Extra Credit

So I went to the Mimematics show.  It was pretty cool andI got a free cookie.

It was hard to understand some of the acts, like the one on distance had an accordion type thing that said don't touch and measure yourself carefully.  I don't know what that had to do with distance, maybe it meant keep your distance.

The most interesting part was how they taught us how to do some simple mime tricks, like catching a ball in a paper bag, or doing the wall.  They had an interesting act where one got into some air conditioning ventilation and used it to represent a function as stuff that went in came out small.  Unfortunately, he took the wrong thing, and didn't have a miniature version of it, so it kind of threw off the act.

Student Ratings and Review due on 10 December

Definition of limit and proving limits was one of the most important things that we studied.

Before the exam it would be great to go over serjective/injective proofs, as well as strong induction, and most missed questions on past exams.

Post 12.4, due on december 8

The most difficult part of the material was simply the minor details in the proofs of these theorems.  Some of them are not too difficult to understand, but it will be difficult to remember which lemma's to prove for each theorem.  I suppose practice would help a lot.

The most interesting part is how limits are relatively easy to work with since you can add them and subtract them, multiply and divide.  It becomes less difficult if they involve infinity and zero, because what is 0 times infinity.  I remember having a discussion about this in early calculus, but my memory is long gone.

12.4, Due on 5 December

The most difficult part of the material was using the definition of the limit to prove these theorems about functions, especially choosing delta, but it seems very similar to how we choose N to define limits of series and summations.

The most interesting material was how limits can simply be added and subtracted, as well as multiplied and divided.  The other thing was how delta is used to describe a neighborhood.  In calculus and also in physics and chemistry, we mainly use delta to mean partial such as a partial charge or a partial differential.  It is interesting that we use it in two different ways.

Section 12.1, Due 2 December

The most difficult part of the material was proving that a series diverges by using a proof by contradiction.  It does not seem like an easy thing to do.  Other than that, it was simply 12.1 plus some induction, which really is not too difficult.

The most interesting part was the harmonic series.  It is weird to think that it is a diverging series, but it did remind me of a math joke, though it is not the same series.  A mathematician walks into a bar and orders 1 glass of soda, then the next mathematician comes in and orders a half glass of soda, then another comes in and orders 1/4 glass.  At this point the bartender sees a long line of mathematicians and simply pours 2 glasses.  Because of this joke I know this series converges to two.

12.1, Due on 1 December

The most difficult part of the material was proving that a series was divergent.  I understand that there is no limit, but the formatting of the formal proof seems a whole lot more difficult then it needs to be.  Especially deciding which e to choose and so on.

The most interesting part is this remind me a lot of calculus.  I remember in multiple dimensional limits, we learned how to use the definition of a limit to formally show that a limit exists, but it was always easier to show that a limit did not exist by approaching the same point along different lines.

November 25

What I learned in the course:
I learned a lot about sets, and subsets, and cardinality, and functions, and injections, and serjections, and power sets, and primes, and evens, and odds, and divisibility, and integers, and latex, and proofs, and more proofs, and more proofs...

How will they help me in the future:
They will help me to think logically, keep the goal in mind when working to solve something, and sometimes realizing things are much easier if you can prove something else first.

11.5-11.6, Due On Monday November 24

The most difficult material were simply the proofs, especially of the fundamental theorem of arithmetic.  The theorem makes sense intuitively, but using induction is always difficult.

The most interesting part were the tests for divisibility of certain integers.  Those 4 rules seem really useful, especially the second one.  I remember using it as a kid to remember the multiples of 9, 18, 27,36, 45, 54, 63, 72, 81, 90, as summing the digits give you 9.  It is really neat that that extends to 1233, and other super large numbers.

Exam Review

• Which topics and theorems do you think are the most important out of those we have studied?Countability, The SB theorem, and also the Division Algorithm

• What kinds of questions do you expect to see on the exam?                                              Proving theorems, proving whether something is countable, finding bijective functions, finding greatest common denominator.

• 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.                                                                                                                                       Just a bunch of examples of an injective function from an infinite set to a finite set would be great.  That always seems to be hard to do.

11.3-11.4, Due on 19 November

The most difficult part of the material was understanding the proof for the Euclidean Algorithm's lemma.  The Euclidean algorithm proof itself was pretty straight forward though.  I understand how to use the lemma though.  It is not hard to use it.

