The introductory chapter to Apostol’s Calculus 1 is kind of a mishmash of
different topics, but it includes the introduction of several vital topics for the study
of Calculus.
1.
Section I.1: Historical Introduction
This is a really introductory section, read it at least once, but it does not contain
any facts/methods that you really need to know.
2.
Section I.2: Some basic concepts in the theory of sets
We will be working a lot with sets so it is vital you have a rough idea what a set
is, and can work with the notations introduced here.
A set can be viewed as a big bag, which can hold arbitrarily many items, but
holds each item at most once. It gets particularly complicated when the elements
of a set are themselves sets.
The most important notations you should be familiar are
•
x
∈
A
: Element of
•
A
=
{
1
,
4
,
5
}
: Defining a set by listing its elements;
•
A
⊂
B
(and derived notations as
A
⊃
B
,
A
⊆
B
,
A
⊇
B
): Being contained
in;
•
A
=
{
x

Some condition on
x
}
: Definition by defining property;
•
A
∪
B
, and
S
n
A
n
: Union;
•
A
∩
B
, and
T
n
A
n
: Intersection;
•
A

B
=
A
\
B
: Complement;
• ∅
: Empty set.
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 Spring '10
 323
 Real Numbers, Natural Numbers, Supremum, completeness axiom

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