Amplifications of Chapter 1: Sets, Functions and Relations

Remark 1.2 is a little sloppy. Peano's axioms are due to Dedekind. This axiomatization of the natural numbers has it that there is a first number, that every number has a unique successor, and that every natural number (other than the first one) is the successor of a unique natural number. Finally, something is a natural number if and only if it can be shown to be one on the basis of these axioms. In the book we do not give the axioms themselves, but (in technical terms) a model of them. If you like, this is a manifestation of the axioms within set theory.


There are results about Pascal's triangle and the binomial theorem which every working mathematician should have at his or her fingertips. Following on from that the inclusion-exclusion principle should be of great interest. This is about working out the size of a union of finitely many finite sets. This has all sorts of probabilistic and combinatorial consequences.
Inclusion-Exclusionadobe pdf plain text dvi postscript LaTeX2e source.


Some people have requested notes on countability, uncountability, cardinality and so on. dvi
Countabilityadobe pdf postscript LaTeX2e source.


Notes on how functions, preimages, unions and intersections interact (how to abuse function notation and get away with it).
Functions, Preimages, Unions and Intersections adobe pdf plain text dvi postscript LaTeX2e source.


One may characterize both injections and surjections via the existence of a one-sided inverse.
Injections and Surjections adobe pdf plain text dvi postscript LaTeX2e source.


Return to book home page