CS1200 - Discrete Mathematics for Computer Science

Meetings  

• Theme 1 : Describing Sets : Logic & Proofs - 19 meetings

  • Meeting 01 : Mon, Jan 09, 09:00 am-09:50 am

  Why do this course? The course is about sets. Describing them, Estimating their sizes, structure and deriving properties about them. <br> The fun side : the fork-road puzzle.<br> Admin : Activities in the course. Grading.

  References	The discussion about the need for mathematical tools and formal methods is not from a textbook.
  Exercises
  Reading	<a href="https://en.wikipedia.org/wiki/Pentium_FDIV_bug">Pentium FDIV bug</a>.

  Why do this course? The course is about sets. Describing them, Estimating their sizes, structure and deriving properties about them.
  The fun side : the fork-road puzzle.
  Admin : Activities in the course. Grading.
  

  References	:	The discussion about the need for mathematical tools and formal methods is not from a textbook.
  Reading	:	Pentium FDIV bug.

  • Meeting 02 : Tue, Jan 10, 08:00 pm-08:50 pm

  The fork road puzzle. Propositions and propositional calculus. Examples. <br> Admin : TA grouping announcements.

  References	Section 1.1 of [KR] Book.
  Exercises
  Reading	See <a href="http://puzzles.nigelcoldwell.co.uk">more puzzles</a>.

  The fork road puzzle. Propositions and propositional calculus. Examples.
  Admin : TA grouping announcements.
  

  References	:	Section 1.1 of [KR] Book.
  Reading	:	See more puzzles.

  • Meeting 03 : Wed, Jan 11, 01:00pm-01:50pm

  Compound Propositions. Truth tables. Implication. Example translations from English and back.

  References	Section 1.1 of [KR] Book
  Exercises
  Reading	<a href="http://www.math.niu.edu/~richard/Math101/implies.pdf">Implication</a>

  Compound Propositions. Truth tables. Implication. Example translations from English and back.
  

  References	:	Section 1.1 of [KR] Book
  Reading	:	Implication

  • Meeting 04 : Mon, Jan 16, 09:00 am-09:50 am

  Tautologies, contradictions, contingents. Propositional equivalences.

  References	Section 1.2 of [KR]
  Exercises
  Reading

  Tautologies, contradictions, contingents. Propositional equivalences.
  

  References

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  Section 1.2 of [KR]

  • Meeting 05 : Tue, Jan 17, 08:00 pm-08:50 pm

  From Truth tables to compound propositions. Back to implication. Contrapositives. Converses. Inverses. Propositional Arguments. Argument forms.

  References
  Exercises
  Reading

  From Truth tables to compound propositions. Back to implication. Contrapositives. Converses. Inverses. Propositional Arguments. Argument forms.
  

  References

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  None

  • Meeting 06 : Wed, Jan 18, 01:00pm-01:50pm

  Validity of argument forms. Examples and Non-examples. Rules of inferences.

  References
  Exercises
  Reading

  Validity of argument forms. Examples and Non-examples. Rules of inferences.
  

  References

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  None

  • Meeting 07 : Mon, Jan 23, 09:00 am-09:50 am

  Example proofs using rules of inferences. Resolution.

  References
  Exercises
  Reading

  Example proofs using rules of inferences. Resolution.
  

  References

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  None

  • Meeting 08 : Tue, Jan 24, 08:00 pm-08:50 pm

  Predicate Logic. Syntax and semantics. Predicates and Propositional Functions. Domain of discourse. Interpretation. Truth in predicate logic.

  References
  Exercises
  Reading

  Predicate Logic. Syntax and semantics. Predicates and Propositional Functions. Domain of discourse. Interpretation. Truth in predicate logic.
  

  References

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  None

  • Meeting 09 : Wed, Jan 25, 01:00 pm-01:50 pm

  Examples translating English sentences to predicate logic. Logical equivalences in predicate logic.

  References
  Exercises
  Reading

  Examples translating English sentences to predicate logic. Logical equivalences in predicate logic.
  

  References

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  None

  • Meeting 10 : Mon, Jan 30, 09:00 am-09:50 am

  Nested Quantifiers. Rules of inferences in predicate logic. Building valid argument forms. Examples.

  References
  Exercises
  Reading

  Nested Quantifiers. Rules of inferences in predicate logic. Building valid argument forms. Examples.
  

  References

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  None

  • Meeting 11 : Tue, Jan 31, 08:00 pm-08:50 pm

  Fallacies in deductions. Examples. Axioms and Proof Systems. The case of natural numbers. Mathematical Proofs.

  References
  Exercises
  Reading

  Fallacies in deductions. Examples. Axioms and Proof Systems. The case of natural numbers. Mathematical Proofs.
  

