You might also want to include some symbols like the symmetric difference, relative complement and aleph-null.
Has anyone else seen I used for irrationals?
No, but neither have I seen "I" represent the "set of imaginary numbers".
There does not appear to be a regularly used symbol for the irrationals.
I believe the standard is R-Q. Q with a horizontal line on it, or R\Q.
(Problem) At the end of the first chapter and beyond, the book is slightly over my head. I only cover a few pages a day at my current pace because I have to really think about the material and I have other responsibilities. Sometimes I need to look up other references to understand certain things. At my current pace it will take me 10-12 months to finish this book. Is the payoff from reading and understanding this book worth the effort or should I read other books?
The first chapter does not even require much algebra. I think the issue is that you haven't adapted to the style of writing in higher level maths books (although this is really nothing compared to something like Rudin). You'll have to get used to re-reading multiple times and filling in the gaps on your own. Maybe, as I said before, you should supplement it with Apostol to help you.
This book is certainly worth it, no matter how long it takes. It sets you up perfectly for a rigorous multivariate calculus course and real analysis.
lol yep I'm not good enough. Well at least Spivak is going to be too much of an uphill battle for me to keep progressing for now. I have not fully mastered elementary algebra since occasionally I run into a basic problem where I get stuck and routinely get bewildered on brilliant.orgs more difficult ones. Does anyone know any rigorous algebra books? Is gelfand's algebra good enough for this? I haven't read it. I would like the ones that use an axiomatic approach. You know the ones that give the definitions, theorems and then prove them. Then they have the corrolaries and all that good stuff. Also I think it's best that I go through an introductory proof book. Do you know any great introductory ones?
I think you may be underestimating yourself a little. The only preparation you need for Spivak/Apostol is Lang's "Basic Mathematics" along with some precalculus.
If you are looking for an algebra book, Gelfand and Sullivan are two good options. You would be hard pressed to find an algebra book that is in the form Definition-Theorem-Proof (I don't see why you would want to either) but those two books proof enough theorems.
Going through a whole book on proofs might be a little too much. Just search some basic notes on it online. However, if you do need a book somewhere down the line, a great one is "How to Prove It: A Structured Approach" by Daniel Velleman.
By the way, you might also want to check out Apostol's calculus book. In some aspects, it is a little better than Spivak. For example, Apostol covers Linear Algebra, puts an emphasis on history and covers integral calculus before differential calculus.