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.
- State and instantiate the definitions of "divides", "divisor", "factor", and "multiple" and use correct notation to work with these concepts.
- Explain what a "proof" is and what it takes for a proof to be correct.
- Explain the concepts of "undefined terms" and "axioms" and give examples of these in real life.
- Explain the terms "conjecture", "theorem", "proposition", "lemma", and "corollary".
- Construct a know-show table for a conditional statement using a combination of forward and backward steps.
- Convert a completed know-show table into a well-written English paragraph proof that adheres to MTH 210 writing guidelines.
- State what it means for two integers to be "congruent modulo n".
- Determine the truth value of propositions involving divisibility, integer congruence, and other basic arithmetic concepts. If a proposition is true, construct a correct proof. If not, demonstrate a working counterexample.
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:
- Integer divisibility (8:53)
- Direct proof involving divisibility (8:50)
- Integer congruence (9:28)
- Reducing an integer modulo n (9:44)
- Proofs involving integer congruence (8:01)
- 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).
- Work Preview Activity 2, part 1 only.
- 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.
- What questions or comments do you have about the content in Section 2.4?
- Please write up your responses to Practice exercises 1–3. Please use software to type up your responses this time; no handwritten work will be accepted. If you want to use LaTeX, see the Piazza thread on “Learning “LaTeX” about using LaTeX. There is a LaTeX template on Blackboard near this Guided Practice assignment that has a sample truth table you can modify. Hand this paper in at the beginning of class on September 17.
- For Practice exercise 4 (your questions or comments), there is a discussion thread set up on Piazza for this item. Go to Piazza, find the thread, and leave your question or comment no later than 8:00am on September 17.
Leave questions or comments on Piazza and make sure to tag your questions with #gp3.1.