Sets IV: ordered pairs, Cartesian products and powers, subset and powersets, including cardinality

Logic IV: negating statements

Proofs I: setting up a proof by contradiction

The new groupwork has been posted. I will go one more week before remixing groups, but as always, contact me if you have trouble scheduling 2 hours or any other problem. If you have 6 people in one group, consider breaking into two groups of 3.

For today’s class:

We will mostly focus on proofs by contradiction today. (The Logic IV and Proofs I badges are relevant to the study of proof by contradiction, so they will begin to be available on the badges quizzes.)

Bring your homework about “threeven” that was assigned in the previous daily post.

Read Hammack, page 111 (the first page of Chapter 6), and from page 113 to the end of Section 6.2 on page 116. In a couple spots, there’s a bit of notation from Boolean Logic in this chapter, because Hammack assumes we have covered Chapter 2 before getting to Chapter 6. Consider it an exercise in adaptive reading — just work around the notations you don’t know. We will be working on Boolean Logic soon, and for now skipping over his reference to it still leaves a very useful reading.

Study for the Badges Quiz. My advice: look your scores up on canvas, look over your returned quiz, and decide on 2 or 3 badges you’d like to focus on getting full credit for this week. (Study a few excellently, instead of all of them passably.) To study, it is helpful to find the relevant material in Hammack and do exercises (odd answers are in the back) to brush up on the concepts. If you have questions about the badges quiz material, I can take them up at the beginning of class Monday before we do proof by contradiction.

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.

WEDNESDAY WE WILL HAVE A LATEX WORKSHOP IN DUAN G116. NOT OUR REGULAR ROOM!

The room is a computer lab but I’m concerned it won’t have enough computers to accommodate everyone (the school couldn’t get me anything better). Therefore please bring a laptop if you have one. The only thing you need is web access, no need to prepare. I will give an introduction and overview and then I will debug and walk around to help as you work through the worksheet. If you are a True Master of LaTeX already, you may skip this class, or come with your LaTeX questions, but please email me to let me know if you plan to skip.

General announcements:

Group homework will be weekly. Make time to meet for two hours. I won’t exceed this. If you don’t finish in two hours, but made a decent effort during the two hours, then call it a day and pack up — that’s ok. If you have any trouble with groupwork, contact me privately if you wish and I will try to help. This includes scheduling issues, personality conflicts, or anything else. Your next group work is due Friday. See Grading.

Proof quizzes. Our first one was Friday. It was returned on Monday. Please pull it out of your bag now and take a look at my comments. The comments explain how to improve your grade for next time, so make sure you understand. If you don’t, please ask me before/after class or in office hour. See Grading.

Badges quizzes. Our first one was Monday. Remember that you don’t need to do everything on the quiz. Each section is a “badge” which you will have many chances to earn. You may wish to choose two per week to study for and work hard at, instead of always attempting everything. Your grades will be updated in canvas so you can check them before the next quiz and know which badges to work on. See Grading.

Daily posts. By now you have realized that you must check this website for daily posts and that I will rely on you to have done the homework between each class.

Monday in class we proved that the square root of 2 is irrational. Here’s an enjoyable video from Numberphile that recaps the proof. (If you’ve never seen Numberphile, you are missing out.)

I will post all the handouts from class etc. under Resources, so if you miss something, you can find it there.

Checklist for Wednesday:

Read the comments on your quiz which was returned Monday, and write out a proof that you believe addresses those comments. Suggestion: staple this new proof to the old quiz and keep for your records. Depending on your confidence in this exercise, see me in office hour (Wed/Thur 1-2 or by appt) if you feel unsure. I’m happy to look it over.

Complete the handout from class today, including the part on powersets which was rushed in class. To check your answers in the table, the correct answers read, from bottom to top, “NUMBERS RULE THE UNIVERSE.” If you don’t understand why one of your answers is wrong, ask me!

Read Hammack, 1.5, 1.6 and 1.7. Section 1.7 is about Venn Diagrams. These are a sort of “visual aid” to the ideas of 1.5 and 1.6. Read actively, as always. Make an outline only if you’ve found that a useful exercise in past.

Do as many exercises from 1.5, 1.6 and 1.7 as you need to feel confident.

