No groupwork this week (nothing due on Friday Mar 16). It’s a week off!

Please fill out the new Midsemester Feedback Survey (anonymous) on canvas. In particular, this is your chance to weigh in on how we should change the group work system, if it needs any changing.

The badges quiz will cover Logic, Proofs, and also (new) Functions I and II.

Please review the notions of function, domain, codomain, surjective and injective, as covered in class. You can also find these in Hammack. Make sure you come Monday having reminded yourself what each of these words means.

By popular demand, here is the link to the registrar Spring 2018 exam schedule. I don’t know why it isn’t showing up in my.cu.edu as in previous semesters.

This coming week again, you may earn a missing Sets badge by coming to office hour with a study sheet, as described in detail in this old post (scroll down).

Please continue to study induction, as the proof quiz next Friday will be an inductive proof.

The proof quiz will be a proof which is easiest by contrapositive. (You have freedom to choose a proof method, but contrapositive will be your best bet.) To study for this, spend time with Hammack, Chapter 5. The proof will be similar to Chapter 5, p. 110, Exercises 1-13. Also, here are solutions to the Set Up Contrapositive worksheet.

Read Hammack, Section 5.3, which has some writing advice.

Please complete the Functions Worksheet we just started in class, by Monday. We will take this up on Monday.

Please keep studying induction, it is a tricky topic and you must find your peace with it as a method — we will test it next week.

Finish the weekend’s homework, if you didn’t have time to.

Read Hammack, Section 2.7, and do the odd exercises and compare to the back.

Read Hammack, Section 2.9, and do the odd exercises and compare to the back.

Correction: In one of the classes, I called both symbols “universal quantifiers”. That’s incorrect. The “for all” symbol is a universal quantifier, and the “there exists” is an existential quantifier. To cheer you up.

Consider coming to office hour to earn a missing sets badge. The current hours (although they don’t generally change) are listed under “About” on the navigation bar above. Below is more information on this opportunity.

HOW TO EARN a Sets Badge in Office Hour:

You may earn a missing sets badge (maximum one per week) by coming to office hour (you can email if the regular times don’t work) and doing the following:

Arrive with a one-page study guide to the content of the relevant badge. This should include definitions of the relevant terms, and examples and non-examples, as well as a list of tricky confusions and their resolutions. You have freedom of design, but I want to see a solid effort to master and present the material, not a scribbled summary. (You can write it by hand if you can produce something as tidy and organized as LaTeX.)

In office hour, after discussing the study guide and any clarifications needed, I will assign you a tricky badges problem, and you will solve it on the board or at the desk in front of me.

If you do these things well, you will earn your missing badge.

Late addition: you can now find notes and video of my introduction to induction.

Read about Induction in Hammack, pages 154-164. Read actively, as always.

There are lots of induction exercises in Hammack, Chapter 10 (page 169-171). The answers to odd ones are in the back of the book. Please do as many of these as you need to feel you are mastering the skill. For example, 1, 3, 5, 13, 17.

Recall our group work theorem: The complete graph on n vertices has (1/2)*n*(n-1) edges. Write a nice proof by induction for yourself (it is possible your group did this; if so, write it for yourself again, without reference to your groupwork — the purpose of any exercise is the engagement with the process).

Study for Monday’s badges quiz as needed. It will include all the Logic and Proofs badges. In particular, Proofs III is setting up a proof by induction — we’ll practice in class on Monday.

Note: this is potentially a decent amount of work. Do several hours over the weekend and if you don’t finish all this, come back to it during the week to finish.

Look ahead: We’ll have a proof by induction on our next Friday quiz. During the week we will spend more time on induction, and possibly some other topics.

Your quiz will be a constructive existence proof, and will involve rational/irrational numbers. Study appropriately (see your groupwork, and read Hammack 7.3-7.4). Here are a couple similar types of proofs appropriate to study by:

For every irrational x, there exists an irrational y such that xy = 1.

For every two rational numbers x and y, there is an irrational between x and y (we did this one before).

Every rational number can be written as a product of two rational numbers.

Please compare your Quiz #5 to the solutions and contact me if you want to discuss.

Please think about how to write an inductive proof that all trees on n vertices have n-1 edges. Here is a handout with the relevant definitions and the question. Please attempt to write a proof and bring it to class on Wednesday. Even if you aren’t sure you are doing it right, having made the attempt will prepare you to get the most out of class.

Announcement: There’s a Math Club Talk on Wednesday night. This week it is about computational complexity: how hard is it to compute certain things, like factoring.

Also, some people asked about how to include a picture in latex. The short story is to make sure your preamble where you include packages includes the package:

\usepackage{graphicx}

And then include a picture where you want it in your document by putting (in text mode, on its own line, surrounded by empty lines):

\includegraphics[width=3in]{mypicture.jpg}

You can find more info about this online, for example here. You can adjust the width to make your picture bigger or smaller. You can include jpg, png or pdf files, among others. You can put it inside a “figure” environment if you want it to “float” and position itself with a caption (see link above).

Next week (Monday Feb 26) will be the last opportunity for Sets badges on the quiz. There may be an opportunity to earn one you are missing in office hour later (but you can’t do them all in office hour), so buckle down and study Sets.

To do for Friday:

Prepare for group presentation day.

Prepare for your proof quiz. It will be a reasoning/counting proof similar to the proof we did in class on Wednesday. See the Wednesday class notes on Graph Definition and Handshake Lemma. Study this proof so you understand it very well, and you will be prepared for the quiz.

Spend some time thinking about the problems on our Graph Theory Warmup handout. That is, solve the Warmup problems, and give some thought to the Exploration problems while you are eating dinner, having a shower, or falling asleep (all of which are excellent times to think about math).

Study for sets badges for Monday if applicable (see above).

In class on Monday, we covered truth tables and logical implication, including the notions of contradiction and tautology. We used a handout of truth-tables and played a logic game (instructions and cards).

For Wednesday: (links below are coming shortly)

Fill in the truth table handout. Lines 3 and 4 (each a single table) are each a demonstration (proof, actually!) of a certain logical equivalence. Write down what the logical equivalence is, and then compare to the solutions.

Read Hammack, Chapter 2 up to the end of 2.5, doing the exercises. Keep going if you like. This formalizes the logic we have done intuitively. I will do a mini lecture in class on this material, and then take it a bit further.

Make sure you have sorted out your new groups.

The badges quiz will cover all the previous badges PLUS Logic I and Logic V. The Sets I-IV badges will appear perhaps two more times (including today). So get on them this week if you don’t have them yet! (Canvas is now up to date — note 2=full, 1=partial 0=no credit.) Otherwise focus on getting Logic IV and Proofs I. Then with your spare time, if available, try the new ones.

I have created a 7-minute recap of my advice on how to negate sentences. Watch this whenever you feel you need it for studying. It will be listed under resources. In particular, I suggest you watch it before tackling the Logic IV badge.