10.1, Due November 3

Difficult: I always find it much more difficult when we start talking about infinities. For example, the introduction of this chapter discusses that the set of all natural numbers has the same cardinality as the set of all the perfect square natural numbers. This statement just blew my head off. I'm with Galileo on this one; I can definitely feel his hesitancy working with such odd things.

The one part I don't understand is how equivalence classes relate to numerically equivalent sets. Instead of a number, like [1], do we now say [A], which are all the sets that have a bijective function with set A?

Interesting: I thought it was neat how the first theorem used collections of sets, relations, functions, and equivalence relations all together. That's an official blast form this course's past!

Exam 2 Review, Due October 31

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

Principle of Mathematical Induction and Strong Principle of Mathematical Induction

Reflexive, symmetric, and transitive properties

Theorem: [a]=[b] if and only if aRb

Functions- Domain, codomain, range, surjective, injective, bijective, composition

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

Proof by MPI/SPMI.

Knowing the properties of relations

Prove a relation is an equivalence relation

Proving functions are bijective

• What do you need to work on understanding better before the exam? 

Reminder of SPMI problems

Equivalence class problems 

Well-defined vs Well-ordered

(I've got the hang of recent chapters, it's just about getting those rusty gears from chapter 6 and 8 moving again!)

9.6-9.7, due on October 29

Difficult: Whoa, big concept chapters! A lot to take in! First off, the proof that inverse functions are functions if and only if the functions are bijective was a lot to take in. If we're going to need to prove that on our own, it would be helpful to review it again in class. Also, are rational and linear the only functions we can easily find inverses of? It mentioned we can't find inverses for some polynomial functions, but are there any other ones we can do?

For permutations, do we always use natural numbers that come right in a row? Or could me use B= {2,5,8}, or even B={a,b,c} to find permutations? Also, what do we do with them...? Or just look at them since is it just a concept for now?

Finally, at the end it said "permutations satisfy the the properties of closure." What is a property of closure?

Interesting: Well, the one thing I do like about permutations is that they are easier to write than listing out all the commas and parentheses. I also thought it was interesting how we can make all functions inverses by cutting off the codomain that we don't use, but I guess that's kind of like what we did for the inverse trig functions.

9.5, Due October 27

Difficult: For some reason, it always throws me off that g ∘ f is g(f(x)), since I always think that the letter that comes first should be the function we do first, so that kept throwing me off but I think I've got it figured out now.

Also, it started talking about how ran A only needed to be a subset of B' for (g∘ f):A--> C to be defined. Well B' does NOT mean the derivative of B, correct? Because that would be MEGA confusing.

Interesting: At first, this concept was totally tricky. However, it made a lot more sense with the nice picture with circles and arrows, along with the examples of finite sets. I also never knew that composition could be associative, yet not commutative. I thought that was fancy!

9.3-9.4, Due October 24

Difficult: Alright, now I've got a lot of concepts floating around in my head that I need to work on solidifying. So we have three main special relations that we've learned: one-to-one, onto, and bijective. I understand most of them visually, like on a graph, but I need to understand them conceptually now. So, if we have to prove something is bijective, we must prove it is both one-to-one and onto? Several examples of proving these two proofs would be helpful.

I didn't understand completely about why theorem 9.4 is false for infinite sets, but I assume we don't need to completely understand that yet, since we haven't discussed infinite sets with these properties yet.

Interesting: It's interesting thinking about these properties with finite sets. I can always picture a graph on the real Cartesian plane to fulfill the properties, but when we start talking about equivalence sets, it's more difficult to picture/conceptualize, but easier to prove.

9.1-9.2, Due October 22

Difficult: So let me make sure I have this correct, the difference between codomain and range is that the codomain is all possible values of y, while the range is the only ones that get used? What does the symbol that is an o with a vertical slash through it mean? I just think it looks nice, but does it actually mean something specific? Also, if B={1,2}, would it be bad to represent the set of all functions from A to B by 2^A? (Since B is supposed to be {0,1} according to the last line of chapter 9.2)

Interesting: There is definitely a lot of interesting notation/vocabulary used and learned, such as image and mapping. I also thought the evolving definition of the word "function" over time was interesting as well, though the really old ones are really quite abstract and difficult to understand.

8.6, Due October 20

Difficult: So this all sounds great and dandy, I'm just not so sure about when sums or products are not well-defined. I mean, I guess the example in the book at the end of the chapter just completely changed the definition of what multiplication (redefined the function) and then called it not well-ordered... Sounds like cheating to me. Haha

But it would be great to go over proofs of well-defined. But from what I see, It looks like we just added another layer. Well defined --> Equivalence Classes --> Relations --> Modulos--> Divisiblity and then all back out!

Interesting:  Man, we use modular arithmetic all the time! Clocks is a great example, both minutes and hours are mod 60! Days of the week work in mod 7, but I guess days in a month gets messed up because it's (mod 28, 30, or 31). But still cool nonetheless!