Sets
CHAPTER 1
The purpose of this chapter is twofold: to provide an introduction to, or
review of, the terminology, notation, and basic properties of sets, and,
perhaps more important, to serve as a starting point for our primary
goal-the development of the ability to discover and prove mathematical
theorems. The emphasis in this chapter is on discovery, with particular
attention paid to the kinds of evidence (e.g., specific examples, pictures) that
mathematicians use to formulate conjectures about general properties.
These conjectures become theorems when the mathematician provides a
rigorous proof (methods of proof start in Chapter 4).
The information on set theory contained in this chapter is important in
its own right, but the spirit of discovery-proceeding with caution from
the particular to the possibly true general, which we emphasize in discussing
sets-applies to all areas of mathematics and is indeed what much of mathe-
matics is about! We will continue to stress its importance in later chapters,
even as we concentrate increasingly on the mechanics of theorem-proving.
The formal development of set theory began in 1874 with the work of
Georg Cantor (1 845- 19 18). Since then, motivated particularly by the dis-
covery of certain paradoxes (e.g., Russell's paradox, see Exercise lo), logi-
cians have made formal set theory and the foundations of mathematics a
vital area of mathematical research, and mathematicians at large have in-
corporated the language and methods of set theory into their work, so that
it permeates all of modern mathematics. Formal, or axiomatic, set theory is
not normally studied until the graduate level, and appropriately so. But
the undergraduate student of mathematics at the junior-senior level needs
a good working knowledge of the elementary properties of sets, as well as
facility with a number of set theoretic approaches to proving theorems. As
stated earlier, our treatment of the latter begins in Chapter 4. Here we en-
courage you to develop the habit of making conjectures about potential