  References

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  None

  • Meeting 12 : Wed, Feb 01, 01:00 pm-01:50 pm

  Direct proofs. Indirect Proofs. Examples.

  References
  Exercises
  Reading

  Direct proofs. Indirect Proofs. Examples.
  

  References

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  None

  • Meeting 13 : Tue, Feb 07, 09:00 am-09:50 am

  Proof by Contradiction. Examples. Proof by exhaustive cases. Non-constructive Proofs.

  References
  Exercises
  Reading

  Proof by Contradiction. Examples. Proof by exhaustive cases. Non-constructive Proofs.
  

  References

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  None

  • Meeting 14 : Wed, Feb 08, 01:00pm-01:50pm

  More example of the non-constructive proofs. Uniqueness proofs.

  References
  Exercises
  Reading

  More example of the non-constructive proofs. Uniqueness proofs.
  

  References

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  None

  • Meeting 15 : Mon, Feb 13, 04:00 pm-04:50 pm

  Axioms for Natural Numbers. Mathematical Induction. Structure of the proof. Examples.

  References
  Exercises
  Reading

  Axioms for Natural Numbers. Mathematical Induction. Structure of the proof. Examples.
  

  References

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  None

  • Meeting 16 : Tue, Feb 14, 08:00 pm-08:50 pm

  (4pm - 5pm) Creativity in Induction. Knife fight game. Lines in the planes, regions, can be colored with 2 colors.

  References
  Exercises
  Reading

  (4pm - 5pm) Creativity in Induction. Knife fight game. Lines in the planes, regions, can be colored with 2 colors.
  

  References

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  None

  • Meeting 17 : Wed, Feb 15, 01:00pm-01:50pm

  Coloring of maps. The coloring problem. Graphs as an abstraction. <br> Strong Induction. Factorization example. Demonetization example.

  References
  Exercises
  Reading

  Coloring of maps. The coloring problem. Graphs as an abstraction.
  Strong Induction. Factorization example. Demonetization example.
  

  References

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  None

  • Meeting 18 : Mon, Feb 20, 09:00 am-09:50 am

  Root-2 is irrational - proof using strong induction. Tower of Hanoi problem.

  References
  Exercises
  Reading

  Root-2 is irrational - proof using strong induction. Tower of Hanoi problem.
  

  References

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  None

  • Meeting 19 : Wed, Feb 22, 01:00pm-01:50pm

  Fallacies in induction proofs. <br> Combining proof techniques. Infiniteness of prime numbers.

  References
  Exercises
  Reading

  Fallacies in induction proofs.
  Combining proof techniques. Infiniteness of prime numbers.
  

  References

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  None

• Theme 2 : Sizes of Sets : Finite & Infinite - 22 meetings

  • Meeting 20 : Thu, Feb 23, 04:00 pm-04:50 pm

  Sets, Relations, Domain, Codomain, Functions, Partial Functions, One-one, Onto, Image, Pre-image, Bijection Examples. Representing functions using graphs. Notion of cardinality of sets.

  References
  Exercises
  Reading

  Sets, Relations, Domain, Codomain, Functions, Partial Functions, One-one, Onto, Image, Pre-image, Bijection Examples. Representing functions using graphs. Notion of cardinality of sets.
  

  References

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  None

  • Meeting 21 : Mon, Feb 27, 09:00 am-09:50 am

  Review of Cardinality. Notion of infinities. Cardinality of infinite sets. Deceiving examples. Cardinality of even numbers, Odd numbers equal to Natural numbers.

  References
  Exercises
  Reading

  Review of Cardinality. Notion of infinities. Cardinality of infinite sets. Deceiving examples. Cardinality of even numbers, Odd numbers equal to Natural numbers.
  

  References

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  None

  • Meeting 22 : Tue, Feb 28, 08:00 pm-08:50 pm

  Countable and countably infinite sets. Any infinite subset of N has the same cardinality as N, using well ordering principle. Proof of well-ordering principle using mathematical induction.

  References
  Exercises
  Reading

  Countable and countably infinite sets. Any infinite subset of N has the same cardinality as N, using well ordering principle. Proof of well-ordering principle using mathematical induction.
  

  References

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  None

  • Meeting 23 : Wed, Mar 01, 01:00pm-01:50pm

  A is a subset of B. If B is countable then A must be countable. In search of bigger infinities. Cardinality of NxN is same as N. Composition of Injections.

  References
  Exercises
  Reading

  A is a subset of B. If B is countable then A must be countable. In search of bigger infinities. Cardinality of NxN is same as N. Composition of Injections.
  

  References

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  None

  • Meeting 24 : Mon, Mar 06, 09:00 am-09:50 am

  If A,B are countable, then A union B is countable and AxB is countable. Conclude that Z, Q are countable sets.

  References
  Exercises
  Reading

  If A,B are countable, then A union B is countable and AxB is countable. Conclude that Z, Q are countable sets.
  

