Bridge to Abstract Mathematics

The Bridge to Abstract Mathematics is the first course of abstract nature in mathematics provided at the Indiana University East. The bridge section begins with the set theory, the study of “collection of objects”, where objects can be anything – mathematical ideas, numbers, specific items, and so on. We can organize or define numbers in a different way. Some commonly used and well known sets are the set of natural numbers , the set of integers , the set of rational numbers , the set of “real” numbers , and the set of complex numbers .

Then, we begin to relate the objects included in sets. Sets can be linked across different and multiple sets, but also on its own, treating itself as another set. Then, the notion of relation evolves into function. Roughly speaking, when the relation begins to be formally defined, it becomes a function.

The last part is counting, formally defined using the term cardinality. In fact, cardinality or the cardinal number of a set refers to the number of elements in a given set. Indeed, there can be an empty set without any element; finite sets with countably many number of elements; denumerable sets with countably many infinite number of elements; and uncountable sets with uncountably many infinite number of elements.

Karki said in his syllabus the course objective is to familiarize students with basic proof techniques and basic abstract concepts in mathematics. Therefore, students are expected to be able to not only state definitions, but also write rigorous proofs.

Personally mathematics of abstract nature was something unfamiliar, and if possible, the author wanted to avoid. However, on successful completion of this course, the author found himself to have become a more logical person, as the beauty of this course, the author claims, is honing the ability to make a sound argument with mathematical rigor. The readers should follow suit.