It is crucial that you are checking the website between classes and keeping up with homework tasks between classes, whether graded or not. (Hint: this Friday I may check for completeness.)

**General announcements:**

**THIS THURSDAY OFFICE HOUR IS CANCELLED**due to an unavoidable conflict. Please email if you’d like to arrange to meet.- Friday is another
**group presentation day**. Please put your group member names and section number on your PDF. - Friday will also have a
**proof quiz**. It will be a “direct proof” (in Hammack’s parlance), similar to the examples of Section 4.3, and to exercises 1-7, 10-12, 19-20 of Chapter 4, and to the proofs from the first groupwork assignment. - Note: office hours will end 15 minutes early Feb 8th.
- Groupwork submitted on canvas must from now on be typset, by LaTeX or Word or any other method. But
*not*handwritten. - I hope you all enjoyed the LaTeX tutorial on Wednesday. You can access a LaTeX overview and links, including the sample file, on the navigation bar at left. Learning LaTeX will be very helpful for your undergraduate career!
- If the badges quiz confused you, or worried you, remember that you have multiple attempts at each badge. You can check your current badge earnings in canvas. Please read Grading again for details.

**Checklist for today’s lecture:**

- Prepare for group presentation (see above).
- Study for proof quiz (see above).
- We have spent some time on boring old even and odd integers. Your task for Friday is to invent a notion of “threeven” (i.e. divisible by 3) and two types of non-threeven-ness (i.e. two distinct ways an integer can fail to be threeven). In other words,
- The numbers 0,3,6,9,… are threeven
- The numbers 1,4,7,10,… are one type of non-threeven
- The numbers 2,5,8,11,… are the other type of non-threeven

You can get creative in giving them names. What I’d like you to do is:

- Give formal definitions of the three types of integers
- Give a formal proof that if is threeven, then is threeven. Hints:
- You may find it helpful to read some of Hammack, section 6, namely the first page and the example at the bottom of page 115.
- You may find it helpful to first prove that if is
*not*threeven, then is*not*threeven. You can then use this as a lemma if you like (you can revisit the notion of a ‘lemma’ on page 88 of Hammack). - If you are still unsure, but have given it a good effort, write up what you have as incomplete ideas.