  References

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  None

  • Meeting 25 : Tue, Mar 07, 08:00 pm-08:50 pm

  Uncountability. Cantor's diagonalization. More examples of uncountable sets.

  References
  Exercises
  Reading

  Uncountability. Cantor's diagonalization. More examples of uncountable sets.
  

  References

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  None

  • Meeting 26 : Wed, Mar 08, 06:00 am-06:00 am

  Diagonalization in set form - there cannot be an surjection from a set to its power set. Infinitely many types of infinities. Equivalent definitions of countability.

  References
  Exercises
  Reading

  Diagonalization in set form - there cannot be an surjection from a set to its power set. Infinitely many types of infinities. Equivalent definitions of countability.
  

  References

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  None

  • Meeting 27 : Thu, Mar 09, 04:00 pm-05:00 pm

  Counting & Finite Sets. Product rules, Sum rule, Permutation and Combinations.

  References
  Exercises
  Reading

  Counting & Finite Sets. Product rules, Sum rule, Permutation and Combinations.
  

  References

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  None

  • Meeting 28 : Tue, Mar 14, 08:00 am-08:50 am

  Combinatorial identities. Double counting. Examples.

  References
  Exercises
  Reading

  Combinatorial identities. Double counting. Examples.
  

  References

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  None

  • Meeting 29 : Wed, Mar 15, 01:00pm-01:50pm

  Proof by Bijections. Number of even sized subsets and number of odd sized subsets are equal. Formal proof of bijectivity. Associated Binomial identity.

  References
  Exercises
  Reading

  Proof by Bijections. Number of even sized subsets and number of odd sized subsets are equal. Formal proof of bijectivity. Associated Binomial identity.
  

  References

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  None

  • Meeting 30 : Thu, Mar 16, 04:00 pm-04:50 pm

  Number of paths (which use only east and south edges from a point) of a grid from top left to bottom-right corner of an nxn grid. <br>Problem of counting the number of diagonal avoiding paths. The bijection from number of balanced paranthesis to diagonal avoiding paths.

  References
  Exercises
  Reading

  Number of paths (which use only east and south edges from a point) of a grid from top left to bottom-right corner of an nxn grid.
  Problem of counting the number of diagonal avoiding paths. The bijection from number of balanced paranthesis to diagonal avoiding paths.
  

  References

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  None

  • Meeting 31 : Mon, Mar 20, 04:00 pm-05:00 pm

  Catalan Number and the proof by Bijection.<br> Multi-choosing and associated bijections.

  References
  Exercises
  Reading

  Catalan Number and the proof by Bijection.
  Multi-choosing and associated bijections.
  

  References

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  None

  • Meeting 32 : Wed, Mar 22, 01:00pm-01:50pm

  More on Combinatorial Identities. Principle of Inclusion-Exclusion. Derangements.

  References
  Exercises
  Reading

  More on Combinatorial Identities. Principle of Inclusion-Exclusion. Derangements.
  

  References

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  None

  • Meeting 33 : Thu, Mar 23, 04:00 pm-04:50 pm

  Pigeon Hole Principle and generalizations. Proof and a few simple applications.

  References
  Exercises
  Reading

  Pigeon Hole Principle and generalizations. Proof and a few simple applications.
  

  References

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  None

  • Meeting 34 : Mon, Mar 27, 09:00 am-09:50 am

  More Applications of PHP. Erdos-Szekeres Subsequence Theorem.

  References
  Exercises
  Reading

  More Applications of PHP. Erdos-Szekeres Subsequence Theorem.
  

  References

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  None

  • Meeting 35 : Mon, Mar 27, 04:00 pm-05:00 pm

  More on PHP. Dirichlet's Approximation principles for irrational numbers.

  References
  Exercises
  Reading

  More on PHP. Dirichlet's Approximation principles for irrational numbers.
  

  References

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  None

  • Meeting 36 : Wed, Mar 29, 01:00pm-01:50pm

  Principle of inclusion exclusion. Example of "prime looking numbers" General formulation and proof of the principle using combinatorial arguments. An application. Derangements. Formulation as a PIE problem.

  References
  Exercises
  Reading

  Principle of inclusion exclusion. Example of "prime looking numbers" General formulation and proof of the principle using combinatorial arguments. An application. Derangements. Formulation as a PIE problem.
  

  References

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  None

  • Meeting 37 : Thu, Mar 30, 04:00 pm-04:50 pm

  Counting the number of derangements using inclusion-exclusion principle. <br>Formulating Counting problems as recurrences.

  References
  Exercises
  Reading

  Counting the number of derangements using inclusion-exclusion principle.
  Formulating Counting problems as recurrences.
  

