deductions/derivations using rules of inferences. Four examples. References [KR] Book Section 1.5. Exercises Reading Example deductions/derivations using rules of inferences. Four examples.References : [KR] Book Section 1.5. Resolution refutation. Examples. Axiomatization of Proposition logic. Limitations of Propositional Logic. Need for a more powerful language. References [KR] Book Section 1.5. Exercises Reading Resolution refutation. Examples. Axiomatization of Proposition logic. Limitations of Propositional Logic. Need for a more powerful language.References : [KR] Book Section 1.5. Predicate Logic. An informal introduction to quantifiers and their informal semantics. References [KR] Book Section 1.3 Exercises Reading Predicate Logic. An informal introduction to quantifiers and their informal semantics.References : [KR] Book Section 1.3 Translating English sentences to predicate logic. Use of function symbols and examples. References [KR] Book Section 1.3. For some examples, refer class notes. Exercises Reading Translating English sentences to predicate logic. Use of function symbols and examples.References : [KR] Book Section 1.3. For some examples, refer class notes. Syntax for predicate logic. Well-formed formulas. The parse tree. Free variables, Scope of the quantifier, bound variables. References [HR] (relevant section uploaded to moodle) Section 2.2 Exercises Reading Syntax for predicate logic. Well-formed formulas. The parse tree. Free variables, Scope of the quantifier, bound variables.References : [HR] (relevant section uploaded to moodle) Section 2.2 Semantics of Predicate Logic. Models. Environment (variable assignments). Truth of a formula under a model. Examples of model which satisfy a formula and model which do not. References [HR] (relevant section uploaded to moodle) Section 2.4 Exercises Reading Semantics of Predicate Logic. Models. Environment (variable assignments). Truth of a formula under a model. Examples of model which satisfy a formula and model which do not.References : [HR] (relevant section uploaded to moodle) Section 2.4 More examples of models. Validity checking problem (undecidabile). Satisfiability problem. Arguments in predicate logic. Valid argument forms. Rules of inferences - UG, UI, EG, EI. References [HR] (relevant section uploaded to moodle) Section 2.4<br> [KR] Section 1.5 Exercises Reading More examples of models. Validity checking problem (undecidabile). Satisfiability problem. Arguments in predicate logic. Valid argument forms. Rules of inferences - UG, UI, EG, EI.References : [HR] (relevant section uploaded to moodle) Section 2.4

[KR] Section 1.5Examples of valid arguments in predicate logic. Using them to establish logical equivalences. Axiomatization of predicate logic. Axioms, rules of inferences, proofs, theorems, completeness, soundness. References [KR] Section 1.5 Exercises Reading Examples of valid arguments in predicate logic. Using them to establish logical equivalences. Axiomatization of predicate logic. Axioms, rules of inferences, proofs, theorems, completeness, soundness.References : [KR] Section 1.5 Tutorial References Exercises Reading TutorialReferences : None How to develop a theory based on predicate logic. Attempts on Theory of natural numbers, Theory of graphs. Peano's axioms. References Class Notes. Exercises Reading How to develop a theory based on predicate logic. Attempts on Theory of natural numbers, Theory of graphs. Peano's axioms.References : Class Notes. Mathematical Proofs. Proof techniques: Direct proof. The structure of a proof.Indirect proof or Proving the contrapositive. Example(s). Proof by contradiction. Examples. Proof by cases. Examples. References Exercises Reading Mathematical Proofs. Proof techniques: Direct proof. The structure of a proof.Indirect proof or Proving the contrapositive. Example(s). Proof by contradiction. Examples. Proof by cases. Examples.References : None A conjecture and confirming the conjecture by a proof. Existence proofs. Non-constructive proofs. Three examples. References Exercises Reading A conjecture and confirming the conjecture by a proof. Existence proofs. Non-constructive proofs. Three examples.References : None Principle of mathematical induction. Three examples. References Exercises Reading Principle of mathematical induction. Three examples.References : None Strong Induction. Structural Induction. Examples. Well-ordering vs Mathematical Induction. References Exercises Reading Strong Induction. Structural Induction. Examples. Well-ordering vs Mathematical Induction.References : None 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 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 : None References Exercises Reading References : None References Exercises Reading References : None Cardinality. Finite and Infinte subsets. Any infinite subset of N has the same cardinality as N. Countable and countably infinite sets. Uncountability. Cantor's diagonalization. More examples of uncountable sets. Countability of NxN. Union, Cartesian product of countable sets. Diagonalization in set form - there cannot be an surjection from a set to its power set. Infinitely many types of infinities. References Exercises Reading Cardinality. Finite and Infinte subsets. Any infinite subset of N has the same cardinality as N. Countable and countably infinite sets. Uncountability. Cantor's diagonalization. More examples of uncountable sets. Countability of NxN. Union, Cartesian product of countable sets. 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 Completeness of Propositional Logic. Proof of Completeness Theorem. Eliminating the premises. Completeness theorem with empty premises. Reintroducing the premises. Completing the first and the last step. <br> Proofs for the formula from the lines of the truth table. Combining proofs. Deduction theorem (statement).<br> Deduction Theorem. Completing the proof of completeness theorem References Exercises Reading Completeness of Propositional Logic. Proof of Completeness Theorem. Eliminating the premises. Completeness theorem with empty premises. Reintroducing the premises. Completing the first and the last step.

