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.