Final Exam, due on December 10

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

1. Logic Tables

2. Direct Proof (Format)

3. Proof by Contrapositive (Format)

4. Proof by Contradiciton (Format)

5. Proof by Mathematical Induction (Format)

6. Proving an Equivalence Relation

7. Congruence Modulo n

8. Proving a Bijective Function

9. Schroder-Bernstein Theorem

10. Section 12.4

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.

1. Direct Proof (Format)-EXAMPLE

2. Proof by Contrapositive (Format)-EXAMPLE

3. Proof by Contradiciton (Format)-EXAMPLE

4. Proof by Mathematical Induction (Format)-EXAMPLE

5. Proving an Equivalence Relation

6. Congruence Modulo n

7. Proving a Bijective Function

8. Section 12.4

12.4, due on December 8

DIFFICULT
So far this has been one of the easier chapters out of this book.  Although these proofs look to be longer and more tedious too.

REFLECTIVE
Again this is stuff I remember back in 112 days, and as a tutor I am helping 112 students understand it now.  They were fairly straight forward then and are probably now.

Faster than a calculator | Arthur Benjamin | TEDxOxford - Make-Up Blog Entry

DIFFICULT
It was difficult to understand how he matches words with numbers and uses them in his thinking process.  Just thinking of doing more than 2 digit math in my head gives me a headache.

REFLECTIVE
Mnemonics are how I do most of my memorizing, but not on the same scale that Arthur Benjamin uses them.  We did talk about Arthur Benjamin when we were talking about Modular things, but I fail to see the correlation here.

12.3, due on December 5

DIFFICULT
I am not understanding the term Deleted Neighborhood.  Like I've said before I've never understood the point of Delta-Epsilon Proofs in 112, so this should be a good lecture tomorrow.

REFLECTIVE
I've understood the material in this chapter, and I've seen this material in 112 before.  So this section shouldn't be too rough.

12.2, due on December 3

DIFFICULT
Infinite Series were never easy for me.  So this whole section might be a little rough.  The steps taken to prove the results are a little difficult for me to understand,just like section 1, was a little hard to understand too.

REFLECTIVE
Math 113 just finished up Chapter 11 which talks about infinite series.  The P-Test comes to mind when thinking of Result 12.10.  We are again using things we've learned previously to help with more complex problems.

12.1, due on December 1

DIFFICULT
I never liked Delta/Epsilon proofs in calculus because I didn't know their purpose, I also found them to be quite hard.  Being a lower division math tutor I still found them hard to explain to students just because I never knew their purpose, so this chapter will help me in the future.

REFLECTIVE
I always wanted to know how Delta/Epsilon proofs proved limits in calculus.  I work in the Math Lab as a lower division tutor, Math 112 and 113 students are finishing up their individual sequences and series sections, so I should be well versed in at least this section.

Reflective Questions, due on November 25

WHAT HAVE YOU LEARNED IN THIS COURSE?

I have learned a great deal of things in this class, although when a test comes around I tend to forget all the things I have learned.  I love taking notes in class and understanding 80% of it.  Some of the cool things I have learned were logic and logic tables, cardinalities and denumerability, mathematical induction was fun, and this prime stuff in Chapter 11 is great too.

HOW MIGHT THESE THINGS BE USEFUL TO YOU IN THE FUTURE?
I really don't know how this class would be useful in my future.  I guess the use of logical thinking in problem solving in civil engineering would be useful, it might also be annoying.  I'm sure this class will be very useful in the future, I have just yet to see that yet.

11.5-11.6, due on November 24

DIFFICULT
The only thing that will be rough will be the material found in section 5.  This whole prime number thing is kind of hard to grasp, where at the same time it isn't that bad.  We'll see how the final does though.

REFLECTIVE
These sections are highly reminiscent of elementary school days.  I remember doing all sorts of math things pertaining to finding multiples of 2, 3, 4, 5, etc... Like the rules found on page 258.

Exam III, due on November 21

DIFFICULT
Theorem 10.1, Theorem 10.3, Result 10.5, Result 10.7, Theorem 10.13, Theorem 10.14, Theorem 10.19, Corollary 10.20, Lemma 11.1, Theorem 11.2, Theorem 11.3, Theorem 11.4, Theorem 11.7, Theorem 11.8.
What does countable mean?

