Blog Post, due on October 1

Which topics and theorems do you think are the most important out of those we have studied?

I feel like the methods of proving things are the most important. Contradiction, Induction, Direct Proof, etc.

What kinds of questions do you expect to see on the exam?

I'm expecting lots of proofs and maybe a couple definitions. I feel like it would be unfair to ask us to prove something outside the range of what we've done (number theory, set theory etc) just because there are little tricks that you have to know to be able to work the proof.

What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

Definitely mathematical induction. I feel like we talked about it and didn't really work any problems. So while I don't have any specific problems or anything, I would like to hear about what we need to know about it for the exam.

7.1-7.3, due on September 28

I don't have much intelligent to say about these sections. 7.1 was hard for me because I didn't really follow the importance of it. It was almost like a history lesson or a long, long definition about the difference between a theorem and a conjecture. Section 7.2 is not really anything new, just a bunch of simple statements and applying different qualifiers, all of which seemed like we've covered before. Section 7.3 got a little more interesting.

I enjoyed 7.3, especially the option of proving if it is false or true. However, I realize that this could be a lot tougher for proofs since we don't necessarily know that it is true. It will require greater analysis before the proof to decide how to proceed.

5.4 - 5.5, due on September 26

I usually use this space to kind of write my questions and things I didn't understand. I hope that helps and/or is what you're looking for. This whole idea of uniqueness, what kind of properties can this entail? I assume that when we ask if there is a unique root on an interval, that means it is the only one in the interval? Would uniqueness also expand to parity? For instance, there is only one number contained in the set S that makes some function even.

I find it neat that we're using the things that we learned about odd and even numbers and proofs in the principles of disproving existence statements. Is there ever an end of things to prove? I appreciate that the book added in a different viewpoint at the end. Every disproof is really just a proof of the of the negation.

5.2-5.3, due on September 24

The most difficult portion of this text was the actual explanation of contradiction. I understand all the examples and proofs, but when symbols are thrown in it gets hairy. The notation of -> Contradiction threw me for a loop, but as I look at it a second time it does make sense. Another question arises with Theorem 5.16. It states an assumption "We may further assume that a/b has been expressed in lowest terms". I assume that we assumed this just because the nature of the proof. How would we know to do something like this? Is there a guideline?

The prisoner problem was also interesting. Myself, I would just assume that the other two prisoners were dumb and wouldn't follow that logic to find out that their dot was red, but the point was clear. The review likewise was interesting. However we have also read about induction (ahead of the book) so I wonder how that compares to the others. It would be nice to see another table like that with induction in it.

4.5-4.6 and 6.1, due on September 21

The principle of mathematical induction was tough. It made sense, but many of the proofs are a little tough to follow. I hope we can review this in class, especially Theorem 6.2. I think that they're proving this by contradiction, however it states that if the theorem is false, then it satisfies conditions 1 and 2 (first case is true and the implication is true) but then it states that there's still positive integers for which P(n) is false. If the initial conditions are satisfied by assuming the end is false, you don't have a proof. One of the 2 conditions needs to be false. Also the concept of well-ordering could be explained a little better. I'm not sure how being well-ordered leads to the concept of induction. 

The first two sections, 4.5 and 4.6, were very basic, but useful. Those proofs were short and clever, but nothing really new. Induction is a neat principle, one that I've also had a little experience with. But the proofs are super long! Also mathematically intensive! Good thing I like math. I also liked the way to compute sums super quickly with the little story thrown in there too. I always wondered how people did those big numbers so fast. It's just neat that we can prove stuff like that and then use it!

4.3-4.4, due on September 19

These sections were very straightforward and there wasn't a lot that was too difficult to understand. We basically took the techniques that we had been applying with integer-related proofs and applied them to the real numbers and sets. The one thing that did stump me for a minute was the proof of theorem 4.17 involving absolute values. I was concerned about the use of the inequality in the proof but then I realized that we pulled it from the previous page in the statement of the proof. Another thing that did come up was stating contrapositive and the use of loss of generality. The book states that stating contrapositive isn't necessary, however in class you instructed us that we might want to do it. Regarding the loss of generality, the book just stated WLOG and then assumed one of the possibilities. You told us in class that we might want to write something like, the proof follows the same steps. Are we going to get knocked points if we don't, even though the book says it's not necessary?

I thoroughly love set theory. It's something pretty familiar to me; In linear algebra and the IMPACT boot camp we did a lot of proofs involving different sort of sets. What's different now is I actually understand it! It is almost weird to me but I enjoyed these proofs. I thought they were pretty cool but also elegant.

Additional Questions:

• How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?

Homework assignments don't really take me that long. I usually get through all of them in under an hour. Yes, the reading and lecture have been super helpful in preparing me. I feel like there's usually one that I don't quite understand, but if there is I ask about it at the start of class.