The most interesting part is how Euclid came up with this so many years ago.  I wonder if he ever had to prove it like we prove it, or if his proof looked like something else.  I looked up more on Euclid and he seems pretty interesting as a Greek living in Alexandria. It says his textbook was used up to the late 19th century, so his reasoning is pretty sound.

11.1-11.2,due 16 November

The most difficult part of the material was the proof for the division algorithm. I did not understand it the first time, but after reading over again it made more sense. Using the algorithm itself was not too difficult to understand.  I think I have been using it since elementary school.

The most interesting part of the material is that we could actually prove something that just seems obvious.  I had never though about trying to prove something so simple, but we need to if we are ever going to use it.

10.5, Part 2 due on 14 november

The most difficult part of the reading was still wrapping my head around the proof.  It is going to take me some time to understand it, but it makes since that something that is bigger than the natural numbers is equivalent to the real numbers, but that is not to say that all sets bigger than the natural numbers is equivalent to the real numbers which they showed by looking at the power set of the real numbers.

I found it interesting that the Axiom of Choice and theorem b were equivalent.  I guess the book does not go into that much detail.  It was also interesting to see a German presenting a "defective" proof, and it took us over 50 years to finally come up with the correct one.

10.5 Part 1, due 11 November

The most difficult part of the reading was the proof for 10.17, and how they got for the method they used.  It seems they used a proof by cases and defined a subset as an example, but because this example turned out to be bijective we know there exists a bijective function from A to B.

The most interesting part was simply the restriction.  If we can't make something one to one, we will simply cut it in half so that it is one to one.  Restrictions especially in calculus can seperate things into different integrals making the math a whole lot easier.

10.4 Due on November 10

The most difficult part of the material was the new terminology for the cardinality of the natural numbers and the real numbers.  I have never seen an aleph null before, and the term continuum is pretty new, or at least now I know how to spell it.  The proof by contradiction was not too difficult, but I need to get used to using them for abstract infinite sets.

The most interesting part is that there is no largest set.  No matter how big a set is, even if it is infinite, there will always be a set with a larger cardinality.  So there is something bigger than our universe, and something even bigger than that and so on and so forth.

10.3, Due on November 6

So the reading was not too difficult as it is simply the opposite of section 10.2.  By far the most difficult part was understanding the proof that an open interval is not a denumerable set.  I understand that it has to do with something being expressed as .40000 and .39999 and that not making it onto, but not sure the stuff in between.

The most interesting thing of course was simply that .40000 and .39999 are the same number.  That kind of blew my mind, but it makes sense.  By that reasoning is .40000000...1 also the same number?

10.2, due on November 4

To be honest the material was really easy, simply because we already went over most of it on Monday.  The most difficult part though is simply thinking that something can be a subset of something else but still retain the same cardinality.  Weird things happen at infinity.

The most interesting part of the reading was of course the square type function used to count the ordered pairs.  I wonder who came up with that and if there is a way to express the function whiteout simply showing a pattern.

10.1, Due on November 3

The most difficult part of the material was looking up Theorem 9.11 and Corollary 9.8 to review those theorems, but after reviewing them the proof for theorem 10.1 made a lot more sense, which it did not at first.

The most interesting part of the material was the theorem that if there exists a bijective function from A to B and A R B, then R is an equivalence relation.  This may seem trivial for finite sets, but when applied to infinite sets, this is really useful.  This also is because infinite sets are said to be able to have the same cardinality.

9.6-9.7, Due on 29 October

The most difficult material in the reading was that on permutations.  I understand the concept that the permutations are simply reordering the numbers, but it still maps to the original domain.  Once the book started using matrices though it reminded me of linear algebra.

The most interesting part was how the book worded inverses.  I remember seeing these functions but we would always set them up as matrices in order to find the inverse.  I do not remember exactly how we found the inverse that way.  I do remember in high school just switching y and x and that seemed to always work.

9.5, Due on October 27

The most difficult part of the reading was the proof on a composition being injective and subjective.  I have been doing so much studies in other courses, that I barley remembered the meaning of those two terms as they are pretty new to me, but the proof did not seem to be that difficult.

The most interesting part was all of the applications in calculus including the product rule and the chain rule.  I thought it was cool how they were mentioned and was hoping to see more, but this book probably does not cover calculus.

9.3-9.4, Due on 23 October

The most difficult part of the material was understanding why the cardinality of A had to be bigger than the cardinality of B for a function to be one to one from A to B.  After re-reading the section it became obvious that if B was less than A, then there would have to be repeating values and it would no longer be one to one.

