Review, due April 14, 2014

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

      I think that the Schroder-Bernstein Theorem.  The idea of implications and biconditionals are probably most important.  I like the fundamental theorem of arithmetic and the division algorithm.  I could see proofs by induction being very useful.

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 think just getting the calculus proofs down would be useful.  Anything will be helpful there.
What have you learned in this course? How might these things be useful to you in the future?

      I realize to a fuller extent, the importance of being specific and precise in language, particularly in math.  If we don't use precise language, we are prone to mistakes and problems, even if we don't foresee them.  I will be more precise in my language from now on.

12.5, due by April 10, 2014

      Continuity all seems to make sense.  It seems that the only addition here for proofs is to show that the limit exists and that the limit is equal to the value of the function at that point.  I think this section should be most straight forward so long as I can keep delta-epsilon proofs straight.
      This is a useful topic though because it connects limits with the actual function to see it the limit that we predict is included where we think it should be.  I wonder though how we would work off of limits approaching from one side or from another.

12.4, due by April 9, 2014

      Most of the proofs there seemed quite difficult to follow.  I think I need to go back to section 12.3 again to see if I really understand the proofs there.  They still feel a little complicated and I can't seem to keep my variables straight.  I'm sure it will work out after I've done the homework problems for section 12.3.
      This section makes proving the limits of many things so much easier.  I especially liked the proofs at the end of the section which proved generalized limits of polynomials by induction.  It just makes things so much easier.

12.3, due April 7, 2014

      I think delta-epsilon proofs make sense, they just take a little more getting used to with keeping track of the numbers.  As always, I will have to do plenty of practice problems.
      Back in calculus they felt backward for the reason that you can only do them if you have a sense of what the limit is and there never seemed much reason to prove it otherwise.  Though now it makes a lot more sense why we would want to approach limits from a delta-epsilon perspective.

12.2, due by April 4

      The most difficult part will probably be making a good guess for what the pattern is for the formula of the sum.  I will live in hopes that I will not be assigned problems that are too hard.  Otherwise it should work out.
      I enjoy the proofs about series.  I like the idea of making a lemma so that it just becomes a problem of finding the limit.  I find it impressive how there even exist methods to take series and convert them to a formula that can be solved so much easier to deal with.

12.1, due by April 2, 2014

      The most difficult part was keeping straight N, L, and epsilon and what the significance of each one.  When I walk myself through the proofs and keep all of those variables straight, the proof is apparent so I just need practice.
      I like the proofs in calculus.  It is interesting to see that the tools we have to work with can be applied to calculus very easily when given clear definitions.  I like the idea that these proofs can be done in terms of integers and such.

Review, due March 31

• Which topics and theorems do you think are the most important out of those we have studied?
• The Schroder-Bernstein Theorem
• The Fundamental Theorem of Arithmetic
• gcd
• The Division Algorithm
• Cardinality of denumerable and nondenumberable sets
• What kinds of questions do you expect to see on the exam?
• I expect to see a lot of definitions, and proofs
• Specific proofs I expect to see are ones using the Schroder-Bernstein theorem, which should be pretty straight forward.  I hope not to see one where I need create a bijective function that from a subset of the real numbers to the real numbers but I will be prepared.  I expect to see some questions on relative primes and the division algorithm.  It should be a fun exam.
• 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 need to understand problems that involve uncountable sets and using the decimal expansion to prove things.  I think 10.36 would help me understand things better.

11.5-6, due March 28

      The proof for the fundamental Theorem of Arithmetic seemed a little complicated.  I will need to review the theorem again before we review it in class.  I followed almost all of it but when dealing with p_1*p_2*p_3*...*p_s = q_1*q_2*q_3*...*q_t.  The part after that was a little complicated.
      I enjoyed reading about the different ways to determine divisibility by 2^n, 9, and 11.  I may have known those in the past but found them interesting and useful as I read about them this time.

11.3-4, due March 26

      Though the proofs are clear for the greatest common divisor, it seems as if the same thing was being stated twice when discussing the 2nd definition of greatest common divisor.  I don't see what the distinguishing part is about the other definition such that it is worth restating differently since we already proved everything to go into that definition.
      I like the Euclidean Algorithm as it is naturally recursive.  I decided that I will write a simple program in C++ to solve for the greatest common divisor.  I am a little frustrated as I do not yet know the conventions for recursion in C++ but I am sure I'll figure it out.

11.1-2, due March 24

      I found it difficult to follow the Division Algorithm proof the first time I read it.  Though being more careful to not the notation, I found it much easier to follow and may be able to replicate it.
      I liked how the division algorithm was related to congruency modulo n.  The division algorithm makes more sense and seems more significant when understanding that connection.

