COT3100H Honors Introduction to Discrete Structures Spring 2006

Week #1: (Jan 10, Jan 12) - Sets, Relations, Functions

1. Outline:
   • Sets (sets, operations with sets, natural numbers, mathematical induction, finite and infinite sets, recursion)
   • Relations and functions
   • Closures
   • Relational systems
   • Boolean algebras
2. Lecture Slides, Notes, and References:
• Lecture Notes #1, pp. 1-30
3. Supplementary Reading:
• Chapters 2, 3, and 4 of [Grim2004], or Chapters 1 and 3 of [Rose2003], or Chapters 1 and 2 of [Ande2004]

Week #2: (Jan 17, Jan 19) - Sets, Relations, Functions

1. Outline:
   • Sets (sets, operations with sets, natural numbers, mathematical induction, finite and infinite sets, recursion)
   • Relations and functions (relations, equivalence relations, functions and operations, order relations)
   • Closures (closures, structural induction, definitions by induction, definitions by recursions)
   • Relational systems
   • Boolean algebras
2. Lecture Slides, Notes, and References:
• Lecture Notes #1, pp. 31-53
3. Supplementary Reading:
• Chapters 2, 3, and 4 of [Grim2004], or Chapters 1 and 3 of [Rose2003], or Chapters 1 and 2 of [Ande2004]
4. Homework Assignments:
• Assignment #1
• Due: February 2, 2006

Week #3: (Jan 24, Jan 26) - Sets, Relations, Functions

1. Outline:
   • Sets (sets, operations with sets, natural numbers, mathematical induction, finite and infinite sets, recursion)
   • Relations and functions (relations, equivalence relations, functions and operations, order relations)
   • Closures (closures, structural induction, definitions by induction, definitions by recursions)
   • Relational systems (partially ordered sets, lattices)
   • Boolean algebras
2. Lecture Slides, Notes, and References:
• Lecture Notes #1, pp. 54-93
• The Dragon Curve - Doc
• The Dragon Curve - Demo
3. Supplementary Reading:
• Chapters 2, 3, and 4 of [Grim2004], or Chapters 1 and 3 of [Rose2003], or Chapters 1 and 2 of [Ande2004]

Week #4: (Jan 31, Feb 2) - Computational Introduction to Number Theory

1. Outline:
   • Divisibility. Prime numbers
   • Greatest common divisor
   • Congruences
   • Euler's function
   • Congruential equations
   • Chinese remainder theorem
   • Asymptotic notation
   • Applications to cryptography
2. Lecture Slides, Notes, and References:
• Lecture Notes #2, pp. 1-16
3. Supplementary Reading:
• Chapter 4 of [Grim2004], or Chapter 2 of [Rose2003], or Chapters 3, 7, and 10 of [Ande2004]

Week #5: (Feb 7, Feb 9) - Computational Introduction to Number Theory

1. Outline:
   • Divisibility. Prime numbers
   • Greatest common divisor
   • Congruences
   • Euler's function
   • Congruential equations
   • Chinese remainder theorem
   • Asymptotic notation
   • Applications to cryptography
2. Lecture Slides, Notes, and References:
• Lecture Notes #2, pp. 17-26
3. Supplementary Reading:
• Chapter 4 of [Grim2004], or Chapter 2 of [Rose2003], or Chapters 3, 7, and 10 of [Ande2004]
4. Midterm Exam #1:
• Midterm Exam #1, February 7
5. Homework Assignments:
• Assignment #2
• Due: February 28, 2006

Week #6: (Feb 14, Feb 16) - Computational Introduction to Number Theory

1. Outline:
   • Divisibility. Prime numbers
   • Greatest common divisor
   • Congruences
   • Euler's function
   • Congruential equations
   • Chinese remainder theorem
   • Asymptotic notation
   • Applications to cryptography
2. Lecture Slides, Notes, and References:
   • Lecture Notes #2, pp. 27-34
   • Section 7 of Lecture Notes #2
3. Supplementary Reading:
   • Chapter 4 of [Grim2004], or Chapter 2 of [Rose2003], or Chapters 3, 7, and 10 of [Ande2004]

Week #7: (Feb 21, Feb 23) - Computational Introduction to Number Theory

1. Outline:
   • Divisibility. Prime numbers
   • Greatest common divisor
   • Congruences
   • Euler's function
   • Congruential equations
   • Chinese remainder theorem
   • Asymptotic notation
   • Applications to cryptography
2. Lecture Slides, Notes, and References:
   • Lecture Notes #2, pp. 27-34
   • Section 8 of Lecture Notes #2
3. Supplementary Reading:
   • Chapter 4 of [Grim2004], or Chapter 2 of [Rose2003], or Chapters 3, 7, and 10 of [Ande2004]

Week #8: (Feb 28, March 2) - Semigroups and Monoids

1. Outline:
   • Definitions and examples
   • Free semigroups and monoids
   • Word semigroups
   • Cyclic semigroups
   • Variable-length codes
   • Huffman codes
2. Lecture Slides, Notes, and References:
   • Lecture Notes #3, pp. 1-18
3. Supplementary Reading:
   • Chapter 9 of [Ande2004]

Week #9: (March 7, March 9) - Semigroups and Monoids

1. Outline:
   • Definitions and examples
   • Free semigroups and monoids
   • Word semigroups
   • Cyclic semigroups
   • Variable-length codes
   • Huffman codes
2. Lecture Slides, Notes, and References:
   • Lecture Notes #3, pp. 19-32
3. Supplementary Reading:
   • Chapter 9 of [Ande2004]
4. Midterm Exam #2 -- March 7

Week #10: (March 13-17) - Spring Break

Week #11: (March 21, March 23) - Semigroups and Monoids

1. Outline:
   • Definitions and examples
   • Free semigroups and monoids
   • Word semigroups
   • Cyclic semigroups
   • Variable-length codes
   • Huffman codes
2. Lecture Slides, Notes, and References:
   • Lecture Notes #3, pp. 19-41
3. Supplementary Reading:
   • Chapter 9 of [Ande2004]

Week #12: (March 28) - Semigroups and Monoids

1. Outline:
   • Definitions and examples
   • Free semigroups and monoids
   • Word semigroups
   • Cyclic semigroups
   • Variable-length codes
   • Huffman codes
2. Lecture Slides, Notes, and References:
   • Lecture Notes #3, pp. 19-51
3. Supplementary Reading:
   • Chapter 9 of [Ande2004]
   • Time-varying Codes

Week #12: (March 30) - Groups

1. Outline:
   • Definitions and examples
   • Subgroups. Lagrange's theorem
   • Cyclic groups
   • The group Zm*
   • The discrete logarithm problem
   • Applications to cryptography
2. Lecture Slides, Notes, and References:
   • Lecture Notes #4, pp. 1-12

Week #13: (April 4, April 6) - Groups

1. Outline:
   • Definitions and examples
   • Subgroups. Lagrange's theorem
   • Cyclic groups
   • The group Zm*
   • The discrete logarithm problem
   • Applications to cryptography
2. Lecture Slides, Notes, and References:
   • Lecture Notes #4, pp. 13-
   • Section 6 of Lecture Notes #4

Week #14: (April 11, April 13) - Rings and Fields

1. Outline:
   • Definitions and examples
   • Ring homomorphisms
   • Characteristic of a ring
   • Finite fields
   • Applications to cryptography
2. Lecture Slides, Notes, and References:
   • Lecture Notes #5, pp. 1-18
   • Section 5 of Lecture Notes #5