If people do not believe that mathematics is simple,
it is only because they do not realize how complicated life is.
John von Neumann (1903–1957), Hungarian-American mathematician
This chapter provides exercises from three different mathematical topics (Number Theory, Abstract Algebra, and Real Analysis) that will test your proof-writing skills. To succeed in advanced math classes, you will need to be able to solve problems like these.
Up to this point, our valid deductions have been called “theorems,” but mathematicians usually reserve this name for the ones that are particularly important, and apply some other name to the others. The terminology allows some flexibility, but here are general guidelines:
- Any valid deduction can be referred to as a “result.”
- A theorem is an important result.
- A proposition is a result that is not sufficiently important to be called a theorem.
- A corollary is a result that is proved as an easy consequence of some other result.
- A lemma is a minor result that is not interesting for its own sake, but will be used as part of the proof of theorem (or other more significant result).