10.5 part 2, due 21 March 2013

      In the proof of Theorem 10.19, it felt as if not every subset was represented in the function f.  Consider for instance {42}.  I do not see how the f(x) ever would result {42}.  I'm not sure what I am missing.  I was also a little confused in the proof for the Schröder-Bernstein Theorem.  I must not be keeping my notation straight.
      I find it interesting that the cardinality of the real numbers is numerically equivalent to the cardinality of the power set of the natural numbers.  I thought that Corollary 10.20 was a nice thought too.

10.5 Part 1, Due 19 March 2014

      I did not understand the proof to Theorem 10.17.  With the introduction of new sets and other information, I lost track of what was going on.  The theorem makes sense but the proof is still a little fuzzy.
      I like the idea of mapping the elements of one set to a subset of that same set.  It starts to make comparing cardinalities more exciting.  I am looking forward to seeing what ideas we can take from this idea.

10.4, due March 17, 2014

      I did understand the proof for Theorem 10.15.  It makes sense by intuition but the proof itself seemed unclear because we defined a function f_S and f_T.  I do not understand yet why.
      I liked the idea that we have proven that there is no largest set.  I also found the Continuum Hypothesis interesting but I'm not sure I understand its significance.  Only that it is an interesting idea, that no set has a cardinality that falls between the the cardinality of the natural numbers and the real numbers.

10.3, due by March 14, 2014

      The most difficult part was the proof strategy when the quadratic equation was used.  It didn't quite make sense why they chose the sign they did.  Suppose it was because the "-" did not fall into the correct range where the "+" sign did.
      I like that we are starting to get into visual representations of the math that we are doing.  It helps me to better grasp what is going on in the section.  I find it exciting to be able to start making more functions like that.

10.2, due by March 12

Though we went over it in class the proof to find that the subset of a denumerable set is denumerable.  I just need to read it a little more carefully.
I found it very interesting that the set of all rational numbers is denumerable but I suppose it makes sense considering it is a relation on the integers and that is a subset of the cartesian product of the integers, which is denumerable and a subset of a denumerable set is denumerable.  It seems this denumerable thing is starting to make a lot more sense.

10.1, due March 10, 2014

      The most difficult part is understanding what was being discussed at the beginning about Galileo. I am not sure I understand the infinite sets and proper subsets of it but I am sure I will by the end of the chapter.
      I thought it was interesting that we have not yet defined cardinality of infinite sets but we have started to get into equivalence of infinite sets and talking about equivalence classes.  I think that could be interesting.

Review, March 7, 2014

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

      The topic of proof by induction will be very important to know.  Also understanding equivalence relations will be useful.  Also proving in the generalized case that the composition of functions is onto or one-to-one given certain properties of the functions that compose it.

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

I expect to see true of false questions related to whether or not a function has properties of bijective, injective, surjective, transitive, symmetric, reflexive, etc.  I expect to see questions about proving by induction a formula that maps a sum and mostly things we have covered so far.

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

      I do not yet understand how to do proofs with equivalence classes.  I am not sure what is sufficient for those proofs and for proof of compositions of function.

• Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

I would like to see Number 22 on the practice midterm 3 worked out.

9.6-9.7, due March 5, 2014

      The most difficult part of this will be keeping straight the definition we have established for the inverse function.  It still follows that you just swap the elements of the ordered pairs of a function but to keep the formal definition in sight will certainly be the challenge.
      I like that the composition of two inverse functions results in the identity function.  It makes senses when I go back and think of how I learned it in high school but it is rather exciting to learn it in this newer context of sets, cartesian products and such.

9.5, due by March 3

      I am still not comfortable proving that functions are surjective or injective.  Sitting down and working through proofs will be the way to learn it better.  Otherwise the material seems pretty straight forward.
      The idea of the composition of functions is so important for calculus and in defining what taking the image of of an image is.  I remember using that idea for chain and quotient rules for calculus.  It is such a useful topic.

9.3-9.4, due 28 February 2014

      I did not understand the idea with bijective functions of equivalence classes.  I do not clearly understand how the proofs work and why we can just switch to the namesake element of the equivalence class when the proof wants to prove it on the entire equivalence class.  If I understood that better, I think the proofs with them would make more sense.
      I like the idea of injective and surjective functions.  In high school I remember discussing one-to-one functions but I remember wondering why we did not discuss onto functions and if they had any significance.  I like to the thought that they are worth discussing but I am looking forward to see why.

