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.