so I'm starting set theory and was directed to Paul Halmos' book. It's great and I really like it, but it's a bit lacking in exercises. I know that verification is a big part of the book (as stated in the preface),but I want to do more exercises so that I can really master the concept. Are there any online problem sets that follow the order shown in the book. Any other types of books are good as well!
Thanks!
On
This book has "exercises" scattered in the body of the text as well as the exercises at the end of the chapters, for example on page 18 "here are some easy exercises on complementation...", page 23 "Here is a non-trivial exercise ....", page 27 "Exercise: for each of these three possible properties", page 28 "Exercise: Show that $X/R$ is indeed a set ...", etc.
Patric Suppes' "Axiomatic Set Theory" is a great book with proper exercises.
study chapter 1,2 to really get a good glimpse at the 7 axioms of ZF and language of logic, very recommended.
chapter 2 starts developing set theory from axiom of extensionality and axiom schema of replacement which enable us to prove the existence of the empty set.
now we have a problem of proving existence of other sets, so here come axioms of pairing and union...
each section is followed by good exercises to practice proving things in set theory.
chapters 3, 4 deal with relations and functions and cardinality of sets.
chapter 8 is difficult.
To summarize if you really want to learn howto prove theorems in mathematics and understand set theory, this is the book for you.