Introduction to Numbers and Set Theory

Abstract

Here we will give the main definitions and concepts of sets, the concepts of surjective, injective, and bijective functions, countability and uncountability of sets, cardinal number, and the cardinality of the continuum. These are very important in mathematical analysis and in other fields of the mathematical sciences. For this reason, different theorems connected with the concepts of countability and uncountability of sets are proved and it is shown that the cardinality of the set of functions defined on a closed interval is greater than the cardinality of the continuum. Furthermore, the Dedekind completeness theorem for the set of real numbers is proved. Finally, some of the topological properties of metric spaces are treated.