186 CHAPTER 4 Abstract Algebra
- Operations on subsets of a given set: intersection (∩), union,
(∪),difference(−), andsymmetric difference(+) of two sub-
sets of a given set - Set-theoretic constructions: power set(2S), andCartesian prod-
uct(S×T) - Mappings(i.e., functions) between sets
- Relationsandequivalence relationson sets
Looks scary, doesn’t it? Don’t worry, it’s all very natural....Before we launch into these topics, let’s get really crazy for a mo-
ment. What we’re going to talk about isnaiveset theory. As opposed
to what, you might ask? Well, here’s the point. When talking about
sets, we typically use the language,
“the set of all ...”Don’t we often talk like this? Haven’t you heard me say, “consider the
set of all integers,” or “the set of all real numbers”? Maybe I’ve even
asked you to think about the “set of all differentiable functions defined
on the whole real line.” Surely none of this can possibly cause any
difficulties! But what if we decide to consider something really huge,
like the “set of all sets”? Despite the fact that this set is really big,
it shouldn’t be a problem, should it? The only immediately peculiar
aspect of this set—let’s call itB (for “big”)—is that not onlyB ⊆ B
(which is true for all sets), but also thatB ∈B. Since the set{ 1 }6∈{ 1 },
we see that for a given setA, it may or may not happen thatA∈A.
This leads us to consider, as did Bertrand Russell, the set of all sets
which don’t contain themselves as an element; in symbols we would
write this as
R = {S|S6∈S}.This setRseems strange, but is it really a problem? Well, let’s take a
closer look, asking the question, isR∈R? By looking at the definition,