The most interesting part of the material was simply the identity function expressed that way.  I am so used to seeing it as x=x or something similar.  I cannot wait to use this function in proofs, but it seems almost trivial.

9.1-9.2, Due on 21 October

The most difficult part of the reading was understanding what an image is.  I eventually realized it is just the value that you get when you plug something into a function.  The way the book stated it was not the most clear to me.  Example 9.1 was also a little difficult to understand.  I thought they were talking about related functions, when they were just giving a list of examples.

The most interesting thing was mapping.  I remember hearing mapping a lot especially when doing triple integrals.  We would either map a function to a plane, or map it to polar coordinates so that we could do the integral.  It wasn't too hard once we got some practice with it.

8.6, Due on October 19

The most difficult part of the material, was the principle of what makes something well defined and the proof of n greater than or equal to 2.  I think I understand the principle that the equivalence classes need to be equal, but I am not very confident in my ability to prove it.

The most interesting part of the material was the principle of closed to addition and multiplication.  I remember seeing those principles before, but I am happy now to have a way to define them using these new equivalence classes.

8.5, Due on 17 October

The reading was not all that long and pretty straight forward.  The most difficult part was understanding the different equivalence classes for modules and following their work on how to get there.  The classes make after looking at it a while though.

The neatest part of the material was that modulos are reflexive, transitive, and symmetric, which is interesting.  At just a glance it would not seem so.  I still do not understand why they are useful though to be honest.  The book said something about a division algorithm that we will study later.  I suppose I will just have to wait.  I just know how to prove that they are equivalent.

8.3-8.4, Due on October 15

The most difficult part of the reading was the section over equivalence classes.  The proofs for transitive, symmetric, and reflexive really are not all that new, but the analysis of equivalence classes took a while to understand.  I am interested in what they are useful for as I understand they are partitions.  I assume we will learn later.

The most interesting part of the material that I got to read was simply the new definition of equivalence as something that is reflexive, transitive, and also symmetric.  I had never though of equivalence in that way, but it makes sense.  I also thought it was interesting they used the word class in equivalence class.  I do not recall anything being called a class in math, and now that I think about it, it is not often used in physics or chemistry.  In biology of course we have kingdom, phylum, class, genus, order, family, species, but that is not a very mathematical definition but more of an arbitrary classification.

8.1-8.2, Due on October 15 (originally due on October 13)

The most difficult part of the reading was understanding that the empty set is a relation because it is a subset of A cross B but it means that not a single element in A is related to B.  Class really did clarify this though.  Other than that it was difficult to understand the notation for domain and range.  I understand what they mean, but the notation looks unfamiliar.

The properties of reflexive, transitive, and symmetry are really important especially in proving things.    If I relate it to chemistry, many reactions are reflexive, only some are symmetric if the equilibrium is 50/50, and practically some are transitive where they can take two steps to reach a final product or only one step, but for the most part they are neither symmetric or transitive.

6.4, Due on 10 October

So 6.4 was difficult to understand.  It seemed very similar to the principle of mathematical induction except there is another step added in where you define where a variable i lives.  It made sense until they made an assumption by the recurrence relation.  In the end, it think it just means showing an added case.

This obviously applies to live because recursion is used a lot by computers and calculators to calculated things numerically. The shorter the recursive formula, the more efficient the calculator, and the quicker that things are processed.  I think that this is the relationship.

6.2, due on 8 October

The most difficult part of the reading was understanding the step with factorials.  It took me a while to remember what a factorial was, and why the book could take the step it did.

The most interesting part of the reading was that it really was not that much different from 6.1.  I wondered why the reading was so short this time.  Most of the examples were proofs that involved the induction theorem and it was pretty neat to be able to reinforce that concept.

6.1, due on 5 October

The most difficult part of the reading was of course all of the algebra.  I have not done common denominators with variables since high school algebra 2.  I look forward to improving my algebra.

The most interesting part was of course the proof by induction.  I remember a Calculus course where after the AP test, we had nothing to do, so our instructor taught us about logical deduction and induction, and told us that induction was almost always better than deduction, but we learn most things through deduction in our every day life.

Midterm 1 Review Due on 3 October

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

I feel that the most important topics are the different types of proofs, as well as the proofs that are dealing with real numbers.  It may just be my opinion but determining whether something is even or odd is useful for learning the form of proofs, but I do not see it too applicable in my life.

