chapterI - The introductory chapter to Apostols Calculus 1...

The introductory chapter to Apostol’s Calculus 1 is kind of a mish-mash 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.