Got a Math Minor. Where should I start?

[PeteyCoco]

I just got a math minor appended to my physics major. I'm registering for courses next week, but I'm having a bit of difficulty choosing math courses. None of them have prerequisites, but the prof I met with today did say that some would be very difficult without having developed mathematical maturity. The courses I am choosing between are:

Analysis I
Textbook: Introductory Real Analysis - Dangello & Seyfried
Mathematical rigour: proofs and counter-examples; quantifiers; number systems; Cardinality, decimal representation, density of the rationals, least upper bound. Sequences and series; review of functions, limits and continuity

Abstract Algebra I
Textbook: Abstract Algebra - Dummit & Foote
Introduction to the ring of integers and the integers modulo N. Groups: definitions and examples; sub‑groups, quotients and homomorphisms (including Lagrange’s theorem, Cayley’s theorem and the isomorphism theorems). Introduction to the Cauchy and Sylow theorems and applications

Now my mathematical maturity is pretty weak in terms of proofs, but I do well in my physics courses where vector cal is used (I'm finishing up Griffiths' Electrodynamics and can tackle most problems in it). Would these courses be too much for someone new to formal mathematics like me?

My other option is to take Linear Algebra I & II next year as a way to improve my mathematical abilities before moving up to these courses:

Linear Algebra I
Textbook: Linear Algebra - Friedberg, Insel, & Spence
Matrices and linear equations; vector spaces; bases, dimension and rank; linear mappings and algebra of linear operators; matrix representation of linear operators; determinants; eigenvalues and eigenvectors; diagonalization.

I've covered most of the material mentioned in this blurb, but it was in a course for physicists and we used Elementary Linear Algebra by Anton. Should I just ease my way into math, starting with Linear Algebra I & II? There's no time pressure.

 

[micromass]

Can I ask you some questions in order to judge how far you are?

1) Can you prove that a natural number $n$ is odd if and only if $n^2$ is odd?

2) Can you prove that $2^n \leq n!$ for $n\geq 4$ and natural.

3) Prove or give a counterexample: If $f:X\rightarrow Y$ is a function and $A,B\subseteq X$ then $f(A\cap B) = f(A)\cap f(B)$.

4) Prove or give a counterexample: For sets $A,B,C$ holds that $A\setminus (B\cup C) = (A\setminus B) \cap (A\setminus C)$

5) Formulate what it means that a function $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and prove that $f(x) = 10x$ is continuous.

 

[heatengine516]

Being better at linear algebra can never hurt, especially for quantum mechanics. That would be my first choice.

 

[JaredEBland]

I had a course with Anton and I really liked it. I know we didn't cover as many theoretical things in linear as I'd like. If your linear for physicists course covered everything you mentioned for linear, I would vote for the introduction to proofs class. If you didn't do all of those mentioned in the linear description along with linear vector spaces, change of basis, eigenvalues and vectors, and the Graham-Schmidt process, then I'd suggest a course in linear. I have found it indispensable.

I wouldn't tackle abstract algebra until you've had an introduction to proofs (required for the "pure math" classes at my institution).

 

[Vahsek]

Can I ask you some questions in order to judge how far you are?

1) Can you prove that a natural number $n$ is odd if and only if $n^2$ is odd?

2) Can you prove that $2^n \leq n!$ for $n\geq 4$ and natural.

3) Prove or give a counterexample: If $f:X\rightarrow Y$ is a function and $A,B\subseteq X$ then $f(A\cap B) = f(A)\cap f(B)$.

4) Prove or give a counterexample: For sets $A,B,C$ holds that $A\setminus (B\cup C) = (A\setminus B) \cap (A\setminus C)$

5) Formulate what it means that a function $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and prove that $f(x) = 10x$ is continuous.

How would questions #3 and #5 help you judge how far he/she is? The OP clearly stated he did not take analysis and is not very good with proofs.

You should have asked only questions #1,2 and 4, which are fairly doable by a beginner IMO.

 

[micromass]

How would questions #3 and #5 help you judge how far he/she is? The OP clearly stated he did not take analysis and is not very good with proofs.

You should have asked only questions #1,2 and 4, which are fairly doable by a beginner IMO.

He might have seen epsilon-delta proofs in calculus. And #3 is standard set theory.

 

[Vahsek]

He might have seen epsilon-delta proofs in calculus. And #3 is standard set theory.

Sorry, my bad. (I never saw epsilon-delta proofs before).

