Mona EC Chapter 1

Assignment: Chapter 1 of Edwin Connell's Group Theory book http://www.math.miami.edu/~ec/

Comments:

Good questions!

1. Ah, Bad notation on the prof's part. The ' stands for complement. So consider the rationals in the space of the reals. (rationals)' is just the set of all real numbers that are NOT rationals. For example, pi. In this case, where Q is the rationals, Q'=R-Q. Ok, question, but we won't keep this one.

2. The complex plane is actually just the complex numbers represented by the cartesian plane, R x R. For a complex number a + bi, the a corresponds to the x coordinates and b corresponds to the y coordinates.

3. Yes, that is why we have A X A.

4. Equivalence classes explained here http://www.youtube.com/user/pvuong87#p/u/16/Urm5PzfVwHE.

5. The projection maps, in this case, are two types of functions. The first function takes in a cartesian coordinate pair (x,y) and spits out x. The second function, likewise, takes in a pair (x,y) and spits out y. So, calling the first function f, we have f( (2,3) )= 2.

6. http://www.youtube.com/user/pvuong87#p/u/11/9rN248OPdNM Explains bijection. But "natural" means, in this case obvious. Very good question that will create good conversation.

7, 8. I can explain these, but they are very valid questions.

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Submitted Work:

Hey Philip, I just finished that packet and after hours of wikipedia-ing and using other fun math websites I was able to answer most of my own questions but here what I had left.

1) on page two where it says theorem, the last two lines before the page break say (A intersect B)'=A' union B' etc.

In the paragraph above it says C'=S-C when defining what that ' means. So is the ' in those two lines referring to the same set, S?

2) The very last line of page two says that R x R= the plane so would R x C be the complex number plane?

3) When speaking of Cartesian products it define them as X x Y={(x,y) x is an element of X and y is an element of Y} so in order for property 2) under relations on page 3 to be true would X have to be the same as Y?

4) On page 5 it says that each element of A is an element of one and only one equivalence class. It doesn't go on to say why nor was I able to figure it out from the information provided.

Page 11 was pretty much totally over my head. I spent a good hour and a half on it but still wasn't quite able to understand it.

5) What exactly is a projection map? (the term is used in the first sentence on page 11)

The rest of the text was all pretty understandable (last couple pages were all number theory stuff haha)

I just had a couple definition question.

What is a natural bijection? (page 12 exercise 2)

What is a characteristic function? (page 12 exercise 4)

What is an index set? (page 2, first actual sentence)