• What has contributed most to your learning in this class thus far?

The best has probably been the reading. The blog post forces me to actually read before class and if there's something that I don't fully understand we go over the reading in the lecture. 

• What do you think would help you learn more effectively or make the class better for you? (This can be feedback for me, or goals for yourself.)

I'm enjoying the class and understanding most of it. I have however heard horror stories about Math 290 tests. We haven't really talked too much about what will be expected of us for the test (as far as during lecture, something like "you will be required to prove similar things like this"). Perhaps this helps students not to single one thing out, but it would be nice to know!

4.1-4.2, due on September 17

I think the toughest part about this section is getting used to the vernacular and notation. For instance modulus was a bit confusing since it was introduced and then thrown around everywhere. Something that I don't quite understand was result 4.12 in the book. I understand how the result was obtained, but if we need to prove one statement or another do we really need to prove both to prove the result by contrapositive? Logically it would seem like we wouldn't need to.

Something that was neat was the use of cases combined with contrapositive. As well that any number can be written as a mod of 2 or 3. This proves useful when we proved a bunch of little properties of products and sums of numbers. All of these properties seem so simple but it's neat to be able to prove them.

3.4-3.5, due on September 14

One thing about this section that was tough was the proof of theorem 3.16. It is a biconditional and so you have to prove it backwards and forwards. However, it seemed like the book only proved it forwards. It took me a little while to wrap my head around it. I realized that it was a proof by contrapositive (it didn't say)  and it still was a proof by cases. 

Proof by cases was pretty neat. It really makes sense logically to examine every case so that we can draw a general conclusion that applies for every case. As well was the introduction of "without loss of generality". I'm a fan of saving time. 

3.1-3.3, due on September 12

1. First of all this was a really neat section. It's nice to get into proofs. One thing that was a little hard to follow was some of the even and odd number proofs. At first I failed to see how (for instance in Result 3.5) the statement "Since -5x-2 is an integer, -5n-3 is an odd integer" proved that -5n-3 was an odd integer. Something along these lines was present in many proofs. Upon closer inspection I realized that we simply needed to show that -5n-3 could be written in the generic form 2y+1 to be an odd integer and what I was missing was that we used the fact that any product or addition etc of integers remained in the integers.

2. I've had some experience with proofs and this section is neat because it explains the theory of how to prove something. For instance, proof by contrapositive. I never understood why we would use it, just that you prove pretty much the opposite and that proves the original. However now with logical equivalence, I can see why and what's more, I'm convinced now that this is an actual proof. Another cool thing along the same lines is I now understand what a trivial proof is and why it is in fact trivial. Likewise this also stems from the logic and the logic tables.

Pages 5-12 of Chapter 0, due on September 10

1. One thing I would like clarification on is when we can use the symbols "for all" or "there exists". The text states that they should only be used when discussing logic. Does this mean only when we're setting up a problem or in the body of the proof or what? As well, do we need to state the meaning of these symbols if we haven't used them before?

2. It was neat learning about a "display". I realized immediately that the text uses this very effectively and conveys the idea simply to the reader. This chapter is very good to read just before we use LateX to compose our homework because it'll be fresh in our minds while learning. The same goes with the majority of everything in this section! It'll be important to learn the LateX commands. I also really enjoy the use of "we" in mathematical context. It helps me not to feel dumb as I read really complicated math.

2.9-2.11, due on September 7

1. Most of the things in this chapter are self-explanatory, they're just an extension of logic. The characterization threw me a little for a loop. The example that it gives in the book is that the the statements: A triangle T is equilateral if and only if T has three equal sides. It states that this is not a characterization but a definition. I'm not sure what the difference is. If we had defined an equilateral triangle as a triangle with three equal angles, then would this statement be a characterization?

2. This semester I'm also in a programming class for engineers and the logic that we have been learning is the exact same as all the logic used in writing code. This is super neat to me because all of the logic and the not statements etc. are all of a sudden actually useful to me instead of just for use of proving higher math (which I'm not planning on taking).

2.5-2.8, due on September 5

1. The toughest part of this reading is honestly the symbols. There was introduced a lot of symbols from last section and now in this section they're everywhere! Another thing that is tough for me is once again the practical application of some of the things that we learned. For instance with example 2.16, it once again refers to the teacher giving the A example. I don't understand the logical equivalence (with this example) of P=>Q and (~P) v Q. ~P would be If you don't earn an A on the final exam and Q would be you receive an A for the final grade, but I don't quite understand how both a logically equivalent.

2. Probably the neatest thing in this section is the idea of logical equivalence. I like how in this section the book identifies how it would be useful. Ofttimes it is difficult to show that one very abstract statement is true hence it is super cool that we can show that it is logically equivalent with a much easier example and then prove that that is true, thus proving the first is true!