Nevertheless, I believe if the OP can tackle only # 1,2 and 4, then he/she should be just fine because an intro to analysis course usually teaches one how to write proofs (albeit rather quickly and carelessly).

 

[micromass]

Sorry, my bad. (I never saw epsilon-delta proofs before).

Nevertheless, I believe if the OP can tackle only # 1,2 and 4, then he/she should be just fine because an intro to analysis course usually teaches one how to write proofs (albeit rather quickly and carelessly).

Sure, not disagreeing. I never said that he needed to be able to answer all 5 flawlessly in order to be able to handle analysis or abstract algebra. I just asked some questions to be able to judge his level better. I asked 5 because perhaps his level is better than he himself realized.
Also, I think #3 is actually easier than #4...

 

[PeteyCoco]

Can I ask you some questions in order to judge how far you are?

1) Can you prove that a natural number $n$ is odd if and only if $n^2$ is odd?

2) Can you prove that $2^n \leq n!$ for $n\geq 4$ and natural.

3) Prove or give a counterexample: If $f:X\rightarrow Y$ is a function and $A,B\subseteq X$ then $f(A\cap B) = f(A)\cap f(B)$.

4) Prove or give a counterexample: For sets $A,B,C$ holds that $A\setminus (B\cup C) = (A\setminus B) \cap (A\setminus C)$

5) Formulate what it means that a function $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and prove that $f(x) = 10x$ is continuous.

The only one of those that I can prove is 4, and that's not off the top of my head (referencing book on my desk). I'm thinking that maybe I could self teach from an introduction to proof writing book this summer to better prepare myself for these courses. What book would teach me the material to tackle problems like those you listed?

 

[TheKracken]

Alright, I totally do not mean to hijack a thread or anything (never done that before.) But I am in pre-calculus right now and I plan to be a math major. Looking at the things you asked to prove, I do feel like I understand how 1 and 2 are true and I think I could explain why. But how would you write a proof on it? I am almost positive math is for me because I love thinking about how these things work underneath it all and I think I would enjoy learning to prove it all.

 

[micromass]

The only one of those that I can prove is 4, and that's not off the top of my head (referencing book on my desk). I'm thinking that maybe I could self teach from an introduction to proof writing book this summer to better prepare myself for these courses. What book would teach me the material to tackle problems like those you listed?

The book "How to prove it" by Velleman is an excellent resource here. It will show you how to prove #1 - #4 (don't worry about #5, it's not necessary to know to start the courses).

I saw analysis has quite a bit on intro proofs, so that might still be worth taking now. Many of it will also be revision from calculus. Don't underestimate the course though.
I think I would advise against abstract algebra at this point, something like linear algebra is much more suitable.

Are you sure there's no intro proofs course at your uni?

 

[micromass]

Alright, I totally do not mean to hijack a thread or anything (never done that before.) But I am in pre-calculus right now and I plan to be a math major. Looking at the things you asked to prove, I do feel like I understand how 1 and 2 are true and I think I could explain why. But how would you write a proof on it? I am almost positive math is for me because I love thinking about how these things work underneath it all and I think I would enjoy learning to prove it all.

The basic idea behind $1$ is to write $n=2k+1$. Then $n^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1$. This is of the form $2m+1$ and thus even.
But I asked for an if and only if, so you need to prove the converse arrow too. The trick here is contradiction. Assume $n^2$ is odd, but $n$ is not odd. Then we can write $n=2k$, but then $n^2 = 4k^2$ is not odd.

I asked $2$ to see if the OP was familiar with the principle of induction. If you know induction, then $2$ is a very easy application of this. Read the following text and then see if you can provide the proof yourself: http://www.people.vcu.edu/~rhammack/BookOfProof/Induction.pdf

 

[PeteyCoco]

There is an introduction to mathematical thinking course that covers construction of proofs, number systems, ordinality and cardinality, etc but it seems to be a course designed for students with no math background. The math majors do not take this course and jump straight into linear algebra and multivariable calc in their first year. The advisor I met with did say that the analysis course had been watered down a little to make things easier, but that it is still a tough course if you don't take it seriously. I'm thinking I should take linear algebra to warm up to the mathematical thinking. I'll add analysis I to my schedule to see if I can handle it, but I'll drop it I I find I'm too lost. Thanks for the book recommendation micromass, I'll be sure to give it a look this summerSent from my iPhone using Physics Forums