Do you mean:
"What are the axioms upon which this branch of mathematics is founded?"
... or do you mean:
"What level of mathematical ability previously attained by a student is necessary in order to be able to study the subject?"
There is absolutely no way to answer your questions.
Whereas, almost everyone can agree what material needs to be included in a calculus textbook and the order in which it is presented is more or less standard.
That is certainly not true for Discrete Mathematics.
I have a collection of discrete textbooks that ranges from one written especially for secondary mathematics teachers to one written especially for computer science majors. Moreover, there is no way to get away from what you have called ‘arbitrary school rules’. If some mathematics department calls for a course in foundations of mathematics, logic & sets, then its discrete mathematics course will more than likely include counting, graph theory, and recursion. On the other hand, fully half of the texts in my collection include all of those topics in one textbook. I would also point out that most texts do not require a calculus background but do label a few problems as “requires calculus”.
You could do worse than get Vol.1 of The Art Of Computer Programming and just work thru the first chapter. Then there's his Concrete Mathematics by him, Graham and Patashnik which is chapter 1 of TAOCP expanded somewhat. If you are as you describe yourself it'll be right up your street. All the prerequisites you need are in there
Great. That is indeed the case. Seems perfect that I get to go right to what I consider the good stuff.
After reading some of the reviews for that book, in many different places, I wonder if it really does deserve the snob association it gets.
I'll let you know how it works out, if you like.
Thanks again, to everyone in this thread and especially Matt who gave me the best possible answer.
I used the following textbooks to study Discrete Mathematics during my classes.
Discrete Mathematics and Its Applications, 6th Edition
Discrete Mathematics with Applications, 3rd Edition
Discrete Mathematics, 5th Edition
Wish that are useful for you.