9.1-9.2, due by February 26, 2014

      The most difficult idea was the one presented in 9.2 but the idea is to produce all possible functions from one set to another.  I can see that this would be possible on finite sets but I do not see how one would start to express this, or even need this for all the sets on an set of infinite size.
      I liked being presented with the different and developing definition of a function.  I can see that the first few differed from ones that followed but I could not quite tell the difference between the last few.  Or at least what the implications of that definition could be.

8.6, due 24 February 2014

      The most difficult part of the reading is knowing if a set is closed under a certain operation.  It seems rather simple but keeping it straight that when elements of a set undergo an operation that the result is in that same set.  I suppose that means that division of integers is not closed as the result could be a rational number.
      I like the idea of addition and multiplication of equivalence classes as it allows us to manipulate and interact with equivalence classes in a more interesting way.  It allows us also to relate equivalence classes to other ones and to do more interesting proofs.

8.5, due February 21, 2014

      The idea of congruency is not new so that was pretty straight forward.  I'm not sure that I could recreate the proof for the symmetric case from memory though.  Remembering how to do the proofs for the different properties of equivalence relations is still a little difficult.
      I liked the significance of the equivalence sets when related to the congruent to modulo relation.  The way to interpret the equivalence sets and determine how many distinct sets should exist, is an interesting idea as well.

8.3-8.4, due February 19, 2014

      I struggled to understand equivalence classes.  I think the difficulty was the new notation without too much notice.  They make sense to me in that when each element is only in one equivalence class and there are no nonempty equivalence classes that a partition is formed.  They also seem to follow rules of sets but I do know know what I am really looking at or what is being communicated to me when reading about equivalence classes.
      I find them interesting as these equivalence classes remind me of topic in linear algebra like vectors and being non-colinear.  It seems to no that understanding these equivalence classes would expand on my understanding of linear algebra.  I also find it exciting that equivalence relations seem to follow patterns when discussing different sets such as lines, number sets and even sets themselves.

8.1-8.2, due February 18, 2014

      The most difficult part to understand was the idea of a relation being transitive.  I did not understand it until I was given examples of specific relations and what it mean for them to be transitive or not.  Being transitive is to say that if a relates to b and b relates to c then a relates to c.  An example of a transitive relation is the divides.  Like if a divides b and b divides c then a divides c for real numbers a, b, c.
      Knowing the properties of certain relations can be quite helpful as one tries to manipulate a given result to prove or disprove it.  One can know useful ideas that we used in induction such as if a > b and b > c then a > c.  But it would be a mistake to say if a is not equal to b and b is not equal to a then a is not equal to a.  Keeping in mind the properties allows for steps to be taken that could not otherwise be taken.

6.4, due by 14 February

      The strong principle of mathematic induction seemed quite simple, though proposing formulae for recursive functions seem a little difficult.  Other than writing out the first few cases and guessing what it might be, then testing it by proof, I'm not sure I see a much better way.  I should be excited to learn what other ways there are.
      This idea continues to expand our ability to use induction.  I like the idea that we can now assume that all elements in a domain from the least element to k can be assumed true in order to prove the implication that the case n = k + 1 is true.  As shown in the book, this will become quite useful when dealing with recursive functions.

6.2, due February 12

      The proof of De Morgan's law for any finite number of sets seemed difficult.  I saw how the proof was done towards the end and it makes sense but the beginning was unclear to me.  I will need to do more practice with these as I prepare for the test.
      I an see that induction is useful as it allows us to now prove results for any well-ordered set.  To be able to induce that something is true over such a large domain is rather exciting.

6.1, due February 10

      I really did not understand the significance of mathematical induction.  It seems to me like for a given sentence you prove that it works for an actual case, then for a generalized case and then for another generalized case proving that it works the same way.  I can see that it could be useful in cases lie that shown in the book about series and sums.  I just do not yet have a clear picture on how it would work.
      I was excited by the example of counting squares.  I have had that problem given to me in the past but I just counted them out.  I am excited now though because I see the generalized case where a square is dimensions n x n, the number of squares is n^2 + (n-1)^2 + (n-2)^2+...+(n-k)^2, where n = k.

Review, due February 7, 2014

      I think that the most important topics are direct proofs, proofs by contrapositive and proofs by contradiction.  It will also be important to review sets as every proof we've done has related in some degree to sets (if only as a domain).

      I expect to see questions about proving implications such as the parity of a number, given another.  I expect to see proofs comparing equality of sets and proofs determining whether something is a rational number given rational and irrational elements of it.

      I need to understand how to evaluate proofs better.  I think it would be helpful to understand expectations for how to evaluate a proof.  Question 5.46 in the book would be helpful to review beforehand.

