Friday, February 2nd

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!

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 $a^2$ is threeven, then $a$ 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 $a$ is not threeven, then $a^2$ 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.