Book/Article recommendation I am a first year Math major in the university, this summer I want to self study and go over some specific subjects.
Firstly, can someone can give a suggestion for a detailed book/article about the construction of Real numbers? I felt like we discussed this subject briefly in my first Analysis course, and want to know more about it.
Secondly, I am planning on studying Set theory , and need a good book for self study. (Undergraduate)
Lastly, I need a good suggestion for Discrete math  (undergraduate) text book for self study.
If by experience someone knows good books, I would be happy to hear your suggestions! 
Appreciate the feedback!
 A: For real numbers I like MAA's book by Henle Which Numbers Are Real, it's very didactic and has lots of exercises. The added advantage is that you can also go over constructions of other number systems with it when you are ready for them, complex, quaternions, etc.
For Discrete Mathematics I like Rosen, well structured and well written, also lots of exercises, but they say "profs love it, students hate it". Apparently, students love Epp, and a free electronic version seems to be available.
Set Theory courses vary a lot across different colleges (naive vs. axiomatic, level of difficulty). Look at the answers to What are good books/other readings for elementary set theory? to select what fits yours the most.
A: I think most discrete math books are junk (e.g. things like Johnsonbaugh's book) aside from Knuth, Oren and Paschnik's Concrete Mathematics - they don't go into enough detail for getting useful things out of them. You're better off with some basic combinatorics book like Van Lint's "A Course in Combinatorics" + Wilf's Generatingfunctionology (free on author's website) and a graph theory text like West's "Introduction to Graph Theory".  
Most intro real analysis books have some construction of the real numbers. "Principles of Mathematical Analysis" by Rudin uses Dedekind cuts while "Elementary Classical Analysis" by Marsden and Hoffman uses sequences. Kenneth A. Ross' "Elementary Analysis: The theory of calculus" is also a nice book, especially for self study which you can use on your own.
I think unless if you want to specifically look at set theory [ i.e. you just want a functional knowledge of whats useful ], the first chapter of  Munkres' Topology 2e is good enough, or the preparation in Rudin, or the preparation from a book like "How to Prove it" by Vellman. However, Halmos has been mentioned for other answers as well. You may be able to kill part of two of your objectives with Kaplansky, "Set Theory and Metric Spaces"
A: For set theory, the following is a good book to start out with:
Naive Set Theory by Halmos, http://books.google.com/books?id=x6cZBQ9qtgoC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false,
Then for discrete, Concrete Mathematics by Graham, Knuth, & Patashnik is good if you have a bit of math background.
As far as construction of the reals, I feel like Baby (Blue) Rudin would have that included.
A: I agree that the best book to start off rigorous Set Theory is Naive Set Theory by Halmos. You can progress to more advanced texts thereafter. And there are two different constructions (that I know of that is) of the Real Numbers. One is the method of Dedekind cuts. In my opinion no book is better than the true classics. Look for Foundations of Analysis by Landau and also read through the exposition by the creator himself Essays on the Theory of Numbers by Richard Dedekind. And the other method is constructing the reals from Cauchy Sequences of Rational Numbers. The Way of Analysis by Strichartz has a fantastic exposition on the first few chapters that is extensive and descriptive - great for a beginner. I never took a Discrete Math course so I'm unable to recommend anything on that. 
Finally, let me give you a piece of advice. I'm guessing you are a bit of a purist who wants to build mathematical knowledge from a perfect sequence justifying each step. That is probably why you are trying to peruse the foundations. But it can be a bit tedious and at times frustrating because the foundation of mathematics is not as concrete as you might imagine. In fact organising it all is one whole subject. Read Halmos' preface. It warns against stubbornness in terms of a perfect learning sequences. Sometimesyou have to allow for "gaps". But you will fill them up as you go along. Hope I helped. 
