CS6030 - Logic and Combinatorics for Computer Science

Meetings  

• Theme 1 : Basic Propositional & Predicate Logic - 8 meetings

  • Meeting 01 : Tue, Aug 26, 05:30 pm-06:45 pm

  Administrative Announcements. Why this course? Overview of the course contents.<br> Propositions, Truth values. Compound propositions and truth tables.

  References
  Exercises
  Reading

  Administrative Announcements. Why this course? Overview of the course contents.
  Propositions, Truth values. Compound propositions and truth tables.
  

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  None

  • Meeting 02 : Tue, Sep 02, 05:30 pm-06:45 pm

  Implication and double implication. Truth tables. Tranlating English statements to propositional logic. Tautology, contingent, and contradictions. Logical equivalences.

  References
  Exercises
  Reading

  Implication and double implication. Truth tables. Tranlating English statements to propositional logic. Tautology, contingent, and contradictions. Logical equivalences.
  

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  None

  • Meeting 03 : Wed, Sep 03, 05:30 pm-06:45 pm

  Logical Equivalences. Satisfiability and applications. Argument forms and valid arguments. Rules of inference for propositional logic. Examples.

  References
  Exercises
  Reading

  Logical Equivalences. Satisfiability and applications. Argument forms and valid arguments. Rules of inference for propositional logic. Examples.
  

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  None

  • Meeting 04 : Tue, Sep 09, 05:30 pm-06:45 pm

  Resolution principle. Fallacies. Predicates and Quantifiers.

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  Exercises
  Reading

  Resolution principle. Fallacies. Predicates and Quantifiers.
  

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  • Meeting 05 : Wed, Sep 10, 05:30 pm-06:45 pm

  Existential and universal quantifiers. Truth and falsity of predicate logic statements. Free and bound variables. Negating a statement in Predicate logic. Logical equivalences.

  References
  Exercises
  Reading

  Existential and universal quantifiers. Truth and falsity of predicate logic statements. Free and bound variables. Negating a statement in Predicate logic. Logical equivalences.
  

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  • Meeting 06 : Tue, Sep 16, 04:00 pm-05:15 pm

  Nesting of quantifiers, Order of quantifiers. Translating English sentences to statements in predicate logic. Rules of inference for predicate logic. Introduction to mathematical proofs.

  References
  Exercises
  Reading

  Nesting of quantifiers, Order of quantifiers. Translating English sentences to statements in predicate logic. Rules of inference for predicate logic. Introduction to mathematical proofs.
  

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  • Meeting 07 : Wed, Sep 17, 05:30 pm-06:45 pm

  From logical deduction steps to "human-friendly" proofs without losing the mathematical rigour. Methods of Proof. Direct Proof and Indirect Proofs. Examples.

  References
  Exercises
  Reading

  From logical deduction steps to "human-friendly" proofs without losing the mathematical rigour. Methods of Proof. Direct Proof and Indirect Proofs. Examples.
  

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  None

  • Meeting 08 : Tue, Sep 23, 05:30 pm-06:45 pm

  Proof by contradiction. Three examples. Conjectures, Propositions, Theorems, Lemmas.

  References
  Exercises
  Reading

  Proof by contradiction. Three examples. Conjectures, Propositions, Theorems, Lemmas.
  

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  None

• Theme 2 : Mathematical Proofs & Techniques - 6 meetings

  • Meeting 09 : Wed, Sep 24, 05:30 pm-06:45 pm

  Proofs by exhaustive cases.Constructive and non-constructive proofs. uniqueness proofs. Fallacies in proofs. Examples.

  References
  Exercises
  Reading

  Proofs by exhaustive cases.Constructive and non-constructive proofs. uniqueness proofs. Fallacies in proofs. Examples.
  

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  None

  • Meeting 10 : Tue, Sep 30, 05:30 pm-06:45 pm

  Review of proof techniques. How to write a proof? Disproving statements. Principle of mathematical induction. Basic examples.

