Discrete Mathematics for someone who has already done Analysis/Algebra?

Question

I graduated with an undergraduate mathematics degree this past May, but I had never taken a Discrete Mathematics course. I took the usual years' worth of Algebra and Analysis.

I am interested in a discrete mathematics text (with challenging problems) that covers all of the topics on the Math GRE Subject Test, i.e.,

Discrete mathematics: logic, set theory, combinatorics, graph theory and algorithms

and won't be boring for me to read, given my math background.

(I have heard that I would be bored in such a class, as I once spoke to a Discrete Math professor about taking her class, and she told me it would be a waste of time.)

Answer 1

What you probably want is the MIT Course Notes for CS Math (Discrete Mathematics). It is used at Harvard and MIT, as well as my local university and it is Creative Commons Licensed (The specific license for the course notes are Attribution+Noncommercial+ShareAlike).

The "book" is: Mathematics for Computer Science

BY:
Eric Lehman, Google
F Thomson Leighton, MIT
Alebert R Meyer, MIT

With your math background though you should be fine without perusing it for the Math GRE as others have pointed out.

Creative Commmons License for Reference

Answer 2

If I had one book to recommend covering the topics you mentioned, it would be

Discrete Mathematics and Its Applications by Kenneth Rosen.

The only problem with it is that it's aimed towards new computer science undergraduates, so it may not be as challenging. As recommended in the comments, you could work through it without looking at any of the proofs and figure them out on your own, and don't do any of the tedious exercises, just the challenging ones (starred and double-starred).

But here are some other recommendations:

1. For Set Theory & Logic
   • Set Theory and Logic by Robert R. Stoll
   • Theory of Sets by E. Kamke
   • Axiomatic Set Theory by Paul Bernays
   • Axiomatic Set Theory by Patrick Suppes
2. For Graph Theory
   • Schaum's Outlines: Graph Theory by V.K. Balakrishnan
   • First chapter of Pearls in Graph Theory by N. Hartsfield & G. Ringel
3. For Algorithms
   • My first recommendation, Rosen's book, should suffice; you shouldn't need more than that, but if you want to practice writing algorithms, I'd recommend checking out project euler and learning some basic python

Answer 3

If your interest is computing The Art of Computer Programming by Donald Knuth covers a wealth of discrete Mathematics. If your interest is more mathematical then try generatingfunctionology by Herbert S. Wilf, and Enumerative Combinatorics by Richard P. Stanley. All of these texts are appropriate for graduate level inquiry. These texts would not be a waste of your time.