Sets IV: ordered pairs, Cartesian products and powers, subset and powersets, including cardinality

We have not covered all this information in class yet, so by working ahead you can get a jump-start, if you feel like it. The idea is that you can choose which questions to attempt. You need not attempt them all, but I like to give you many options. You will get many (but not infinitely many) attempts at each badge, so this is not your only chance.

Please read Sections 1.3 and 1.4 of Hammack. Remember: read actively!

As usual, do as many exercises as you need for Section 1.3 and 1.4 to feel comfortable.

Announcement: I have set regular office hours WED/THUR 1-2 pm. The third hour is “floating” because with 50+ students, no set hours will work for everyone. That means as needed, I will schedule the third to meet the needs of those who ask, and announce the time to everyone. Therefore, please email me if the time above won’t work for you and you’d like to meet that week.

Reminder: groupwork is posted and due Friday. This one may require 2 hours.

If you have any scheduling issues with groups, please contact me.

Friday will be a group presentation day and a FIRST PROOF QUIZ. Your only assigned task is to prepare for these.

A little more info about group presentations and what to expect:

Your group will have handed in (by Friday morning 8 am), a PDF (scan of handwriting is ok as long as it is legible) via canvas of at least the three main proofs assigned in the Groupwork Assignment (and possibly the extra one too), or whatever you have accomplished. I will assign a small grade for groupwork but it is assessed on a complete/partial/incomplete rubric. Errors are fine. In fact, I hope you make interesting errors, because that’s where learning happens.

In class, you will hand in a paper copy (handwritten is fine) of the Groupwork Report (you can find a blank copy under Resources). This is just a brief accounting of what happened in your group. It is very helpful for me to know how things went, whether there were sticking points and confusions, how long it took, etc. This should only take a few minutes to complete.

In class, I will use my computer to load up the PDFs from canvas. I will load up a proof and ask for the presenter from that group to come to the front and explain what their group wrote. The class will then discuss writing and reasoning and offer constructive criticism and praise. This is meant to be a safe environment for errors, and I ask you to be thoughtful in your comments. In fact, I will pick proofs that have interesting errors so we can all learn.

Some more information about the first proof quiz:

This is a bit of a dry run. Remember (from Grading), that we drop just less than half of the proof quizzes. But we need to get started so I can see where the class is.

The quiz will ask you to prove something similar to the proofs we have seen. In particular, it will ask for a proof like:

If n is even, then n^2 is even.

If n is odd, then 2n is even.

If n and m are odd, then n + m is even.

etc.

You may wish to examine my proof grading rubric, available under Resources.

The group presentations will be a great way of studying for the quiz, as we will discuss writing and reasoning for similar proofs.

Other comments:

I meant to get a bit farther today than we did; apologies we didn’t get to talk proofs. We will on Friday.

The back of today’s handout on Cartesian products is very interesting material, and I encourage you to wrestle with it on your own time; we may get a chance to come back to it in class in future. Meanwhile, Chapter 1.10 in Hammack talks about some of this.

I will set office hours as soon as my schedule settles down, but please just email me to set a time to meet as needed.

Sometime this week you’ll be meeting with your group. Looking ahead, your group should produce a PDF to submit on Canvas by 8 am Friday, and also a groupwork report (handwritten is fine) to hand in in class Friday. (Meet for approximately one hour and get as far as you can in the work, hopefully that will be enough to write the three main proofs as a team.)

No quizzes this week, since we’re just getting started.

Last Friday you were asked as a group to write a proof of the fact that the sum of two even numbers is even. Take the time now, before Wednesday, to write for yourself a nice version of this. We will do another proof in class, so this will be good preparation for seeing that.

Please read Section 1.2 of Hammack. Read actively. Make yourself a brief outline of the topics. Do as many of the exercises as are needed for you to feel comfortable.

If you are just joining the class, please look at the handouts on the Resources page for an update on what we’ve done, and also read the previous daily posts (left navigation bar) and go through the daily checklists there. The first day we did a general warmup activity about polyhedra. On the second day we did a first proof (see the associated handout).

Both days we worked in (the same) groups and shared contact info. Make sure you are part of a study group and have arranged one hour to meet your group outside class before Friday. The groupwork assignment can be found on the Resources page. If you are having insurmountable troubles meeting, consider skyping in. If that really won’t work, or if you don’t have a group, please email me (kstange@math.colorado.edu). Group homework is due Friday at 8 am.