  References
  Exercises
  Reading

  Review of proof techniques. How to write a proof? Disproving statements. Principle of mathematical induction. Basic examples.
  

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  None

  • Meeting 11 : Wed, Oct 01, 05:30 pm-06:45 pm

  More examples of principle of mathematical induction. Creative uses. Proof by wellordering principle. Principle of strong induction.

  References
  Exercises
  Reading

  More examples of principle of mathematical induction. Creative uses. Proof by wellordering principle. Principle of strong induction.
  

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  None

  • Meeting 12 : Tue, Oct 07, 05:30 pm-06:45 pm

  More examples of strong induction. Structural induction. Fibonacci recurrence example.

  References
  Exercises
  Reading

  More examples of strong induction. Structural induction. Fibonacci recurrence example.
  

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  • Meeting 13 : Wed, Oct 08, 05:30 pm-06:45 pm

  Structural Induction. Functions defined recursively. The number of nodes in a full binary tree with n vertices.

  References
  Exercises
  Reading

  Structural Induction. Functions defined recursively. The number of nodes in a full binary tree with n vertices.
  

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  • Meeting 14 : Tue, Oct 14, 05:30 pm-06:45 pm

  Relations, Functions - definitions, examples, representations. Composing two functions. Injective functions. Examples and representations.<br><br> Surjections and Bijections. Examples of proving that a given function is (a) well-defined, (b) an injection, (c) a surjection.

  References
  Exercises
  Reading

  Relations, Functions - definitions, examples, representations. Composing two functions. Injective functions. Examples and representations.
  
  Surjections and Bijections. Examples of proving that a given function is (a) well-defined, (b) an injection, (c) a surjection.
  

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  None

• Theme 3 : Cardinality, Counting and Combinatorics - 10 meetings

  • Meeting 15 : Wed, Oct 15, 05:30 pm-06:45 pm

  Definition of cardinality of sets. Are there different infinities? Concentric circle example. Natural numbers and even numbers.

  References
  Exercises
  Reading

  Definition of cardinality of sets. Are there different infinities? Concentric circle example. Natural numbers and even numbers.
  

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  • Meeting 16 : Tue, Oct 21, 05:30 pm-06:45 pm

  Finite sets and infinite sets. Search for different infinities. Countable and countably infinite sets. Cardinality of Integers, Rational Numbers. All of them equicardinal with natural numbers.

  References
  Exercises
  Reading

  Finite sets and infinite sets. Search for different infinities. Countable and countably infinite sets. Cardinality of Integers, Rational Numbers. All of them equicardinal with natural numbers.
  

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  • Meeting 17 : Wed, Oct 22, 05:30 pm-06:45 pm

  The of real numbers is uncountable. Cantor's diagonalisation argument.

  References
  Exercises
  Reading

  The of real numbers is uncountable. Cantor's diagonalisation argument.
  

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  • Meeting 18 : Mon, Oct 27, 06:00 pm-07:00 pm

  Programs and "Halting Programs". Motivating example. Demonstration of a computational task that cannot be achieved by "Halting Programs". Diagonalization in action.

  References
  Exercises
  Reading

  Programs and "Halting Programs". Motivating example. Demonstration of a computational task that cannot be achieved by "Halting Programs". Diagonalization in action.
  

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  • Meeting 19 : Tue, Oct 28, 05:30 pm-06:45 pm

  Counting & Finite Sets. Product rules, Sum rule, Permutation and Combinations. Combinatorial identities. Double counting. Examples.

  References
  Exercises
  Reading

  Counting & Finite Sets. Product rules, Sum rule, Permutation and Combinations. Combinatorial identities. Double counting. Examples.
  

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  • Meeting 20 : Wed, Oct 29, 05:30 pm-06:45 pm

  More examples of double-counting. Proof by Bijections. A simple example.

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  Exercises
  Reading

  More examples of double-counting. Proof by Bijections. A simple example.
  

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  • Meeting 21 : Thu, Oct 30, 04:30 pm-05:45 pm

  More proof by bijections. The number of even-sized subsets and the number of odd-sized subsets are equal. Formal proof of bijectivity. Associated Binomial identity. Pigeon Hole Principle and generalizations. Proof and a few simple applications.

