MAT 242: Introduction to Proofs

Spring 2025

Exam 2 information

1. Allowed materials

• Notes and textbooks are not allowed on exams.
• laws.pdf will be provided for you.
• Electronic devices (calculator, laptop, smart phone) are not allowed on exams.

2. How to study

Please see tips from first exam.

3. Exam content

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

• proof techniques:
  • new techniques not yet tested: proof by contradiction, mathematical induction (including strong induction)
  • (surely direct proof and indirect proof may very well naturally come up)
  • I may or may not tell you which technique to use
  • I may ask you to choose and justify which technique is most appropriate.
• elementary number theory: prime numbers, congruence, gcd, the Well-Ordering Principle, proofs.
• proving non-conditional statements: if and only if statements, quantified statements.
• proofs about sets: proof about subset or equality; proof via definitions or using fundamental set properties.
• disproof: by counterexample.
• pay attention to example proofs in the text, some are quite famous: irrationality of $\sqrt2$, infinitude of primes, prime divisibility property, Fundamental Theorem of Arithmetic (existence).

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.