Proofs for the formula from the lines of the truth table. Combining proofs. Deduction theorem (statement).

Deduction Theorem. Completing the proof of completeness theoremReferences : None **Theme 2 : Sizes of Sets : Counting & Combinatorics**- 14 meetings- Meeting 28 : Mon, Sep 23, 09:00 am-09:50 am
- Meeting 29 : Tue, Sep 24, 08:00 am-08:50 am
- Meeting 30 : Wed, Sep 25, 12:00pm-12:50pm
- Meeting 31 : Fri, Sep 27, 11:00 am-11:50 am
- Meeting 32 : Tue, Oct 01, 08:00 am-08:50 am
- Meeting 33 : Fri, Oct 04, 11:00 am-11:50 am
- Meeting 34 : Mon, Oct 07, 09:00 am-09:50 am
- Meeting 35 : Wed, Oct 09, 12:00pm-12:50pm
- Meeting 36 : Fri, Oct 11, 11:00 am-11:50 am
- Meeting 37 : Mon, Oct 14, 09:00 am-09:50 am
- Meeting 38 : Wed, Oct 16, 12:00pm-12:50pm
- Meeting 39 : Fri, Oct 18, 11:00 am-11:50 am
- Meeting 40 : Mon, Oct 21, 09:00 am-09:50 am
- Meeting 41 : Tue, Oct 22, 08:00 am-08:50 am

Back to logic : Applying the proof techniques that we learned. Soundness Theorem for Propositional Logic. Proof by induction. Completeness theorem (statement). References Exercises Reading Back to logic : Applying the proof techniques that we learned. Soundness Theorem for Propositional Logic. Proof by induction. Completeness theorem (statement).References : 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. 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 References Exercises Reading References : None References Exercises Reading References : None References Exercises Reading References : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None **Theme 3 : Structured Sets : Relational & Algebraic Structures**- 11 meetings- Meeting 42 : Wed, Oct 23, 12:00pm-12:50pm
- Meeting 43 : Fri, Oct 25, 11:00 am-11:50 am
- Meeting 44 : Mon, Oct 28, 09:00 am-09:50 am
- Meeting 45 : Tue, Oct 29, 08:00 am-08:50 am
- Meeting 46 : Wed, Oct 30, 12:00pm-12:50pm
- Meeting 47 : Fri, Nov 01, 11:00 am-11:50 am
- Meeting 48 : Mon, Nov 04, 09:00 am-09:50 am
- Meeting 49 : Tue, Nov 05, 08:00 am-08:50 am
- Meeting 50 : Wed, Nov 06, 12:00pm-12:50pm
- Meeting 51 : Mon, Nov 11, 09:00 am-09:50 am
- Meeting 52 : Wed, Nov 13, 12:00pm-12:50pm

To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None **Evaluation Meetings**- 6 meetings- Meeting 53 : Mon, Aug 26, 09:00 am-09:50 am
- Meeting 54 : Tue, Sep 03, 08:00 am-08:50 am
- Meeting 55 : Mon, Sep 30, 09:00 am-09:50 am
- Meeting 56 : Tue, Oct 15, 08:00 am-08:50 am
- Meeting 57 : Fri, Nov 08, 11:00 am-11:50 am
- Meeting 58 : Mon, Nov 18, 09:00 am-12:00 pm

To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None Endsemester Examination References Exercises Reading Endsemester ExaminationReferences : None