To Understand, i.e., to Know How to Apply

Abstract

When have we really understood a proof, an argument, a computational technique? This question is fundamental to every serious student. Here we propose one possible answer: when we are able to apply the above proof, argument, computational technique in a (slightly) different mathematical setting. In each exercise of this chapter, we propose the solution of a problem or the proof of a theorem (say Problem I or Theorem I). Then the reader is asked to use the idea, the reasoning, the techniques just learned to solve another problem or to prove another theorem (called Problem II or Theorem II). Some hints and comments are given.