To do for Wednesday:
- Please fill out your Proof by Contrapositive Setup Sheet, and your Proof by Induction Setup Sheet. We will take them up in class.
- 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.
- If you missed last monday/wednesday’s induction lectures, here is a quick summary: PDF Presentation of Trees having n-1 edges and accompanying video.
- 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.
Over the weekend, please:
- 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.
- A few induction-related webcomics: here and here.
Welcome to the month of March! For Friday:
- Group presentation day — be prepared.
- 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.
Have a great weekend, all! For Monday, please:
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:
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):
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).
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, 2.6, again. At the bottom of page 50 are a list of useful logical laws. Please become familiar with these by using them to simplify statements. Here is a worksheet you should complete for practice and bring to class.
- Learn the meaning of the word converse on page 44 of Hammack.
- Read Hammack, Chapter 5 until the end of 5.1 (pages 102-105). From this, learn the meaning of the word contrapositive.
- Take a look at these solutions to Proof Quiz #4, and compare to your proof. (Grading note: I haven’t decided what to do about proof quiz #3 yet.)
- In class Wednesday, we will take up the homework above, and then do something fun and educational.