MAT 241: Discrete Mathematics

Fall 2022

Final exam information

1. Exam format

The final exam is comprised of two portions:

1. Mandatory portion.
   • This will focus on new material that has not been tested before (see below).
   • Think of it as a 5th midterm exam.
   • You should plan to spend up to 70 minutes on this portion, as usual.
2. Optional portion.
   • Bonus opportunity to recover some points for Units 1 and/or 2.
   • If you want to re-test Unit $N$, where $N\in\{1,2\}$, then:
     • Review and master the topics of that unit (previous topics).
     • On the exam, there will be a section clearly marked as Unit $N$.
     • In that section will be a couple of problems (similar to the harder ones on Exam $N$).
     • Do those.
     • If you do well, I will use their average to replace your Exam $N$ score.
     • But this is capped at 90%.
     • If you don't do well, I will NOT lower your Exam $N$ score (so you have nothing to lose).
   • Since the grade replacement is capped at 90%, if you got above 90% on Exam $N$, you should not bother re-testing Unit $N$.
   • This benefits all kinds of students:
     • If you consistently did well on midterm exams, you need not worry about re-testing.
     • If you had a fluke on one or two exams, this is your chance to make it up.
     • If you consistently did poorly on all midterm exams, you should focus your attention on one or two units and master that material.
   • If you object to this bonus opportunity, simply don't re-test and your grade won't be affected.

2. Allowed materials

• Notes and textbooks are not allowed on exams.
• tables1.pdf will be provided for you.
• Electronic devices (calculator, laptop, smart phone) are not allowed on exams.
  • Leave answers such as $3\cdot7\cdot2+2^3-6$ as is; do not simplify.
  • Do not leave answers in terms of $P(-,-)$ or $C(-,-)$; simplify to factorials or binomial coefficients.
  • Simplify $\binom7r$ for $r=0,1,7$; these do not require calculators.
  • Leave answers such as $7!$ or $\binom73$ as is; do not simplify.

3. How to study

Please see tips from first exam.

Get enough rest during finals week and the weekend before. Create a list and schedule of what you will do each day, including enough sleep in your schedule. Start studying early enough so you can take breaks; reward yourself after working hard.

Practicing by solving problems is much more effective than reading over solutions. There are plenty of review questions in the textbook. Going over homework/exams and correcting mistakes is also a good idea.

4. Exam content

The exam covers everything we have done. While the exam is cumulative, the mandatory portion will focus on what was not already tested. Here are some topics we emphasized since the last exam:

• theory of computation (Chapter 9 and Section 11.3.1)
  • context-free grammars, parse trees (11.3.1).
  • regular languages: three faces of regular.
• applications* in number theory (3.4)
  • *in this section, the emphasis is on using—not proving—these famous and useful theorems.
  • (extended) Euclidean Algorithm to calculate GCD and inverse modulo $m$ (3.4.1).
  • linear congruence, Chinese Remainder Theorem (3.4.4).
  • Fermat's Little Theorem (3.4.5).
  • RSA encryption (3.4.6).

As usual, please note that this document is not a contract. I may have inadvertently left something off that ends up on an exam question. Moreover, I will not be able to test all of this material given the time limitations of the exam. I will have to pick and choose some subset of it.