
Instructor: Pat Morin, 5177 HP, morin@scs.carleton.ca

# Course Objectives

A second course that is designed to give students a basic understanding of Discrete Mathematics and its role in Computer Science. Computers handle discrete data rather than continuous data. The course presents an overview of some of the major theoretical concepts needed to analyze this type of data.

# Office Hours

Day AM/PM Staff Time Location
Monday AM Kirk Zhen 8:30-10:30 4125 Herzberg
Monday PM Xiaoying Liu TBA 4125 Herzberg
Tuesday AM Pat Morin 9:00-11:00 5177 Herzberg
Tuesday PM Abdullah Alchihabi 2:40-4:40 4125 Herzberg
Wednesday AM Shelly Wang 10:00-12:00 4125 Herzberg
Wednesday PM Alexander Johns 2:40-4:40 4125 Herzberg
Thursday AM Alexa de Grandmont 9:00-11:00 4125 Herzberg
Thursday PM FX CC 1:30-3:30 4125 Herzberg
Friday AM Zoltan Kalnay 9:00-11:00 4125 Herzberg
Friday PM Mehrnoosh Javarsineh 13:00-15:00 4125 Herzberg

# Important Dates

Sunday Sep 22, 23:55 Assignment 1 due (in cuLearn)
Sunday Oct 13, 23:55 Assignment 2 due (in cuLearn)
Sunday Nov 17, 23:55 Assignment 3 due (in cuLearn)
Friday Dec 6, 23:55 Assignment 4 due (in cuLearn)
Thursday Oct 17 Mid-term exam for Section A (in class)
Friday Oct 18 Mid-term exam for Section B (in class)

# Sample Exams

Here are some sample final and midterm exams that you can use to study from.

# Assignments

• Assignment 4 was due on Friday December 6, before 23:55. Sample solutions for Assignment 4 are now available as a PDF file that was created from this LaTeX file.
• Assignment 3 was due on Sunday November 17, before 23:55. Sample solutions for Assignment 3 are now available as a PDF file that was created from this LaTeX file.
• Assignment 2 was due on Sunday October 13th, before 23:55. Sample solutions for Assignment 2 are available as a PDF file
• Assignment 1 was due on Sunday September 22nd, before 23:55. Sample solutions for Assignment 1 are now available as a PDF file that was created from this LaTeX file. For study purposes, here are two similar assignments (with solutions) from Fall 2018 and Winter 2019

• Late assignments will not be accepted.
• You can type your solutions, or write them by hand and scan them (for example, using a scan app on your phone or using a real scanner).
• Solutions written-up in LaTeX are preferred, but not strictly required. In case you want to learn LaTeX, here is a tutorial. Learning LaTeX is a useful exercise, since many programs (including Microsoft Word) now use LaTeX for typesetting formulas.
• Each assignment must be submitted as one single PDF file through cuLearn.

Assignments 25%
Mid-term exam 25%
Final exam 50%

# Textbooks

We will be using the following free (libre and gratis) textbooks. The first one is the primary textbook for this course. The second contains supplementary and background material:

# Lecture topics

You should already be familiar with the following topics from COMP 1805: basic logical reasoning, sets and functions, proof strategies (direct proof, proof by contradiction, proof by induction), Sigma-notation for summations, basic graph theory, Big-Oh, Big-Omega, Big-Theta. You may take a look at Chapter 2 of the textbook and do some of the exercises at the end of that chapter. Review the relevant parts of Lehman et al if you are still struggling.

• Sep 4/5: Introduction
• Course overview.
• Chapter 1 in the textbook: Ramsey Theory, Sperner's Theorem, Quick-Sort.
• Sep 6: Pigeonhole sampler
• Sep 10/11: Counting (1)
• Product Rule, Bijection Rule, Sections 3.1, 3.2.
• Sep 12/13: Counting (2)
• Complement Rule, Sum Rule, Inclusion-Exclusion, permutations, binomial coefficients, Sections 3.3, 3.4, 3.5, 3.6.
• Sep 17/18: Counting (3)
• Binomial coefficients, Newton's Binomial Theorem, combinatorial proofs, Vandermonde's Identity, Pascal's Triangle, Sections 3.6, 3.7.
• Sep 19/20: Counting (4)
• Sections 3.7 and 3.8.
• How many strings can be obtained from MISSISSIPPI, counting solutions of linear equations, Section 3.9.
• Sep 24/25: Pigeonhole Principle
• Section 3.10
• Excercise 3.84.
• Sep 26/27: Recursion (1)
• Recursive functions, Section 4.1.
• Fibonacci numbers, Section 4.2.
• Counting 00-free bitstrings
• Counting $aa$-free strings over $\{a,b,c\}$
• Counting $ab$-free strings over $\{a,b,c\}$
• Oct 1/2: Recursion (2)
• Proof that $f_n = (\varphi^n - \psi^n)/\sqrt{5}$
• Euclid's algorithm, Section 4.5.
• Oct 3/4: Recursion (3)
• MergeSort, Section 4.6.
• Oct 8/9: Randomization and probability
• Anonymous broadcasting: Dining Cryptographers, Section 5.1.
• Probability Theory: Probability spaces, sample spaces, probability functions, Section 5.2.
• Basic rules of probability, Section 5.3.
• Oct 10/11:
• Finishing up Section 5.3
• Uniform probability spaces, Section 5.4.
• Exercise 5.11, which is known as the Newton–Pepys problem.
• Oct 15/16:
• Midterm review and slack time
• Oct 17/18:
• Midterm exam
• Oct 29/30:
• The Birthday Paradox (section 5.5)
• Find the big box (section 5.6)
• Oct 31/Nov 1:
• Let's Make a Deal, the Monty Hall Problem, Section 5.7.
• Conditional probability, Section 5.8.
• Anil's kids, Exercise 5.40, the remarkable set B.
• Nov 5/6:
• Independent events, Section 5.11.
• Exercise 5.81.
• Nov 7/8:
• Section 5.12, in particular, the probability of a circuit failing, Section 5.12.3.
• Choosing a random line in a file, Section 5.13.
• Nov 12/13:
• Infinite probability spaces, Section 5.15, Exercises 5.85 and 5.91.
• The law of total probability
• Nov 14/15:
• Random variables, Section 6.1.
• Independent random variables, Section 6.2.
• Expected value, Section 6.4.
• Linearity of expectation, Section 6.5.
• Nov 19/20:
• Indicator random variables, Section 6.8, Exercise 6.57.
• Expected running time of Insertion-Sort, Section 6.9
• Largest elements in prefixes of random permutations, Section 6.8.2.
• Nov 21/22:
• Estimating the harmonic number, Section 6.8.3.
• Expected running time of Quick-Sort, Section 6.10.
• Nov 28/29:
• Geometric distribution and its expected value, Section 6.6, Exercise 6.35.
• Exercise 6.59 (the Coupon Collector's Problem)
• Binomial distribution and its expected value, Section 6.7.
• Nov 26/27: The Probabilistic Method
• Finding large bipartite subgraphs, Section 7.1
• Graphs with no large clique or independent set, Section 7.2
• Jaccard distance satisfies triangle inequality, Section 7.4
• Dec 3/4: Exam review
• Dec 5/6:
• No class (for either section)