Chapter 1 Sets
All subjects have to start from somewhere, and we’ll start our work at sets. The authors believe that you, the reader, will have all seen some basic bits of set-theory before you got to this text. We hope we can safely assume 4 that you have at least some passing familiarity with sets, intersections, unions, Venn diagrams (those famous overlapping circle pictures), and so forth. Based on this assumption, we will move quite quickly through an introduction to this topic and do our best to get you to new material. We really want to get you proving things as quickly as possible.
Set theory now appears so thoroughly throughout mathematics that it is difficult to imagine how Mathematics could have existed without it. It might be surprising to note that set theory is a much newer part of mathematics than calculus. Set theory (as its own subject) was really only invented in the 19th Century — primarily by Georg Cantor 5 Really mathematicians were using sets well before then, just without defining things quite so formally.
Since it (and logic) will form the underpinning of all the structures we will discuss in this text it is important that we start with some definitions. We should try to make them as firm and formal as we can.
Assumptions can be dangerous, and in general we will avoid them, or at least do our best to be honest with the reader that we are making an assumption.
A mathematician we will discuss in much more detail in footnotes much like this much later in the text when we get to the topic of Cardinality in Chapter Chapter 12. We will also try to reduce the overuse of the word “much” as much as possible.