REFLECTIVE
Result 10.2, Theorem 10.8, Corollary 10.10, Theorem 10.11, Theorem 10.12, Theorem 10.15, Theorem 10.18, Theorem 11.4.

I feel like I understand what is happening in class, but when I sit down to write my exam, my mind goes blank.  So any strategies specifically for this exam would be helpful, I really want to do about average on this test if anything.

How do your final grades work?  Like if a student attends every class, and takes notes on all those lectures, does all the homework, and all the pre-class readings, but scores below average on all the exams, what can that student expect his final grade to be?  I just want to know if our final grade is dependent on our best effort.

11.3-11.4, due on November 19

DIFFICULT
The Euclidean Algorithm is a little hard for me to comprehend, but I am sure it will be covered in lecture.  Is this an iterative process?

REFLECTIVE
Greatest Common Divisor is something I haven't cared about since primary school, back then it was easy to understand so I an hopeful it still will be.  Theorem 11.8 is also intuitive enough to understand.  Lemma 11.9 is fairly straight forward too.

11.1-11.2, due on November 17

DIFFICULT
These sections seem to be a piece of cake compared the last section of  chapter 10.  Although since this is chapter 11 and since up till now sections have depended on prior sections, I can only assume chapter 11 will get harder

REFLECTIVE
The theorems in section 1 & 2 are pretty understandable and intuitive.  I remember doing this prime number stuff back in primary school.  I really liked the equivalence classes before, although that didn't reflect in my exam score.

The rest of 10.5, due on November 14

DIFFICULT
I'm not grasping the Axiom of Choice, I need it in layman's terms.  Theorem 10.19 is mind blowing, I don't usually find math interesting, but when I do my mind usually ends up being blown.  Corollary 10.20 is also interesting.

REFLECTIVE
Theorem A and B are are easy to understand and pretty intuitive.  I'm interested in seeing how Theorem 10.19 and the Corollary pans out in class, once they are verbally explained to me.

10.5 up to Theorem 10.18, due on November 12

DIFFICULT
I'm not fully understanding this Restriction business, why we need it and what help is provides and how it correlates with Theorem 10.18.  Trying to understand Theorem 10.18 is a little odd.  I hope that this will be clarified in lecture tomorrow.

REFLECTIVE
Some things are familiar from earlier sections in the chapter.  We have talked about Cantor earlier this book when we talked about denumerability.  This section seams to be pretty difficult, good thing we are spending two lectures on it.

10.4, due on November 10

DIFFICULT
Again this single section that was assigned to us is pretty intuitive.  The Continuum Hypothesis is difficult and not so clear.

REFLECTIVE
Theorem 10.14 is pretty understandable.  Theorem 10.15 is also pretty intuitive.  The definition Smaller Cardinality is also easy to understand.

10.3, due on November 7

DIFFICULT
Theorem 10.3 is a little bizarre, but ever since that one lecture my idea of countable has been skew.  All the rest of the theorems are pretty intuitive.

REFLECTIVE
I haven't been using many other proof tools.  So it is neat to see a proof by contradiction in this chapter.  My favorite technique is proof by contradiction.  The theorems in this section seem pretty intuitive.

10.2, due on November 5

DIFFICULT
The idea behind Denumerability is a little intense.  Maybe I am not thinking deep enough, but I think I understood most of section 2.

REFLECTIVE
We talked about section 2 on Monday.  After reading through the section, I think we covered all of it.

10.1, due on November 3

DIFFICULT
As all of these tools are being put together, the thing that will remain hard will be knowing when to use one or the other, and remembering them all.

REFLECTIVE
Obvious there are concepts in this section from chapter 9.  As I have come to accept, this book is organized in such a way that you learn things that help you solve things in later chapters.  Showing that things were bijective in chapter 9 weren't too hard.

9.6-9.7, due on October 29

DIFFICULT
Does Bijective also mean Inverse?  Since in order for a function to have an inverse it has to be bijective.  I am not sure what is happening in Section 9.7.  I'm having a hard time visualizing what 9.7 is talking about.  Things make more sense after lecture anyways to me anyway.

REFLECTIVE
Seeing inverse functions again.  Section 9.6 seems pretty straight forward and easy to understand.  Chapter 9 seems to be a pretty intuitive chapter, probably because it is familiar material.  I kind of understand what a Permutation is although I don't think I've ever seen it before.

