Proof Quiz will be a proof that a function is/is not injective/surjective. Read Hammack Section 12.2. Examples 12.4 and 12.5 are ideal models, as are exercises 2-10 (where “verify” means “prove or disprove” (disprove means prove the negation).
Reminder that individual homework is due. The assignment is under Resources (above) under Groupwork. (But you can do it individually.) Hand in on canvas.
You can still earn badges in office hour Wed/Thur, as explained previously.
In class, we will continue talking about functions and cardinalities.
Keep thinking about the relative cardinalities of the natural numbers, integers, rationals and reals. Think about the cartesian product of the integers with itself, too.
Read Hammack, Section 12.2. Do some exercises from 12.2 — maybe 5 of them.
Get a start on the Assignment for this week, which you’ll hand in on canvas. It’s listed under the Groupwork section of Resources. For more info on how this individual assignment replaces a groupwork, see the post on Monday.
Thank you for your feedback on the midsemester survey! It was very very helpful — thanks for writing so much. Here’s what we will do:
We will have two more groupwork assignments with presentations, due/presented on April 13th and 27th. We will mix up groups one more time beforehand.
On the non-groupwork weeks, I will still assign some proofs for everyone to write up individually and hand in on canvas. You will be allowed to work with others, if you like. They will also be handed in on canvas and I will assign a simple grade based on completeness, but can’t provide detailed feedback via canvas. I can give one-on-one feedback by request in office hour.
On presentation days, I may choose from individual assignments and group assignments for presentations.
For Monday’s class:
I have posted an assignment (under groupwork on Resources) due for hand-in on canvas next Friday. You may work with others if you like, but your write-up should be your own. Some suggestions:
Look for other students in MARC, 6-8 pm Tuesday, Wednesday and Thursday. (I’m just suggesting a general time, there’s no implied guarantee or requirement here.) Note: MARC tutors don’t specifically cater to 2001, but it’s still a good place to meet.
Contact others via canvas.
Continue to work with your group or old group if desired.
Monday’s badges quiz will include the same topics as last time plus some Synthesis. Synthesis badges are usually harder: they involve interesting combinations of your knowledge of other badges.
If you’re having trouble negating “if-then”, take a look at the video (see also Hammack 2.10). If you’re having trouble with quantifiers, read Hammack, Chapter 2.7 and 2.9 and do exercises there.
The following badges will appear only two more times (last time is the Monday after break): Logic I,II,IV, Proofs I,II.
Note: You can still earn Sets badges in office hour before break (see this old post for details). After break, you can earn Logic/Proofs badges in office hour for a few weeks (but Sets will be finished).
I suggest that over the weekend you get a start on the assignment, and prepare for Monday’s badges quiz.
Too bad it isn’t Pi Day forever. I hope you all got some free pie.
The quiz will be on induction. Please study for it, with the goal of understanding the main idea behind induction. In the past few weeks of posts/materials you will find reminders and suggestions for problems/readings/study.
No presentations, since no groupwork. We will, however, work on proofs.
If you haven’t already filled out the anonymous midsemester feedback survey on canvas, please please do so.
Late addition! Wednesday is Pi Day! Some activities:
Please join your math community for free fruit pie and pi-related activities in the foyer of the Math building (near Math 175) from 9am until 4pm.
The next undergraduate MathClubtalk for this Spring 2018 semester will be given by Dr. Robin Deeley, this Wednesday, March 14 from 5 – 6pm in MATH 350.Title: Shift dynamics Abstract: Dynamical systems were introduce by Poincaré in his study of the behavior of the solar system. I will discuss a number of dynamical systems coming from linear algebra and calculus before moving to an introduction to shift spaces and their dynamics. The concept of sensitivity to initial conditions will be emphasized. Free pizza-pi(e) and drinks will be provided.
Please fill out the anonymous midsemester feedback survey on canvas.
Review the terms on your Second Functions Worksheet, and make sure you have completed Problems 1 and 2 and feel confident about the answers. In class Wednesday, we will finish this worksheet, so bring it back to class.
Read Hammack, Chapter 12 up to the end of 12.2 (pages 196-204). This covers functions as we have been discussing them. Note that Hammack uses the term “relation” in Definition 12.1 on page 197. This really just means “subset of the Cartesian product”, as we discussed in class. (No need to read Chapter 11 right now.)
Do some 12.1 and 12.2 Exercises — whatever you feel is interesting.
Do at least one inductive proof — I suggest Chapter 10, Exercise 10.
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.
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.