5.4-5.5, due February 5

      Existence proofs seem rather straight forward.  I found it difficult to intuitively see the methods for beginning existence proofs, especially those relating to rational and irrational numbers.  It took me a little while to understand the example that there exist 2 rational numbers a and b, such that a^b is rational.  I will need to get more comfortable with these proofs, as usual, by experience.
      I enjoy the thought of disproving existence statements.  It requires finding contradictions in the existential case or proving more generalized "for all" statements.  I can see that would begin to be quite a powerful tool.

5.2-5.3, due on February 3

      The proof by contradiction was a little difficult to wrap my mind around.  I found though that it was quite a bit simpler than the formal definition made it sound.  One simply assumes P is true and Q is false or assumes the entire implication is false and then solves for a contradiction.  It makes sense that one would be best off stating that this method will be used for a given result.
      I enjoy adding methods of proofs to my arsenal of proof solving tools.  I can see that the proof by contradiction can be useful in situations where a direct proof or proof by contrapositive is so much more difficult.  This is useful to understand as there must be many situations where this is the simplest method to use.

4.4-4.5,5.1, due January 31

      The proofs involving Cartesian products of sets seemed a little more difficult.  It seems as if keeping all the different elements straight will be quite difficult.  However, the idea is fairly straight forward.  One need take the variables separately and remember what sets the individuals are elements of.  Of course, with practice, it will continue to become more clear.
      I find the counterexamples to be very useful.  I was in a debate earlier today, the opposing side used a generalization in a certain domain.  I thought of a counterexample that disproved their statement.  Though the counterexample did not prove their whole argument as false (because debates are much more subjective than mathematics) I felt logically validated for my use of a counterexample.

4.3-4.4, due January 29

      I found it difficult to understand the proof that verified a particular equality or inequality.  In the example, Result 4.16, I did not understand why (2x - 3y) squared is greater than or equal to zero was used.  I then realized that it came from working the result backwards until the equality could be shown in a way that was known to be true.  Because (2x - 3y) squared is always greater than or equal to zero, it is a great starting point for a direct proof.  It is now much more clear how to verify an equality or inequality.
      I found it interesting to begin working with proofs within sets because it allows us to solve much more general proofs involving sets of numbers rather than just individual number.  I like that we are expanding the tools that we are using to prove statements and expanding the situations in which those tools can be used.

• How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?
• I have typically spent half an hour to three quarters of an hour on reading and another hour on homework problems.  When I wrote my second assignment in LaTeX, I spent about three hours completing it.  I do not anticipate them taking so long as I improve my skills in LaTeX and as I modify the template so that I do not have to redo so much in the assignments every time.
• What has contributed most to your learning in this class thus far?
• I have found that writing about the most difficult part of the reading has helped me to synthesize the material better.
• 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 think I would learn more if I could see where this will be applied in the various topics within math.  I sure as we continue through the year it will become clearer.

4.1-4.2, due January 27, 2014

      The most difficult part of the reading was understanding the proofs involving congruence of integers.  Once I understood what the "congruent to... modulo" statement meant mathematically, I realized how simple the proofs were but until that point, it did not make much sense.
      I was rather excited when seeing the statements about how for a given integer x, 2 divides (x - 0) or 2 divides (x -1).  Also 3 divides (x - 0), 3 divides (x - 1) or 3 divides (x - 2).  It seemed exciting because for the general case of an integer n, n divides (x - 0), n divides (x - 1) ... n divides (x - (n - 1)).  This could be applied when dealing with proofs involving sets.  I am excited to deal with proofs of more general cases.

3.3-3.5, due January 24

      The most difficult part was understanding proof by contrapositive.  I understand that "if P, then Q" is logically equivalent to "if not Q, then not P."  The confusion come when I try to keep straight which statement I am trying to prove true with a given result.  I am sure that I will understand it better with practice.
      I am impressed by section 3.5 on proof evaluation.  It was not until then that I realized that one proof could prove a variety of results so long as the results are logically equivalent.  Keeping this idea in mind could be particularly useful when employing different techniques to prove results.  It is also useful to be able to think critically about a given proof to determine whether it is a convincing proof or not.

3.1-3.2, due January 22

      The reading on proofs was interesting.  The most difficult part was when I skipped a page explaining the proof method to prove an integer is positive or negative.  Going back, realizing I missed a page, I read the idea.  It took me a few readings to realize what they were trying to say.  For an even number, the method is to show the formula can be written to as an arbitrary integer, multiplied by 2.  For an odd integer, it is to show an integer being multiplied by 2 then adding 1.  Understanding that made the following proofs make much more sense.
      I find the method shown to prove that an integer is positive or negative is very interesting.  I'm not sure how many significant proofs there are that would require this but it is good to understand the logic behind it.  This idea has expanded my knowledge on the functionality of substitution because in the end the part that is an integer could be expressed as a single variable.