9.5, due on October 27

DIFFICULT
The proofs we've had to write in this chapter have been challenging just because they are simple.  The Theorems and Corollary that are present in this section are kind of confusing.  Also just remembering definitions will be a difficult task.

REFLECTIVE
I'm really liking this chapter.  Learning different concepts about topics that are already familiar to me.  I just feel like chapter 9 is leading up to something bigger, and now we are just seeing a partial view of the whole.

9.3-9.4, due on October 23

DIFFICULT
The Onto functions in 9.3 are difficult for me to understand, and I think its because I am unclear of what the difference between codomain and range is.  I don't understand what a Bijective Function is or what its difference is between an injective function is.

REFLECTIVE
Section 9.3 talks about one-to-one functions, which have been discussed in calculus.  They are functions that pass the vertical and horizontal line tests.

9.1-9.2, due on October 22

DIFFICULT
The difficult parts of these two sections will be the definitions, and remembering them all.  I also found that the first section contains a lot of math language that is hard to follow for the untrained eye.

REFLECTIVE
These sections seem pretty clear that we are talking about the same functions we were taught about in other math classes, with just a little more words to do it and with more definitions.  I am interested to see how the lecture goes tomorrow morning.

8.6, due on October 20

DIFFICULT
This section seemed pretty self explanatory.  The only things that were different for me to understand are closed under addition and multiplication, but I assume those will be explained in class.

REFLECTIVE
I find this equivalence class stuff pretty interesting, how they set up a Partition.  This section was very similar to what we have seen in this chapter, so it was pretty easy to understand.

8.3-8.4, due on October 15

DIFFICULT
The transitive relationship is nebulous to me.  In class on Monday it was really hard for me to understand, especially with the group activity.  I would like some examples that help clarify that topic, it seemed like everyone else understood it, so maybe I will go to your office hours.

REFLECTIVE
These sections are just appendages to what we have been talking about on Monday.  So besides the things discussed in the difficult section, I feel pretty good about the material discussed here.

8.1-8.2, due on October 13

DIFFICULT
Section 8.1 was pretty intuitive.  Although since these sections are using sets, they might be harder than what I initially thought.  I think the hardest part of these two sections will be remembering all the definitions and how to apply them.

REFLECTIVE
Section 8.1 was pretty intuitive.  Although there are quite a bit of definitions, they should be pretty easy to understand after lecture tomorrow morning.

6.4, due on October 10

DIFFICULT
Again the topics discussed in 6.4 look very similar to those found in the previous sections.  Section 6.1 and 6.2 were very similar, and it looks as if 6.4 is also very similar.

REFLECTIVE
I've seen these topics and such discussed in 6.1 and 6.2.  Although now they can be applied to recursive series.

6.2, due on October 8

DIFFICULT
This section is more of the same as last section was, just with more applications of Induction.  This stuff doesn't seem to difficult to understand, although I haven't done the homework due Wednesday yet.

REFLECTIVE
This is the same stuff we talked about with the other professor on Monday.

6.1, due on October 6

DIFFICULT
The implication P implies P(k+1) might need some explaining and why it's called Mathematical Induction.  I think what I need most out of this section is seeing some different examples.  I don't see how Proof by Induction works.

REFLECTIVE
The idea behind least element isn't hard to understand and is intuitive.  I have heard the Gauss story before.  I have used that ratio that Gauss came up with before too.

Exam 1, due on October 2

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

1.  Understanding truth tables and their applications to proofs.

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

1.  I expect to see a combination of questions that reflect my understanding of sets, logic, contrapositive, direct proofs, counter examples, contradictions, and everything else we've learned.

2.  If this exam is to take the average person 1.5 hours.  I expect to see a section of definitions, and like less-than 10 proofs.

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

1.  I could use some help with anything regarding sets and there operators.

5.4-5.5, due on October 1

DIFFICULT
Existence statements do not seem as complicated as proofs involving sets with various set operators. These existence proofs look very familiar to counterexamples, especially section 5.5.  I think the only difficult thing of this reading block would be the LaTeX document that needs to be prepared, and LaTeX is getting easier to use.  Sections 5.4 and 5.5 seem intuitive.

REFLECTIVE
As stated above existence proofs are very similar to using counterexamples.  Both these sections shouldn't be too different than what we've seen in the past.  They also shouldn't been much more complicated.

