MAT 241: Discrete Mathematics

Fall 2022

Exam 2 information

1. Logistics

• 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.
• Since this class is scheduled within the 70-minute B4 block, you may use the entire 70 minutes.

2. How to study

Please see tips from first exam.

3. Exam content

The exam covers everything we have done, up to and including Section 5.2. (Section 5.3 will be on the next exam.) While the exam is cumulative, we will focus on what was not already tested. Here are some topics we emphasized since the last exam:

• logic and proofs
  • predicate logic: predicates and quantifiers, negating quantified statements, proofs.
  • axiomatic mathematics: entities that do or do not require proofs.
  • elementary number theory: lots of definitions, the Well-Ordering Principle, understand the proof of the Quotient-Remainder Theorem, proofs.
  • proofs: strategies (I may or may not tell you which one to use); pay attention to example proofs in the text, some are quite famous: Quotient-Remainder Theorem, irrationality of $\sqrt2$, infinitude of primes, prime divisibility property, Fundamental Theorem of Arithmetic (uniqueness).
  • induction: including complete induction.
• counting
  • basic counting: permutations and combinations, with or without replacement; binomial coefficients.
  • counting principles: Independent Tasks, Mutually Exclusive Tasks.
  • more complex counting: combinations with replacement, multinomial coefficients, anagrams.
  • combinatorial identities: algebraic and combinatorial proofs.

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.