# Math 221: Discrete Mathematics

### Fall 2019

Monday, Wednesday, Friday 1:00pm - 1:50pm, University Crossings 151

Course Description: This course covers a range of topics in Discrete Mathematics including set theory, induction, counting, number theory, graphs, and cryptography. The goal is to teach students basic methods of mathematical thinking, mainly logical, combinatorial, and algorithmic, which will be of great importance in their future work in pure Mathematics or (especially) in its applications to Computer Science, Engineering, and other areas.
Prerequisites: Math 220 or CS 270 or ECE 200.
Office Hours: Korman 275, Wednesday 11:00am-12:30pm, Friday 11:30am-12:30pm.
Textbooks: We will use a combination of the following texts:
Discrete Mathematics: Elementary and Beyond, by L. Lovász, J. Pelikán, and K. Vesztergombi (Drexel Library online copy)
Pirate This Discrete Math Book, by R. Andrew Hicks (Andrew Hicks is a professor at Drexel who wrote this book specifically for this class.)
• 10% Homework
• 20% Weekly quizzes
• 30% Midterm
• 40% Final
• A: 80-100%
• B: 60-80%
• C: 40-60%
• D-F: 0-40%
• Exam Policy: No books or electronic devices are allowed on the midterm or exam. No collaboration is permitted on the midterm or exam. THERE WILL BE NO MAKE-UPS FOR EXAMS.
The midterm will be in-class on October 23; it will be 50 minutes long.
Students with special exam-taking requirements or time conflicts should contact me by October 4.
Quiz Policy: Quizzes will be given in class most Wednesdays, and will be similar to some of the homework questions due that day. They will be about 15 minutes long. There will be about 8 quizzes total and the lowest quiz grade will be dropped to compute the quiz grade. No books or electronic devices are allowed on quizzes. No collaboration is permitted on quizzes. THERE WILL BE NO MAKE-UPS FOR QUIZZES.
Homework Policy: You may consult each other and the textbook above. List all people and sources who aided you and whom you aided, and write up the solutions independently, in your own language. It is easy nowadays to find solutions to almost anything online. DO NOT consult such solutions until after turning your homework. Solutions to homeworks will be posted on this website and/or discussed in class. Late homeworks will not be accepted.
Please visit this site frequently for new information. Updates to the syllabus and reading assignments, homeworks, homework solutions, and practice exams will be posted here as the course progresses.

## Syllabus

Since we are using multiple textbooks, there will be some overlap with the reading assignments. The most important/relevant sources will be listed first.

Week 1: Sep 23, Sep 25, Sep 27
Set Theory and Functions: read the handout Joy of Sets, Section 1.2 and Theorem 1.3.1 of LPV (the textbook Discrete Mathematics: Elementary and Beyond), and Chapter 4 of Hicks.
Read the handout Mathematical Hygiene. We will discuss some of these concepts throughout the course as needed.
Homework 1 due October 2

Week 2: Sep 30, Oct 02, Oct 04
Induction: Section 2.1 of LPV, Chapter 7 of Hicks
Homework 2 due October 9

Week 3: Oct 07, Oct 09, Oct 11
Pascal's triangle, counting, bijective proofs, binomial theorem : Sections 1.7-1.8, 3.5-3.6, 3.1 of LPV, Chapters 9, 10, 11 of Hicks
Note that we have been using ${{n}\choose{k}}$ for the binomial coefficient n choose k, whereas Hicks uses $C_{n,k}$.
Homework 3 due October 16. It is acceptable to leave binomial coefficients unsimplified.

Week 4: Oct 16, Oct 18
Probability, poker, dice: Chapter 12 of Hicks.
The Bean Machine
Homework 4 do not turn in. (This material will be covered on the midterm, so completing it may be a good way to study for the midterm.)

Week 5: Oct 21, Oct 23, Oct 25
Probability continued: poker, dice, birthday paradox: Chapters 12, 15, 16 of Hicks, Section 2.5 of LPV.
Midterm: October 23. It will be in-class and 50 minutes long.
Try to arrive a couple minutes early to class if possible so we can start exactly on the hour.
The midterm will cover all the material from class up through Oct 21 and homeworks 1-4. The format will be similar to the last quiz, and about three times as long.

Week 6: Oct 28, Oct 30, Nov 01
Introduction to graph theory: vertex degrees, trees, paths, cycles: Sections 7.1-7.2, 8.1-8.2, 13.2 of LPV.
Homework 5 due November 06.

Week 7: Nov 04, Nov 06, Nov 08
Trees, Kruskal's algorithm: Sections 8.1-8.2, 9.1 of LPV.
We will not follow LPV very closely for this topic. Supplementary Wikipedia articles: Vertex degrees, Bipartite graphs, Kruskal's algorithm.
The quiz November 06 will cover probability.
Homework 6 due November 13.

Week 8: Nov 11, Nov 13, Nov 15
Kruskal's algorithm for minimum-cost spanning tree, Euler's formula, platonic solids: Sections 9.1, 12.1-12.3 of LPV
Homework 7 due Nov 20.

Week 9: Nov 18, Nov 20, Nov 22
Number theory: Primes, Euclidean algorithm, modular arithmetic: Chapters 17-18, 22-23 of Hicks, Sections 6.1-6.3 of LPV
Homework 8 due Dec 04.

Week 10: Nov 25
Euler's phi function, Fermat's little theorem, Public key cryptography: Chapters 24-25 of Hicks, Wikipedia article on Diffie-Hellman key exchange
Homework 9 with solutions Do not turn in. This will be good practice for the final.

Week 11: Dec 02, Dec 04, Dec 06
Previous week continued, final review
I will be gone December 2-4 and Elaine will cover class. Monday will be a final review, mainly several practive problems to work on and go over in class. Wednesday will cover Diffie-Hellman key exchange. There will be a quiz in class on December 4. I will not have office hours on Dec 4th, but office hours will be 11:30-12:30 as usual on Dec 6th.
Graph theory practice questions from old quizzes: Old Quiz 5 Old Quiz 6

The Final Exam is on Monday, December 9, in Lebow 241, from 10:30am to 12:30pm. It will cover all the material from class and on the homeworks, with more emphasis on the material from weeks 5-11. The format of the final will be similar to the midterm and about 2-3 times as long.

Homework Help: Math Resource Center (Korman 247)
Important University Policies: