Sections 4.5-4.6 and 5.1, Due on January 31

What did I think was difficult?
Fundamental Properties of Set Operations.  I find that these actual properties are not too bad to understand when approached from a definition standpoint, but I struggle to implement them and completely grasp the concept when they are used in proofs.

I remember learning Commutative laws, Associative laws, distributive laws and De Morgan's Laws back in Stat 240.  At the time, they were difficult for me to understand.  Now, by themselves, they are a cinch.  But when I have to use them in proofs, it's often difficult to remember to use them.  With practice, I feel that I'll get it.  The same thing with Cartesian products.  I feel that I understand the basic examples of cartesian product proofs in the book, but I'm nervous for more complicated versions.

The section regarding contradiction was not too bad for me.  In fact, I want to blog more about it now.

What did I think was interesting?
Counterexamples 

Wikipedia really helped clarify what a counterexample is.

If there is a general rule, say "all x are positive," you just need to find one case where x is negative in order to prove the proof wrong.  This is especially useful with univeral quantifiers.  I think I really like this because I see so often that people make statements that seem univeral.  "Everyone hates me."  "Everyone in my math class is confused."  "No one likes him."  It is dangerous to make such extreme examples!  I think it's so dangerous because all you have to do is find a counterexample that can break your entire argument.  I think when making a proof or argument, you should really think through the counterexamples that people present.  

Counterexamples can be very powerful in proofs.  Look for cases where you know the proof is false.  Be creative in your thinking, and you'll find that the answer is often simple.

Thanks Themistocles.

Sections 4.3-4.4, Due on January 29th

The triangle inequality and the State of the Union

|x+y| < |x| + |y| 

What was difficult for me?

What was originally difficult for me was remembering the properties of Real Numbers that I would later have to use in the proofs.  Some of these properties include:

• a^2 > 0 for every real number a
• If a < 0 and n is a positive odd integer, a^n < 0
• "Of course, the product of two real numbers is positive IFF both numbers are positive or both are negative."

Remembering these facts proved to be very helpful as I moved into solving the actual proofs.  I had a tough time with some of the proofs, especially when fractions were involved.  It was tough for me to see the logic behind multiplying by certain numbers to make the fractions disappear and to make identities and properties more evident.  I feel that I say this a lot, but I believe with more practice I will be able to have a more firm grasp.

Also difficult for me was the use of set notation in proofs.  Just a reminder, there are three types of set notation that are going to be important: intersection, union, difference.  You also have the relative complement of B in A, which is A-B.  You then have the complement of A, which is A with a line over it.  This part was pretty difficult for me, in part due to the use of set notation in proofs.  It was tough remembering properties of sets, and then figuring out how to solve the proof around the set notation.

What did I find interesting?

I am beginning to see that proofs can get more and more complicated.  We had been working with real numbers before, which are pretty simple in terms of their properties.  Now, we are working with real numbers and sets.  I can see how proofs can be applied to many real world applications.  I think of the State of the Union address that President Obama is going to deliver tonight.  Things are not cut and dry in the United States - in fact, I would say they are rather complicated.  This comes as a result of the diversity that is found in the United States, but also comes due to complexities introduced over many years.  I can see how proofs and logical thinking would be helpful in a political environment.  Even though we are only focusing on proofs with real numbers and set notation (which is still uneasy for me), I look forward to more complicated proofs.  Logical thinking is powerful.

Supplementary questions

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

I probably spend an hour to an hour and a half on the homework assignments.  I wouldn't say that the reading prepares me much for the homework assignments specifically, but it provides an introduction to the material in class.  I see that as beneficial.  My mind works well with repetition, so that repetition has been beneficial.  We will see if this trend continues over the course of the semester.

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

The same thing that contributes to my learning in anything - the internet.  When I don't understand something, I look it up on the internet.  A simple google search leads to a vast wealth of knowledge.  You can quote me on that.

What do you think would help you learn more effectively or make the class better for you?

I think the professor does a fine job of teaching the material.  Goals for myself would include the following:

• work out 1 problem from the examples of each section I read.  By doing this, I would be in a much better place to learn during class.  Like I said, repetition is the mother of all learning, and the more I can repeat, the more I'm going to learn.

Thank you Google.

Section 4.1-4.2, Due on Monday January 27th

What was difficult for me?
"For integers a and b with a ne 0, we say that a divides b if there is an integer c such that b=ac.  We write a | b."

The section Proofs Involving Divisibility of Integers was difficult for me to understand at first.  I understand the concept of dividing integers.  The simple proofs did not give me too much trouble.  However, as I continued to read, the problems increased in difficult.    I had to read through the different proofs a handful of times in order to come away with a firm grasp on what they meant and the thought process behind solving them.  What gave me the most difficult throughout the more difficult problems was just the way they thought through solving the proof using the different properties of being a multiple or a divisor.   I believe with more practice I will come away understanding them more.
The second section we were assigned to read, Proofs Involving Congruence of Integers, was also difficult for me.  I did not really understand the concept of being modulo.  I need to read more into that, and I believe with some basic instruction I will come to understand it.

What was interesting to me?
Whenever I read sections like this, I'm fascinated as to how it was discovered.  For example, the following from the reading:

For integers a,b and n > 2, we say that a is congruent to b modulo n, written a = b (mod n) if n | (a - b).

After reading through the following simple example, I found that so interesting.  It also proved to be true.  I don't quite see exactly how it helps, but I'm excited to learn how.  Reading through these problems, I feel like I'm learning to think like a mathematician.  It's a different way of looking at how to solve problem, and I feel that it will help me better solve statistics problems I am trying to code up.

3.3-3.5, Due on Friday January 24

What do I think is difficult?
I first found the idea of a contrapositive difficult to understand.  It just didn't make too much sense to me that P -> Q could be equivalent to   ~Q -> ~P.  I understood that it was a theorem, and I think that in a previous life I would have taken that to be true without blinking twice.  But since I've been in this proofs class, I have learned to question thing and prove them to believe them.

It also didn't make sense that you could do a proof by contrapositive.  I thought to myself, "How could you prove something to be true by only talking about their negations.  With more practice, I believe I will become more comfortable with them.

What do I find interesting?
You know, I've done lots of math classes throughout my life.  I've had lot of different tests in my life that have asked me to prove whether things are true or false.  I've never done too well with these problems, mainly because I haven't known different ways to actually do these proofs.

Now, I've learned contrapositive, where I can prove something is true by proving the contrapositive.  I also learned proof by cases.  I feel that with learning these different ways to do proofs, I will be able to not only do better on tests, but to think through things and find out what is true and false by taking a step back and applying one of these different proving methods.  I look forward to using these in homework and seeing how they work.  I also look forward to learning other direct proving methods.

Section 3.1-3.2, Due on January 22

What was difficult for me?
trivial proof -  "a trivial proof refers to a statement involving a material implication where the consequent, or Q, in P→Q, is always true"
vacuous proof - "a vacuous truth is a statement that asserts that all members of the empty set have a certain property."

The concept of a vacuous proof was difficult for me.  I at first didn't understand how it worked.  So I did what I always do when I don't understand something - I went to the internet.  I found this example on wikipedia that helped make so much sense of this.  "For example, the statement "all cell phones in the room are turned off" may be true simply because there are no cell phones in the room. In this case, the statement "all cell phones in the room are turned on" would also be true, and vacuously so, as would the conjunction of the two: "all cell phones in the room are turned on and turned off"."

So despite the original difficulty in understanding vacuous proofs, I figured it out.  My understanding is that when the "if" part of the implication statement is always false, then no matter when the second part of the implication is, the overall implication is going to be true by the truth values in the truth tables.  I could see this being helpful later on down the road.

What did I find interesting?
From this reading, I really enjoyed reading through the "thought process" of the proof analysis.  I loved the introduction of proof strategy.  It aays this from the book - "from time to time, we will find ourselves in a position where we have a result to prove and it may not be entirely clear how to proceed.  In such a case, we need to consider our options and develop a plan."  This is called a proof strategy.  I like this a lot.  Recently, I read a book called Lone Survivor by a Navy Seal who was a part of an op that went wrong.  They didn't know what to do.  But they weighed their options, developed a plan, and moved forward.  We cannot always be paralyzed by fear.  We must learn how to move forward.  I think that many times I have proofs to solve, and I get so confused and scared, that I freeze up.  With a better proof strategy, I think I can be more effective in tackling these proofs.  I enjoyed reading through the different proof analyses, and the way the book tackled these proofs.  With practice, I can see myself becoming quite proficient at these proofs.

Chapter 0 pages 5-12, Due Jan 17

"So that's what he meant!  Then why didn't he say so?" - Frank Harary, mathematician

What was difficult?
The difficult part of this assignment for me was the amount of small rules that I lived in ignorance of for such a long time that I found it difficult to overcome by bad habits.  I've spent a lot of time mixing words and symbols.  I've used symbols incorrectly.  I also was not aware of "frozen symbols" and therefore have been using them incorrectly for some time.  While these rules were difficult, I believe that just being aware of them now will improve the quality of my mathematical writing.

One of the more difficult parts for me was the use of words such as "any, each, every."  I did not originally see the difference between these words, but upon reading the chapter I was introduced to the vagueness that "any" carries with it.  I understand this, but I believe it is going to be difficult for me to implement its use due to my years of bad mathematical writing practices.

What was interesting?
I think mathematical writing has typically been difficult for me because I never knew what kind of outline to follow, how to plan an outline, or even what an outline for a mathematical assignment was.  In this reading, I loved the part about outlining what you would like to do for your assignment.  I'd like to write out the steps here, and discuss what I liked about each one:

1. Background and motivation
2. The definitions to be presented and possibly the notation to be used
3. The examples to include
4. The results to be presented
5. References to other results you intend to use
6. The order of everything mentioned above

It's important to start from the beginning.  And when a mistake is made, to begin from the beginning.  If you missed something, insert the missing material and start over.

I also really liked the part about writing mathematical expressions on page 8.  Having studied statistics in school, I have experience in LaTex and am looking forward to being able to use it.  I find such things as displays and correct mathematical writing techniques to produce visually pleasing mathematical equations.

The part about using "We" in solving mathematical proofs is great.  I have a friend who tells me that I do this very well - that I include people by using words such as we, let's, us, etc.  

2.9-2.10, Due on January 15

What did I find difficult?
The most problematic part of the reading was quantified statements.  Reading through the examples was tough for me.  You have both the universal quantifier, which is an upside down capital A.  Then you have the existential quantifier which is a backwards capital E.  Universal quantifiers can be read as for every.  Existential quantifiers can be read as there exists.

I think this part of the reading was difficult for me because it was new.  Even after reading through the chapter and the numerous examples, I still have a few questions about these quantifiers.  I understand that "the universal quantifier is used to claim that the statement resulting from a given open sentence is true when each value of the domain of the variable is assigned to the variable."
We also have the existential quantifier, which "is used to claim that at least one statement resulting from a given open sentence is true when the values of a variable are assigned from its domain."

After some reading and explanation, I understand the simple uses of both universal and existential quantifiers, but when the reading got to quantified statements involving two variables, that was difficult for me to wrap my head around.  Then, the book left me nervous, talking about when a quantified statement may contain both universal and existential quantifiers!  Good thing that is not until Section 7.2!

What did I find interesting?
I love these laws:
1. Commutative laws
2. Associative laws
3. Distributive laws
4. De Morgan's laws

I really like De Morgan's law.  For some reason I can picture it really well.  The negation of the disjunction of P and Q is the conjunction of the negation of both P and Q.  I can picture that in my head, and I can see how it would be useful for solving problems.  It's a different way of looking at how to solve problems such as these proofs.  Even more importantly, I think these laws teach a fundamental principle of taking a step back, evaluating the situation at hand, and then seeing if you can see a problem you are facing in a different light.  If you can, you should try and attack it from that different angle.  Just like these laws open up a whole slew of new possibilities in solving mathematical proofs, taking a step back and reevaluating can help greatly with problems we face in life.

I also really like quantified statements despite the trouble they are giving me.  First looking at the symbols, I was confused at how to go about using them.  Then I read more about them and it makes sense.  I see the purpose in changing open sentences into statements, especially in a proofs class such as this. 

2.5-2.8, Due on January 13

What was difficult for me?
After reading this section, the truth tables were a little difficult for me to understand, especially with the biconditional.  (P -> Q)^(Q -> P)  After further reading, my understanding is that it is just a switch, i.e. Q -> is the converse of P -> Q.  If this is true, then the biconditional is P <-> Q.

Then, also difficult for me was the introduction of logical connectives (~,v,^, ->, <->).  I think it was only difficult because it was an introduction to more complicated sentences, these compound statement.  I kind of understood these connectives, but then when we used the truth tables to prove if they were true, it was a little difficult because of the complexity.  I believe with more practice I will improve with these truth tables.  They are new to me.

What did I find interesting?
Life is seldom simple.  A lot of examples in school are simple for the purpose of pedagogy.  But, when looking at things that we experience in real life, we see complex relationships.  Logical connectives and compound statements more accurately reflect scenarios that we see in the world around us.  Especially in working with data, which is what I really like to do.

Lately, I have been working on a project for a company that wants us to do market research.  The company wants to be able to expand and scale their business but they want to do it with quantifiable data in order to make more accurate decisions.  When we first started the project, we assumed that it would be a simple relationship - if we could find one type of data, they would be able to have the necessary information to grow.  But, we found that it was more complex with that.  After learning about these logical connectives, I think I could use them to make connections between the different stats and data types in order to come to better conclusions.  I think I'll give that a try.

2.1-2.4, due on January 9

This section is on logic.  

Google defines logic as reasoning conducted or assessed according to strict principles of validity.

What was difficult for me?
The concept of logic was initially difficult for me.  I have always been good at reasoning through problems and arguments, but putting it down officially on paper and analyzing what exactly logic is was difficult for me.  I didn't understand truth values at first, despite how simply they actually are.  After some further reading and research on the web, I learned that truth value is "the attribute given to an argument that gives information about it's truth or falsehood."  Thanks Google for that clarification.
Also difficult for me were the concepts of disjunction and conjunction.  Maybe it was just given the difficulty of the words, and the intimidation that they gave me.  I couldn't understand what these words meant and how they applied to logic, but, after reading more about it I came to this decisive conclusion:
disjunction is the same as as union.  Either argument, or both, can be correct.  Thinking about it thi way made a lot more sense.
The same can be said for conjunction.  It is the same as an intersect.  Either both arguments must be correct or both must be incorrect.  At least that is my understanding of the concept.  As you can tell, it was difficult for me to understand it.

What did I find interesting and relevant?
I think it's so valuable to be able to think through problems using logic.  It's a proven way to determine whether something is true or false.  I kind of like to think of it as a process where the truth will always come out.  You start by proving one thing right.  You then continue on to the next step, which you build off the first step.  By doing this, you are able to build upon previous steps and arrive at a conclusion that was not clear from the beginning, but due to logical steps, has become clear.

My favorite thing from this reading is the idea of implication.  I think that this is commonly used in arguments and thinking throughout the world.  I see myself constantly thinking about things in this way.  I often think of the consequences, or the implications, that certain decisions will have.  I also see things as a chain reaction - "If this is true, then this is true."

Overall, I think logic is a powerful way to reason through problems.  I look forward to learning more about it.

1.1-1.6, due on January 8

What was the most difficult part of the material for you?
I've previously taken a class, Introduction to Probability (Stat 240), which introduced me to basic set theory.  The class was initially difficult for me, as it seemed to me that set theory was a completely different way of thinking about grouping.  With that said, I ended up doing well and coming to understand set theory well. Thus this simple introduction does not present anything too new.

Yet, there are some things that were unclear in the reading for me, and difficult to understand.  Namely, the new assignment of symbols for different sets:

symbol	for the set of
N	natural numbers (positive integers)
Z	integers
Q	rational numbers
I	irrational numbers
R	real numbers
C	complex numbers

The different groupings of numbers were difficult for me to understand at first, but I ended up understanding it by looking at examples.

Also difficult for me was the concept of differences.  I didn't understand how you could take one set and subtract another set from it.  But after seeing the Venn Diagram for it (which I pasted below), it was a lot more clear.

What was the most interesting part of the material?

Reading through this introduction to set theory and thinking back on past courses I have had on set theory made me think about the interesting way that set theory teaches you to think about grouping.  I've had some experience in segmentation with work that I've done.  With these segments, set theory was very helpful in doing more work and making sense of the different groups.  How? By taking different segments, and sets of segments, and subsets, and complements of these segments.

I think that while the learning of set theory can sometimes seem rote (for me it is difficult to read through examples that do not originally seem to have immediate application to real world problems), the way that simple set theory can be used and applied is powerful.  The different symbols encapsulate important methods of splitting up data and expand your way of thinking about grouping.

Below is an example of different symbols, that even to the student, can be used powerfully for grouping.  These symbols allow you to think and express grouping in ways that are invaluable to solving problems.

Introduction, due on January 8

My name is Nick Pericle and I love math.  Welcome to my journal for learning Mathematical Proofs.  I am very excited to be beginning this journey.  They say a journey begins with a single footstep.  And this is my single footstep.  You can follow my mathematical journey here.

1. I am a junior at BYU and my major in Statistical Science
2. The math classes I have taken at BYU are Math 112, Math 113, Math 313, Math 314.
3. I am taking this class to learn more about thinking like a mathematician in terms of mathematical proofs.  I've always been fascinated with the way mathematicians think, and I want to become more familiar with that.  On a less noble note, I also want to fulfill the necessary credits for a math minor at BYU to supplement my Statistics major.
4. The math teacher who I had was the most effective was my linear algebra teacher.  He taught me by showing me example problems of linear algebra on the board, while simultaneously teaching me methods for solving similar problems.  Thus, he not only taught me how to solve specific problems by working them out on the board, but he taught me how to think through problems, and how to bring together my previous knowledge to evaluate and solve the problem at hand.  This proved to be very effective and I did very well in the class, despite my initial struggles with the material.
5. I took my first steps in Rome and my first word was "church."
6. I am able to come to scheduled office hours and I look forward to it.

Arrivederci professore.