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

I expect to see a lot of proofs and probably some questions about set operations and truth tables.

• 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.
• I still need to work a little bit on the last few chapters about proving something is rational or irrational like for example prove that sqrt(2) is irrational, and also I need to review the set operations in the first few chapters.  Cartesian products are always confusing.  I do not think latex will be on the exam so I should be good there because Latex is simply amazing.

5.4-5.5, due on October 1

The most difficult part of the reading was going through all of the examples.  It took me a while to understand the example about calculus because I have not done calculus in quite some time.  I thought at first they would give a proof of the theorem which I thought would be neat, but they did not.

The most interesting was learning to disprove there exist statements.  That would apply very much to lots of fields like economics, where proving that something does not exist will save a lot of time trying to find out what it is.

5.2-5.3, due on 28 September

The most difficult part of the reading was interpreting the massive amount of text about what a contradiction is.  The concept once I figured it out really was not that hard, but they spent a lot of sentences explaining it.  I need to get more used to math language.

The most interesting was the story about the prisoners.  I could not figure it out on my own which was a bummer, but the explanation was really creative and clever I thought.  Well thats my post for today. Thanks for reading.

4.5-5.1, due on 26 September

The most difficult part of the material was the proofs on the cartesian products.   I had a hard time remembering the properties of cartesian products so I had to go back and review them.

The most interesting was part was section 5.1.  Now we finally get to disprove stuff, and I find it a whole lot easier than writing huge proofs for some of the most simple statements.  I look forward to doing latex homework on this chapter.

4.3-4.4, due on 24 September

The most difficult part of the reading was recognizing that when proving that two sets are equal to each other, we need to prove that they are both subsets of each other.  It is not enough to simply show that one is a subset of the other.  The converse also needs to be proven.

Then neatest part of the reading was proving the triangle inequality.  In math 302, we proved it using the properties of vectors, but here we proved it using the properties of absolute values and real numbers.  In middle school, we "proved" it basically by drawing pictures.

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

I usually spend between 1-2 hours on the homework assignments.  Closer to 2 if I have to use Latex to write it all up.  The reading teaches me a lot, and the lectures help to solidify that knowledge.  Honestly the homework is not that bad once I figure out the patterns in responses.

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

Doing lots of problems.  I learn by practice.

• 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.)

Not forgetting to turn in the homework and having to type up the entire assignment on Latex before midnight.  Also, the midnight deadline is really not good for me, since I have a strong passion for procrastinating doing things until the last minute possible.  This creates late nights for me, and I know it is a habit I need to break.

4.1-4.2, due on September 22

The most difficult part of the material was by far the notation for congruence of Integers.  The concept is not exactly easy to understand, and the reason it's usefulness is also not as easy to understand.  They might as well write n | (a-b), instead of making a=b (mod n).

The most interesting part was where they showed that  how everything divisible by 2 can only have 0 or 1 as a remainder, and then showed that it was similar for n = 3q + 1 or n = 4q + 1, with the number of possible remainders increasing each time.  I look forward to exploring this in more detail in chapter 11.

3.3-3.5, due on September 19

The most difficult part of the material was the last example and trying to find the error in the proof.  It took me a while to catch why the proof was incorrect because they used the same variable in the definitions of n and m.  The other difficult was the need for the contrapositive in proving if and only if.  I am still not sure I understand why it is important, but I assume it is needed.

The most interesting part of the reading, I thought was the separating of things into cases.  I know in previous math courses we would do this all the time especially for x<1 or x>1.  I would love to apply this in other fields and I will certainly look for opportunities.

3.1-3.2, due on September 17

The most difficult part of the reading was understanding the first proof and how they had to factor out  a two.  When I read it the first time, I did not quite catch the fact that they factored the 2, so I was wondering how 3n + 5 proved that 3n +7 was even. I figured it out after reading the other examples.

What I thought was the most interesting part of the reading was the statement that a vacuous result can actually be useful in mathematics.  Personally it seems trivial to prove that the implication is true even though the first statement is false, but this statement made me look forward to further sections and proofs where the vacuous result is useful.  I assume it is not useful alone though.

0, due on September 15

The most difficult part of the reading was understanding the difference between using that and which in a sentence.  Even though both can be correct, which sometimes refers to a single item whereas that could refer to one out of many or exactly one.  I will simply avoid using the word which in mathematical writing.