5.2-5.3, due on September 29

DIFFICULT
Remembering all the tools we learned and brought together and are now talking about in section 5.3 will be the tough part.  Section 5.2 will be only as hard at Proof by Contrapositive, I think.

REFLECTIVE
Section 5.2 also doesn't look to rough at first.  I love visual things that have all the information in one place, like Figure 5.2 in section 5.3.  Section 5.3 will be a great section to go over since there is a coming test this weekend.  I would prefer there to be more of it than section 5.2 in lecture.

4.5-4.6 and 5.1, due September 26

DIFFICULT
The most difficult topics to me in this reading set, would be the Set Operations, for some reason I can just never grasp that.  So between understanding Set Operations and typing things in LaTeX this coming assignment should be fun.

REFLECTIVE
I've used counterexamples in Linear Algebra a couple times so section 5.1.  As for the proofs side of things, they are getting easier the more I do them.

4.3-4.4, due on September 24

DIFFICULT
Section 4.4 I know will be especially hard for me since I didn't quite understand the material back when we first discussed it at the beginning of the semester.  I understand the basics of set operations, but I have a hard time visually understanding it still.

REFLECTIVE
Section 4.3 seemed pretty intuitive and easy to understand.  The section was just like the other proofs that we have done, the only difference is involving Real Numbers.  With this addition of Real Numbers we can be more versatile in writing proofs.

ANSWERS
On average I would say the homework has taken me 1.5-2 hours to complete.  Although the first LaTex assignment took me 4 hours, now this is absurd to me.  If the first LaTeX assignment took 4 hours to complete then the other LaTeX assignments will probably take me longer to complete, I just don't have time to teach myself the language of LaTeX and write proofs.  I love to read, but I just can't follow textbooks or solutions manuals very well.  So after reading I'm not sure how much I can remember.  The lectures however are helping prepare me for the assignments more than reading the subject material before class.  I believe lectures have contributed the most to my learning, then the assignments, then the subject material readings.  I take very detailed notes while in class, I actually think I have over 20 pages thus far, so I think my note taking is good.  I enjoy your lectures and find them easy to understand.  I would just like to see harder examples of proofs in lecture as most of our assignments contain harder/more confusing proofs than the examples shown in class.  I would also like to see more LaTeX syntax in class.

4.1-4.2, due on September 22

DIFFICULT
Section 4.1 I can foresee being difficult, just because of syntax.  Proofs are getting to be easier to understand, through the examples in class and in the textbook.  Section 4.2 material has always been confusing to me.  I've taken a computer programming class in the C++ language and I still have yet to grasp the modulo operator.

REFLECTIVE
Section 4.1 doesn't seem to be too difficult and actually somewhat intuitive. Consequently the material found in section 4.2 I have seen before.  So I am hoping that this next lecture will help clarify a few things.

3.3-3.5, due on September 19

DIFFICULT
After doing the homework due tomorrow, I've concluded that I need more examples of how to prove something.  I just don't think things were clear when you first presented proofs to me in class last Wednesday.  Like the truth table we discussed on Wednesday about P implying Q.  Which of the results yields a Trivial Proof and which a Vacuous Proof.  I clearly understand the objective behind section 3.3, but I feel my Proof-Foundation a little shaky, especially after the assignment due tomorrow.  Like take for example problems 3.1 and 3.2, I had no idea how I arrived at my final answer, or if they were right.  Same thing goes for section 3.4, I understand the objective just not how to arrive at the objective.

REFLECTIVE
The math related in these three sections are intuitive and easy to understand, its just the proofs part that I am having a hard time seeing.  Most of the time following the examples found in the section leave me more confused than when I began.  I have a love hate relationship with mathematics, and this stuff is only making me hate math more, because I don't understand proofs.

3.1-3.2, due of September 17

DIFFICULT
In section 3.1 I feel like I am reading the same things that I read last chapter, just with given names of different truth table values.  What is the definition of the word: vacuous?  I'm sure once section 3.2 is taught in class I will understand it better.  I'm a visually taught person, it is sometimes hard for me to learn things by reading alone.

REFLECTIVE
Result 3.2 and 3.3 are very intuitive as they are just the mathematics that I have always been taught.  I understand most of the examples portrayed in section 3.2.

Mathematical Writing, due on September 15

DIFFICULT
I struggle finding examples to support my arguments.  Writing mathematical expressions might get a little rough, especially trying to find the proper words to use.  I've never liked the idea of putting words with math, but then again my major is Civil Engineering, I hope this class helps with my problem solving skills as an potential Engineer.