Chapter 0, due January 17

      The most difficult part of the reading was determining whether to use "that" or "which" in specific cases.  That is used to introduce a restrictive clause.  This is to specify the exact object being referred to.  Which is used for nonrestrictive clauses.  It seems to be used for giving more information about something.  It is for unnecessary or additional information.
      I thought it was very useful to have the conventions used in mathematical writing expressed clearly.  I had never thought before that it might be confusing to start a sentence with a symbol.  Now that I've read about good mathematical writing, I can be more conscientious about that which I write in this class and in future classes.

2.9-2.10, due on January 15

      This reading was fairly straight forward.  I did not understand why the negation of a quantified statement was being discussed and why during the negation, the inverted "A" was changed to the leftward-pointing "E."  I took a step back and really tried to understand the notation and it became clear that "if all elements x of set S for P(x)" is an statement then the negation would be there exists an element x that is an element of ~P(x) must be the negation.  It is now clear in my mind.
      I was interested by the notation for quantified statements as I often look up information on wikipedia and now I understand what more of the notation means.  This should help me when needing quick information related to math.

2.5-2.8, due January 13

      A difficult element of the reading assignment was determining when a statement is logically equivalent without creating a truth table but I can see that would be time consuming when dealing with large compound statements.  Of course, the construction of a truth table is relatively easy but because I have limited experience with truth tables, I can see how simple it would be to make a mistake or error.
      From taking linear algebra and multivariable calculus, so often I heard the phrase, "if and only if" and understood it as it is not very ambiguous in English.  It is interesting to understand the biconditional in the more mathematical terms.  Understanding this with a formal definition will solidify my understanding of new mathematical ideas better.

2.1-2.4, due on January 10

      The most difficult part of this sections reading was understanding the implication statement.  This is a statement formed from two given statements.  In this one, the statement is expressed "If P, then Q."  It was not clear what this statement meant at all.  I had to talk myself through developing a truth table in English as follows:

"If true, then true" is true.

"If true, then false" is false.

"If false, then true" is true.

"If false, then false" is true.

Looking at this truth table now, I know that I am repeating informations given by the implication statements themselves but it helped me to understand the idea.  After understanding this idea better, I continued on in the reading which confirmed that the understanding I had developed by myself was correct.

      Having had some experience with computer programming I can see the importance of understanding principles of logic.  When I saw the truth tables for one, two and three statements written out in Figure 2.1, it was exciting to think of all the potential outcomes of truth tables for large numbers of statements.  I understand a little better now how computers can become very powerful logic machines by having a large number of switches combined in different combinations.

1.1-1.6, due on January 8

      The most difficult part of the reading was understanding the explanation of indexed collections of sets.  Once an example was presented I had a clearer understanding of the purpose for the set I.  Each element of the set I act as a way to index subsets of the set S.  In fact the union of all sets S_alpha produces the set S.
      The most exciting part of the reading was section 1.6 about the Cartesian product.  I was excited to see that with this product, order matters.  Though I do not yet have a broad understanding of the implications that R x R is the set of all points in the Euclidean plane, I can see that this will be useful when working with functions.  I am excited to learn what there is to study when taking the product of R with itself in n dimensions.  I was also intrigued by pattern that the cardinality of the Cartesian product of 2 sets is equal to the product of the cardinality of the individual sets.

Introduction, due on January 8

What is your year in school and major?

       I am a Sophomore majoring in Chemical Engineering
Which calculus-or-above math courses have you taken?

      I have taken Math 302 and 303 at BYU.
Why are you taking this class?

      I missed taking a math class last semester and thought it would be fun and worthwhile to earn a minor in math.
Tell me about the math professor or teacher you have had who was the most and/or least effective. 

      My least effective math teacher did not engage me as a student or project his interest in math to me.  My most effective math professor showed his competence in the subject as well as relayed to the students how the subject matter related to his work and how it could be applied to other disciplines.

What did s/he do that worked so well/poorly?

      The less effective teacher did not show his passion for the ideas being taught.  He taught the material so that we could complete assignments but never help us to see the big picture.  The more effective one would take time aside from merely explaining the textbook but shared how he felt about the material.  It was interesting as he wold even share the philosophical significance of the models we were discussing.
Write something interesting or unique about yourself.

      I spoke with a British accent until I was 11 years old.
If you are unable to come to my scheduled office hours, what times would work for you?

      I can make it to scheduled office hour times.