Reminder: please contact me ASAP if you have an ADA accommodation, or if you’d like to discuss how I address you, concerns about religious accommodations, etc.

Please read Chapter 1 of Hammack, up to the end of Section 1.1 (i.e. pages 3-7). This introduces the notion of a set. Please read actively. One thing this means is that whenever you come across an assertion, you try to create your own novel examples and non-examples. Every example you should work through yourself (e.g. for Example 1.1, cover the right side of the equation and guess what is there, then compare). Three pages of active reading can take a long time!

Make a summary sheet giving the “cliff notes” version of the reading. This doesn’t need to be detailed; an outline of the big points is the goal.

Do as many exercises for Section 1.1 as you feel is appropriate. If you are finding the reading difficult, then focus on reading for now, or just one topic for now. If you felt you understand the reading, then do all the odd numbered problems (answers in the back), to make sure. Some of them may trip you up!

Note: this assignment, done right, is a challenge to everyone, but for everyone it is a challenge in different ways. There’s always more to understand. (If you really feel everything is easy, and you got all the exercises in an instant, then challenge yourself by inventing interesting questions, corner cases, and trickier problems, or read section 1.10, or ponder how sets could be used to define the integers, or define the real numbers.) Put in an hour of solid effort and get where you get.

For our Friday class, please do the following (mostly easy tasks this time!):

Read all of the pages listed in the top bar of this website: about, goals, syllabus, resources, grading, fun. This is all the info about how the course will run. I expect you to know it without covering it all explicitly in class. Pro tip: all my handouts will all be found on the Resources page.

Understand that this course is unusual in that:

It is run very interactively, with lots of active learning. I expect you to create a supportive environment in all your interactions.

I expect you to check this website and do work for the course between every lecture. I will post announcements and tasks etc. by 1 pm after each lecture and I expect that you are aware of these.

I expect you to meet with a study group once per week outside class and present your solutions on Fridays

We will use a non-standard grading system.

If you have any ADA Accommodations, or other concerns about the above, please talk to me as soon as possible.

Please make sure you have a copy of the text. It is available for free in PDF form (linked also on the left nav bar) or cheaply in paper form at the bookstore.

Please plan to attend class faithfully unless you are contagious or ill etc (see my note about flu on the About page). If you are waitlisted, a spotless attendance record will give you priority as room opens up (and those who do not attend will be administratively dropped). About waitlists: I am not allowed to enroll over the fire limit of the room. Although in past everyone who faithfully attended was able to take the class, I cannot promise.

Please read Chapter 4, intro and Section 4.1, i.e. pages 87-88 of the textbook, Hammack. Read also the bottom half of page 41 (start of section 2.3).

As a way of engaging with this reading material, please find an “if P then Q” type theorem of your choosing. Tip: try googling “awesome theorems” or surfing wikipedia. Choose one that you find interesting and understand the statement of (not from your textbook, and don’t choose something high falutin’ with words you don’t understand). The P is called the hypothesis, and the Q is called the conclusion.

Using the theorem you found, (a) identify the hypothesis and the conclusion; (b) give an example which satisfies the hypothesis and conclusion if possible; (c) give an example which fails the hypothesis and conclusion if possible; (d) give an example which fails the hypothesis and satisfies the conclusion if possible; (e) give an example which satisfies the hypothesis and fails the conclusion, if possible. (Hint: in the example “R” of page 41, the integer 12 is an example which satisfies both hypothesis and conclusion, while 3 fails both the hypothesis and conclusion. The integer 2 fails the hypothesis but satisfies the conclusion.) Bring your work to class. A reminder: I will spot check these tasks for completeness and/or use them in class, but will not generally collect and grade. Make your best effort, but if you can’t complete a task, show me your attempts.

Relax and get settled into your semester. On Friday we will do a first proof.

I am teaching two sections, and your online environment will be combined. I look forward to meeting you all in person, and having a productive and fun semester exploring mathematics. Meanwhile, please look around the website.

Between each lecture this website will have a post describing your tasks before the next lecture. I will post this after class MWF by 1 pm.