A Study on Combinatories in Discrete Mathematics

Abstract

The next two chapters deal with Set Theory and some related topics from Discrete Mathematics. This chapter develops the basic theory of sets and then explores its connection with combinatorics (adding and multiplying; counting permutations and combinations), while Chapter 5 treats the basic notions of numerosity or cardinality for finite and infinite sets.Most mathematicians today accept Set Theory as an adequate theoretical foundation for all of mathematics, even as the gold standard for foundations.* We will not delve very deeply into this aspect of Set Theory or evaluate the validity of the claim, though we will make a few observations on it as we proceed. Toward the end of our treatment, we will focus on how and why Set Theory has been axiomatized. But even disregarding the foundational significance of Set Theory, its ideas and terminology have become indispensable for a large number of branches of mathematics as well as other disciplines, including parts of computer science. This alone makes it worth exploring in an introductory study of Discrete Mathematics.