  References

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  None

  • Meeting 38 : Mon, Apr 03, 09:00 am-09:50 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

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  None

  • Meeting 39 : Mon, Apr 03, 04:00 pm-05:00 pm

  Case when the characterestic equation has repeated roots.

  References
  Exercises
  Reading

  Case when the characterestic equation has repeated roots.
  

  References

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  None

  • Meeting 40 : Tue, Apr 04, 08:00 pm-08:50 pm

  Linear Non-homogeoneous Recurrence relations with constant coefficients. Particular solution. Solution to homogeneous part.

  References
  Exercises
  Reading

  Linear Non-homogeoneous Recurrence relations with constant coefficients. Particular solution. Solution to homogeneous part.
  

  References

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  None

  • Meeting 41 : Mon, Apr 10, 04:00 pm-05:00 pm

  Generating functions. Examples. Solving recurrences using generating functions. <br> Solving Derangements using recurrence relations.

  References
  Exercises
  Reading

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

  References

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  None

• Theme 3 : Structured Sets : Relational & Algebraic - 6 meetings

  • Meeting 42 : Tue, Apr 11, 08:00 pm-08:50 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.
  

  References

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  None

  • Meeting 43 : Wed, Apr 12, 01:00pm-01:50pm

  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.
  

  References

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  None

  • Meeting 44 : Mon, Apr 17, 09:00 am-09:50 am

  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.
  

  References

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  None

  • Meeting 45 : Mon, Apr 17, 04:00 pm-05:00 pm

  Partial Orders, Examples. Comparable and incomparable elements. Graph of a partial order. The Hasse diagram representation. Divisibility relation. Subset relation. 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.
  

  References

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  None

  • Meeting 46 : Tue, Apr 18, 08:00 pm-08:50 pm

  Power set poset of {1,2,3} vs Binary strings of length 3. Two posets with the same structure. Isomorphism. Notion of a total order, and well-order. Well-ordering axiom.

  References
  Exercises
  Reading

  Power set poset of {1,2,3} vs Binary strings of length 3. Two posets with the same structure. Isomorphism. Notion of a total order, and well-order. Well-ordering axiom.
  

  References

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  None

  • Meeting 47 : Wed, Apr 19, 01:00pm-01:50pm

  Maximal and Minimal elements in a poset. Examples. Finite posets will always have maximal and minimal elements. Greatest and Least elements. Examples.<br> Upper Bounds, Lower Bounds, Greatest Lower Bounds, Least Upper Bounds. Lattices. Examples, and non-examples.

  References
  Exercises
  Reading

  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

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  None

• Theme 4 : TuTs, ShoTs, Exams - 11 meetings

  • Meeting 48 : Mon, Jan 16, 04:00 pm-05:00 pm

  Tutorial 1

  References
  Exercises
  Reading

  Tutorial 1
  

  References

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  None

  • Meeting 49 : Mon, Jan 23, 04:00 pm-05:00 pm

  Tutorial 2

  References
  Exercises
  Reading

  Tutorial 2
  

  References

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  None

  • Meeting 50 : Mon, Jan 30, 04:00 pm-05:00 pm

  Tutorial 3

  References
  Exercises
  Reading

  Tutorial 3
  

  References

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  None

  • Meeting 51 : Mon, Feb 06, 09:00 am-09:50 am

  Short Exam 1 : Syllabus Lectures 1-10

  References
  Exercises
  Reading

  Short Exam 1 : Syllabus Lectures 1-10
  

  References

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  None

  • Meeting 52 : Mon, Feb 13, 09:00 am-09:50 am

  Quiz I

  References
  Exercises
  Reading

  Quiz I
  

  References

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  None

  • Meeting 53 : Mon, Feb 20, 04:00 pm-05:00 pm

  Tutorial 4

  References
  Exercises
  Reading

  Tutorial 4
  

  References

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  None

  • Meeting 54 : Mon, Feb 27, 04:00 pm-05:00 pm

  Tutorial 5

  References
  Exercises
  Reading

  Tutorial 5
  

  References

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  None

  • Meeting 55 : Mon, Mar 20, 09:00 am-10:00 am

  Quiz 2

  References
  Exercises
  Reading

  Quiz 2
  

  References

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  None

  • Meeting 56 : Wed, Apr 05, 06:00 am-06:00 am

  Tutorial 8 (Tutorial 7 was sent to the mailing list)

  References
  Exercises
  Reading

  Tutorial 8 (Tutorial 7 was sent to the mailing list)
  

  References

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  None

  • Meeting 57 : Mon, Apr 10, 09:00 am-09:50 am

  Short Exam 2

  References
  Exercises
  Reading

  Short Exam 2
  

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  None

  • Meeting 58 : Mon, Apr 24, 01:30 pm-04:30 pm

  Endsem Exam

  References
  Exercises
  Reading

  Endsem Exam
  

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