Guided Practice for Section 3.1: Direct Proofs

Overview

With this section, we move into the heart of the course by transitioning away from the basic logic and language tools for communicating in mathematics to the actual construction of solutions to mathematical problems. We’ll be looking at direct proofs in this section, which is something we’ve seen and done before. In addition to revisiting this proof technique we’ll be introducing some foundational mathematical content: what it means for one integer to divide another, and the very important notion of integer congruence modulo n. Along the way we will pick up some basic notions of mathematical terminology such as the concept of an axiom and different kinds of results such as propositions and corollaries.

Learning objectives

Resources for reading and viewing

Reading: Read pages 82–95 in the textbook. Remember that “reading” means not only attending to the words on the page but also interacting with the worked-out examples and trying the progress checks. You can try some exercises too if you’re up for it.

Viewing: Watch the following screencasts, which run a total of 44 minutes, 56 seconds:

Practice

  1. Work Preview Activity 1, parts 1, 2, 6, 8, and 10 only. Guidance on format for part 1: For example, the integer 4 divides the integer 68 because 68 = (4)(17).
  2. Work Preview Activity 2, part 1 only.
  3. Take the year in which you were born and call that integer “y”. Take the day of the month on which you were born and call that “n”. Then find the smallest nonnegative integer that is congruent to y mod n. For example, my son was born on January 15, 2009. If he were doing this exercise, he would be looking for the smallest nonnegative integer that is congruent to 2009 mod 15. (That integer would be 14. If you are not sure why, then use this example for practice.) Show your work and explain your reasoning.
  4. What questions or comments do you have about the content in Section 2.4?

Specifications

Questions?

Leave questions or comments on Piazza and make sure to tag your questions with #gp3.1.