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.

Mathematical Morsel I: How to read math books

A note to reader of Math books:

(I like this v much; I am reproducing it verbatim as the advice of G F Simmons, Topology and Modern Analysis, Tata McGraw Hill Publication, India)

Two matters call for special comment: the problems and proofs.

The majority of this problems are corollaries and extensions of theorems proved in the text, and are freely drawn upon at all later stages of the book. In general, they serve as a bridge between ideas just treated and development yet to come, and the reader is strongly urged to master them as he goes along.

In the earlier chapters, proofs are given in considerable detail, in an effort to smooth the way for the beginner. As our subject unfolds through the successive chapters and the reader acquires experience in following abstract mathematical arguments, the proofs become briefer and minor details are more and more left for the reader to fill in for himself. The serious student will train himself to look for gaps in proofs, and should regard them as tacit invitations to do a little thinking on his own. Phrases like “it is easy to see,” “one can easily show,” “evidently,” “clearly,” and so on are always to be taken as warning signals which indicate the presence of gaps, and they should put the reader on his guard.

It is a basic principle in the study of mathematics, and one too seldom emphasised, that a proof is not really understood until the stage is reached at which one can grasp it as a whole and see it as a single idea. In achieving this and, much more is necessary than merely following the individual steps in the reasoning. This is only the beginning. A proof should be chewed, swallowed, and digested, and this process of assimilation should not be abandoned until it yields a full comprehension of the overall pattern of thought.

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Isn’t that beautiful advice for all budding math aspirants ?

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Cheers,

Nalin Pithwa.