Mathematical Morsel III: on sets and functions

Reference: Chapter 1, Sets and Functions; Topology and Modern Analysis, G F Simmons, Tata McGraw Hill Pub.

It is sometimes said that math is the study of sets and functions. Naturally, this oversimplifies matters, but it does come as close to the truth as an aphorism can.

The study of sets and functions leads two ways. One path goes down, into the abysses of logic, philosophy, and the foundations of mathematics. The other goes up into the highlands of mathematics itself where these concepts are indispensable in almost all of pure mathematics as it is today. Needless to say, we follow the latter course. We regard sets and functions as tools of thought, and our purpose in this chapter is to develop these tools to the point where they are sufficiently powerful to serve our needs through the rest of the book.

As the reader proceeds, he will come to understand that the words set and function are not as simple as they may seem. In a sense, they are simple, but they are potent words, and the quality of simplicity they possess is that which lies on the far side of complexity. They are like seeds, which are primitive in appearance but have the capacity for vast and intricate developments.

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