  References
  Exercises
  Reading

  More proof by bijections. The number of even-sized subsets and the number of odd-sized subsets are equal. Formal proof of bijectivity. Associated Binomial identity. Pigeon Hole Principle and generalizations. Proof and a few simple applications.
  

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  • Meeting 22 : Wed, Nov 05, 05:30 pm-06:45 pm

  More Applications of PHP. Erdos-Szekeres Subsequence Theorem. Principle of inclusion-exclusion principle. Example of "prime looking numbers", General formulation and proof of the principle using combinatorial arguments. An application. Derangements. Formulation as a PIE problem. Counting the number of derangements using inclusion-exclusion principle.

  References
  Exercises
  Reading

  More Applications of PHP. Erdos-Szekeres Subsequence Theorem. Principle of inclusion-exclusion principle. Example of "prime looking numbers", General formulation and proof of the principle using combinatorial arguments. An application. Derangements. Formulation as a PIE problem. Counting the number of derangements using inclusion-exclusion principle.
  

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  • Meeting 23 : Tue, Nov 11, 05:30 pm-06:45 pm

  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.
  

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  • Meeting 24 : Wed, Nov 12, 05:30 pm-06:45 pm

  Generating functions. Examples. Solving recurrences using generating functions. Solving Derangements using recurrence relations.

  References
  Exercises
  Reading

  Generating functions. Examples. Solving recurrences using generating functions. Solving Derangements using recurrence relations.
  

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• Theme 4 : Structured Sets - 4 meetings

  • Meeting 25 : Tue, Nov 18, 05:30 pm-06:45 pm

  Structured Sets. Relations. Reflexivity, Symmetry, Antisymmetry and Transitivity. Representing relations using graphs and matrices. Counting the number of reflexive, symmetric relations.

  References
  Exercises
  Reading

  Structured Sets. Relations. Reflexivity, Symmetry, Antisymmetry and Transitivity. Representing relations using graphs and matrices. Counting the number of reflexive, symmetric relations.
  

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  • Meeting 26 : Wed, Nov 19, 05:30 pm-06:45 pm

  Composition of relations. R is transitive if and only if R^n is contained in R for every n. Closure of relations. Reflexive closure, Symmetric closure, Transitive closure. Graph view.

  References
  Exercises
  Reading

  Composition of relations. R is transitive if and only if R^n is contained in R for every n. Closure of relations. Reflexive closure, Symmetric closure, Transitive closure. Graph view.
  

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  • Meeting 27 : Tue, Nov 25, 05:30 pm-06:45 pm

  Equivalence relations and Equivalence classes. Partitions. Examples. Properties of Equivalence classes. Rational numbers as equivalence classes of an equivalence relation.

  References
  Exercises
  Reading

  Equivalence relations and Equivalence classes. Partitions. Examples. Properties of Equivalence classes. Rational numbers as equivalence classes of an equivalence relation.
  

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  • Meeting 28 : Wed, Nov 26, 05:30 pm-06:45 pm

  Partial Orders, Examples. Comparable and incomparable elements. Graph of a partial order. The Hasse diagram representation. Divisibility relation. Subset relation. Examples. Maximal and Minimal elements in a poset. Examples. Finite posets will always have maximal and minimal elements. Greatest and Least elements. Examples. Upper Bounds, Lower Bounds, Greatest Lower Bounds, Least Upper Bounds. Lattices. Examples, and non-examples.

  References
  Exercises
  Reading

  Partial Orders, Examples. Comparable and incomparable elements. Graph of a partial order. The Hasse diagram representation. Divisibility relation. Subset relation. Examples. Maximal and Minimal elements in a poset. Examples. Finite posets will always have maximal and minimal elements. Greatest and Least elements. Examples. Upper Bounds, Lower Bounds, Greatest Lower Bounds, Least Upper Bounds. Lattices. Examples, and non-examples.
  

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