REFLECTIVE
This section is like writing a technical paper, for a technical writing class.  I really liked the content found in the Common Words and Phrases in Mathematics section, I think it will be a great asset to me as I begin writing proofs.  After having taken a technical writing class at BYU, I would like to think I can do well when writing proofs.

2.9-2.10, due on September 12

DIFFICULT
In section 2.10 I'm having a hard time understanding what quantification and an existential quantifier does to help me.  Although maybe I am over-complicating it, because it might be really easy, I tend to do that often.

REFLECTIVE
Section 2.9 was kind of basic, maybe I'm not thinking deep enough.  The section was like the arithmetic laws of commutative, associative, and distributive so it was fairly easy to understand and comprehend.

2.5-2.8, due on September 10

DIFFICULT

I just don't understand the example in section 2.6: 100 is even if and only if 101 is prime, and why it is true.  I found that using a truth table makes these statements quite easy, so maybe when I delve into the next assignment, and after lecture tomorrow, I'll understand it better.

REFLECTIVE

Section 2.5 is just like section 2.4, and I understood section 2.4 more fully after the class discussion on September 8.  Section 2.7 was quite intuitive.  The gospel is a tautology, it is always true no matter the combination and never a contradiction.  We talked about logical equivalence briefly on September 8, and it seemed intuitive, after reading section 2.8 it also seemed intuitive.

2.1-2.4, due on September 8

DIFFICULT
Section 2.2 poses some potential difficulties, for example: ~P : The integer 3 is even.  Remembering the syntax of "~" used in statement notation could be rough.  In section 2.3, I had a hard time understanding when a disjunction was true or false.  In section 2.4 I didn't quite understand the truth table for implication.

REFLECTIVE
Section 2.1, although not mathematics, reminds me of English.  As I was reading section 2.2 I got excited to see what things I will further learn in Math 290.  This chapter seems to be the beginning of the math language of proofs, I guess that's why I got excited when I read through 2.2.  Other than my few questions above, this chapter doesn't seem too tough.

1.1-1.6, due on September 5

DIFFICULT
I found the two terms difference and relative complement confusing.  First, is there a difference between the two terms or are they the same thing.  If they are different, what is that difference?  I tried to consolidate my next issue into a couple of subtopics of the section, but I just couldn't narrow it down to one thing in particular, so I don't understand the entirety of section 1.4.

REFLECTIVE
Section 1.1 through 1.3 were fairly reasonable, especially after our lecture on Wednesday. Section 1.5 Partitions of Sets was an easy section to understand, just because it was very intuitive.  Section 1.6 Cartesian Products of Sets is just like multiplying two matrices together, so that was an easy topic to grasp since I have taken Linear Algebra.

Introduction, due on September 5

Q: What is your year in school and major?
A: I am a Senior majoring in Civil Engineering.

Q: Which calculus-or-above math courses have you taken?
A: I have taken Calculus 1, Calculus 2, Linear Algebra, Calculus 3, and Ordinary Differential Equations.

Q: Why are you taking this class?
A: I'm not going to lie, I am nearing the end of my undergraduate studies at BYU and needed some classes to fill in my 14-credit-scholarship requirement.  Since I only needed Math 290 to minor in math, I though I'd give it a swing.

Q: Tell me about the math professor or teacher you have had who was the most and/or least effective.  What did s/he do that worked so well/poorly?
A: I am a transfer student from Snow College: Ephraim.  My favorite teacher there was Jonathan Bodrero.  He gave us the equations and explained them, and then he did a ton of examples.  He did more examples than theory whenever explaining something new.  That's why I loved him so much as a teacher, he taught me Calculus 1 and 2.  So doing a lot of examples I think is what made him so effective.  In contrast, this other teacher, whose class I later dropped, did too much theory and never enough explaining or examples.  So I think that is what made her less effective.

Q: Write something interesting or unique about yourself?
A: I love and hate math, at Snow College I tutored students, for 2.5 years, in math, physics, and engineering and I loved teaching them.  I loved seeing it make sense for them.  Whereas sometimes I'd be sitting in my own math class and not understanding anything the professor was saying.  I love and hate math, and here I am minoring in Math.

Q: If you are unable to come to my scheduled office hours, what times would work for you?
A: Your office hours work for me.