SECTION 4.1 Basics of Set Theory 187
we see thatR∈Rif any only ifR6∈R! This is impossible! This is a
paradox, often called Russell’s paradox (or Russell’s Antinomy).
Conclusion: Naive set theory leads to paradoxes! So what do we do?
There are basically two choices: we could be much more careful and do
axiomatic set theory, a highly formalized approach to set theory (I
don’t care for the theory, myself!) but one that is free of such paradoxes.
A more sensible approach for us is simply to continue to engage in naive
set theory, trying to avoid sets that seem unreasonably large and hope
for the best!
4.1.1 Elementary relationships
When dealing with sets naively, we shall assume that the statement “x
in an element of the setA” makes sense and shall symbolically denote
this statment by writingx∈A. Thus, ifZdenotes the set of integers,
we can write such statements as 3∈Z, − 11 ∈Z, and so on. Likewise,
πis not an integer so we’ll express this by writingπ6∈Z.
In the vast majority of our considerations we shall be considering
sets in a given “context,” i.e., assubsetsof a given set. Thus, when I
speak of the set of integers, I am usually referring to a particular subset
of the real numbers. The point here is that while we might not really
know what a real number is (and therefore we don’t really “understand”
the set of real numbers), we probably have a better understanding of
the particular subset consisting of integers (whole numbers). Anyway,
if we denote byRthe set of all real numbers and writeZfor the subset
of of integers, then we can say that
Z = {x∈R|xis a whole number}.Since Zis a subset of R we have the familiar notationZ ⊆ R; if
we wish to emphasize that they’re different sets (or thatZisproperly
contained inR), we writeZ⊂ R(some authors^1 writeZ (R). Like-
wise, if we letCbe the set of all complex numbers, and consider also
the setQof all rational numbers, then we obviously have
(^1) like me, but the former seems more customary in the high-school context.