Welcome to Math 299. I will post assignments and announcements here throughout the semester. Check back frequently. Below are links to some resources we will be using in the course.
Below are all of the Training Workouts for Math 299 – Spring 2020 to date. For the most recent Training Workout and additional information see the Math 299 Home Page.
Welcome to Math 299. I will post assignments and announcements here throughout the semester. Check back frequently. Below are links to some resources we will be using in the course.
Give a formal proof that every uniformly continuous function is continuous, i.e., that ‘(UC)$\Rightarrow$(C)’.
Note: You don’t need to know what a function is, or what absolute value means or what a real number is, or any facts about real numbers, or anything from calculus or anything like that. You only need to know about logic and quantifiers to prove this. Also note that you can’t type $|x|$ in Lurch but you can use $abs(x)$ instead if you want Lurch to check your proofs.
We have been proving the basic properties of natural numbers listed in the 26 theorems in section 7 of the lecture notes. We have already proven #2-#6. Prove the following theorems from that list of theorems. Note that if you are proving theorem number $n$ you can use any theorem that precedes it in the list as a rule of inference even if we didn’t prove it (but never a theorem that comes after it in the list).
If nobody loves themself, and everyone loves anyone who is loved by someone they love, then nobody is ever loved by someone they love.
In order to make your extended Spring Break more enjoyable, here is a bonus assignment you can work on optionally. Since this is for bonus points (I’ll count it as one in-class quiz) you don’t have to do it, and I’m making three different options you can choose from (sort of). You can only choose one option. Write them up in Lurch or Overleaf in a separate file, and put it in a folder called ‘Bonus Assignment’ in your Dropbox folder. I don’t want to set a particular due date, so send me an email when you are finished (but the offer expires after this week.)
Type up the following proofs in LaTeX. Use the semiformal proof style we have been using, one statement per line, and a reason stated for each line that needs one, with optional comments where you feel they would help the reader. You can do that by clicking on the Homework Template link below on this page. To hand in your document, download the pdf and put your pdf file in your Dropbox folder with the appropriate filename.
Type up the following proofs in LaTeX. Use the semiformal proof style we have been using, one statement per line, and a reason stated for each line that needs one, with optional comments where you feel they would help the reader. You can do that by clicking on the Homework Template link below on this page. To hand in your document, download the pdf and put your pdf file in your Dropbox folder with the appropriate filename.
Type up the following proofs in LaTeX. Use the semiformal proof style we have been using, one statement per line, and a reason stated for each line that needs one, with optional comments where you feel they would help the reader. You can do that by clicking on the Homework Template link below on this page. To hand in your document, download the pdf and put your pdf file in your Dropbox folder with the appropriate filename. All variables and constants are assumed to be real numbers unless stated otherwise.
You can use any theorems prior to the chapter on real numbers (keeping in mind that theorems about the natural numbers do not automatically extend to theorems about the reals). Other than that you can only use the axioms and definitions about the real numbers, and any theorem you prove first to use in a later proof (of course). But you cannot use any theorems in the problem sections in Chapter 9 in the Lecture Notes.
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