The most interesting part of the reading was all the quotes from the writers and mathematician about writing.  This chapter has really made me more interested in writing, and I am now more aware of how writing logical arguments could help me in my career especially in drafting emails, and also reading critically.

2.9-2.10, due on September 12

The material was fairly straight forward.  I had to read Example 2.25 twice, so I guess I am still not that comfortable working with power sets.  The other difficulty I found is being able to write out the sentences and still make them sound logical.

The most interesting part was the section where they combined the universal and existential qualifiers.  For every..., there exists.  It reminded me of how almost every theorem began in Math 302.  I also personally think for every... sounds better than If...then, and scientists should use the phrase too in their lab write-ups because they are logically equivalent.

2.5-2.8, due on September 10

The most difficult part of the passage for me was the wording in expressing the biconditional.  I feel as though if and only if and also sufficient and necessary seem almost redundant.  The end of the reading tried to justify it, but the explanation was extremely wordy.

The most interesting section, I found to be the reasoning behind the symbol with the three lines or equivalency.  I remember using that symbol a lot especially in chemistry and other math courses and had always assumed it to mean the = sign but just fancier.  Now I understand the meaning as if one is true, then so is the other and likewise if one is false.

2.1-2.4, due on September 8

The most difficult part of the material for me was the new vocabulary and also the truth tables.  There were a lot of writing examples which was new for a math course.  At first the truth table didn't make sense because I was looking at the example which had 4 truths, whereas the table only showed one.  Then I realized that the table was not depicting a ratio, but just that the answer is either true or false.

The reading about the implication in section 2.4 reminded me of the form of a hypothesis if...then that I have used in science course like chemistry.  I wonder if I do use the mathematical expression in a lab report, it may be informal rather than writing it out.  The story reminded me of another course, where the professor will give us an A if we get an A on the final.  He called it the repentance method.

1.1-1.6, due on September 5

The most difficult part of the material I would say is the indexed collections of sets.  It is a new notation and concept which I was not previously familiar with, but it does give a simpler way to express a union of multiple sets which is very useful

The most interesting is the last part about the Cartesian products of sets, especially how sets can be used to express Cartesian coordinates and also the Euclidean plane.  This related directly to the use of sets to describe multidimensional points and then later vectors which included a magnitude. We used these a lot in linear algebra and multi-variable calculus.  It also explains the notation R^2 which came up a lot to express the Euclidean plane or later R^3.





1.1 Describing a Set

• sets contain a small number of elements
• sets without an element is the null set
• we can define a set by  S={x: p(x)} where the elements must satisfy some condition
• different sets include N natural numbers (positive integers), Z integers, Q rational numbers, I irrational numbers, R real numbers, and C complex numbers

1.2 Subsets

• A is called a subset of a set if every element of a set also belongs to the other set. 
• U is called the universal set
• The interval (a,b) = {x ∈ R: a < x < b}.
• Set of all sets is the power set
• A is a proper set of B if A is a subset but not equal to B
• Venn diagrams help visualize sets

1.3 Set Operations

• A U B is adding the sets together
• A ∩ B  is the intersection of the sets
• If the intersection is the null set then they are disjoint
• A−B = {x: x ∈ A and x ∈ / B}. 
• A complement of a set is the universal set U minus the set

1.4 Indexed Collections of Sets

• When finding the union of multiple sets use the notation/ and also intersection with opposite sign

n Ui=1 Ai ={x : x ∈ Ai for some i, 1 ≤ i ≤ n}. 

• I can be used as an index set.  The following is an indexed collection of sets.

USα = {x : x ∈ Sα for some α ∈ I}, α∈I 

1.5 Partition of Sets

• pairwise disjoint means that every two distinct subsets are also disjoint.
• pairwise disjoint where every element belongs to a subset is called a partition.

1.6 Cartesian Products of Sets

• an ordered pair is the set (x,y)
• A cartesian product is when A × B= { ( a , b ) : a ∈ A and b ∈ B } 

Introduction, due on September 5

• This is my second year in school.
• I have taken math 302 and 303.  I have AP credit for 112 and 113.
• I am taking this class because it is a requirement for a math minor.
• Dr. Kuttler was effective because he went through the material quickly so that he could review several times before the test.
• I just got back from a mission in the West Indies. 
• I am available Tuesday and Thursday mornings.