Take Analysis concurrently with Proofs Course? Smart or Stupid Idea?

[ank91901]

Hi all. This is my first post on here. I come here to read what others ask/answer so this was the first place I thought of when this dilemma came up for me.

I am a Math Major and I am for sure taking a proofs course this fall. I have been allowed to take Analysis concurrently with the proofs course, "at my discretion."
My question is whether this is a smart idea or not.
My other option is to put off taking it this fall, and take it in a year. IF (big IF!) I do put it off, I'll be taking it, along with:

• Theory of Probability
• Applied Math or Applied Statistics
• Abstract Algebra II

These are 4 senior level math courses. would this be just as crazy?
Any input would be appreciated.
The book used in that course is "Elementary Analysis: The Theory of Calculus" By Kenneth A. Ross if that means anything to anyone.

 

[micromass]

You're considering quite a tough course load. It would be doable if you are already used to proofs.

So, what is your experience with proofs? Can you solve the following without troubles?

1) Prove $\bigcap_{i\in I}(A_i\cap B) = B\cap \bigcap_{i\in I}A_i$

2) Prove $(1+x)^n \geq 1+nx$ for $x>-1$ and positive integers $n$.

3) Prove that $a$ is even if and only if $a^2$ is even.

How rigorous was your calculus course? For example

1) Do you know the epsilon-delta definition of continuity. Can you use it to prove that $f(x) = x^2$ is continuous?

2) Can you find the limit of the sequence $x_n = 1/n$ and can you prove it rigorously?

 

[ank91901]

Awesome! This is why I love coming here; straight up answers... even if they hurt a little!

Honestly, no. I don't know how to do these things. I guess I will be taking this class in the fall of next year. From what my friends have told me, the proofs course is very rigorous and you leave with a deep understanding of proofs.
So then my next question is whether you, or anyone else, think(s) whether those four courses would be too much as my final semester. I am wanting to graduate next fall, so in the spring I'll be taking:
Applied Math I
Abstract Algebra I
Geometric Systems
Algorithms in Applied Math
My school only offers certain courses in the fall and certain ones in the spring. I'm sure that's common. So if I want to graduate by the end of Fall 2014 I would have to take these classes at these times.

 

[micromass]

Abstract Algebra is not an easy class if you don't know proofs. It's almost as proof heavy as analysis, only the proofs are a bit less complicated. So, if you don't know proofs very well, then you're going to struggle.

I really recommend you to start practicing proofs in the vacation already. Try to be comfortable with sets, functions (like injective, surjective, bijective, inverses), proof techniques. This will help you a lot.

The two other classes should be alright.

 

[ank91901]

Great.
I've started working on "Mathematical Logic" by Stephen Cole Kleene. I saw it was recommended on here by several people so I thought I'd give it a shot. Any other recommendations?

 

[verty]

Great.
I've started working on "Mathematical Logic" by Stephen Cole Kleene. I saw it was recommended on here by several people so I thought I'd give it a shot. Any other recommendations?

My recommendation is to supplement this with a set-theory book, I recommend this one for being extremely clear and motivating:

Suppes - Axiomatic Set Theory

 

[hsetennis]

Something that may ease your apprehension is the fact that Ross's Analysis is a simple and straightforward textbook. If you plan on working ahead in the summer, Ross is very doable for self-study. You may want to check the syllabus to see if your class is doing the "optional" basic topology sections in the book, which can make the class slightly harder.
Also, I've heard good things about Stanford's Coursera course https://www.coursera.org/course/maththink. It may help your transition.

 

[lurflurf]

I think proof classes are stupid. Some students learn proofs naturally. Others (not necessarily less capable) need some extra help, but proof class is not an effective way to achieve that. I would recommend that you try to prove or understand proofs of some things you know and read some proofy book on your level on a subject like number theory, geometry, discrete mathematics, algebra, linear algebra, calculus, or graph theory. Practice understanding proofs and creating them.

Here are a few of my favorites
-prove sqrt(2) is not rational
-prove the primes are infinite
-prove an elementary function like e^x is not a polynomial
-prove some calculus theorems
-prove some sums like n^3 and the binomial theorem