1.2 Background

Notation

Logical Notation:

"there exists."
"for every."
"not"
"implies"

Set Theoretic Notation:

" $\QTR{Large}{x}$ is a member of the set $\QTR{Large}{A}$ "
"all members of $\QTR{Large}{A}$ such that is true"

When the context is clear we write

Mathematical Objects of Interest

The Natural numbers
The Integers
The Rational numbers , a set of equivalence classes which can be represented by or and
The Real numbers (some details to follow)

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The Algebra of Sets

Operations

Given some "universal" set $\QTR{Large}{U}$ and subsets $\QTR{Large}{A}$ and $\QTR{Large}{B}$ , we define:

Union $\QTR{Large}{\cup }$ or
Intersection $\QTR{Large}{\cap }$ and
Complement
There is also the relationship "Subset" $\QTR{Large}{B\ }$

Various Identities:

$\QTR{Large}{\cup }$
$\QTR{Large}{\cap }$
$\QTR{Large}{(A}$ $\QTR{Large}{\cup }$ $\QTR{Large}{\cap }$

More generally, if is some indexed set of sets then
$\QTR{Large}{(A}$ $\QTR{Large}{\cap }$ $\QTR{Large}{\cup }$

More generally, if is some indexed set of sets then

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Products and Maps

We will want to look at products of sets. That is if $\QTR{Large}{A}$ and $\QTR{Large}{B}$ are sets and

If $\QTR{Large}{A}$ and $\QTR{Large}{B}$ are sets, there is some algebra associated with maps ( functions) from $\QTR{Large}{A}$ to $\QTR{Large}{B.}$

Let $\QTR{Large}{B\ }$ be two subsets of $\QTR{Large}{B}$ then

$\qquad$ 1.

$\qquad$ 2.

(see the text, Page 12 Lemma 2.8)
Let $\QTR{Large}{A\ }$ be two subsets of $\QTR{Large}{A}$ then

$\qquad$ 3.

$\qquad$ 4.

To see that relationship 3.is inclusion rather than equality consider the situation where

,and Then and

4. is even more immediate.