my Analysis ( II )

In the Spoken Reals I asked readers to get a copy of Maxwell Rosenlicht's book Introduction to Analysis. I would like to do a new reading of this book both for my own space physics work and to try to make the book into a powerful friend to people who describe themselves as 'hopeless' with respect to both arithmetic and mathematics. I want to do this because I think I see a great power that people could get from this book, if we can overcome the facts that (1) I am taking the book out of the university classroom for which it was written, and (2) I am taking away the prerequisites.

The first of these actions makes the work into self study, but I hope to make the self study possible and rewarding with the materials I have created and will create with the help of others. The second action, taking away the prerequisites, may seem hopelessly ambitious. This is because Rosenlicht says in his preface that "The background recommended is any first course in calculus, through partial differentiation and multiple integrals [...]", but then he immediately continues "[...] (although, as a matter of fact, nothing is assumed except for the axioms of the real number system)." And he treats the real number system in detail in the second chapter, with the prerequisite material on sets covered in the first chapter!

Therefore, my task is to make the first two chapters accessible to my desired audience. After chapters one and two have been mastered, it seems to me that the reader will be well placed to go on to chapters three and four, which Rosenlicht describes as "the meat of the book". I think anyone in my intended audience who gets that far will have gone from being way behind most people to being way ahead. We will only need slightly to modify what Rosenlicht says in his preface: "After this, Chapters V, VI, and VII flow along smoothly, for their substance (elementary calculus) is familiar and the proofs now make sense." For us, the familiarity will come from the sense. Scroll to show contents below.

Topic Index

Begin
(at the beginning)

chapter 1, 1, 2, 3, 4
chapter 2, 1, 2, 3, 4
----- and -----
chapter  3,
chapter  4,
----- and -----
chapter  5,
chapter  6,
chapter  7,
----- and -----
chapter  8,
chapter  9,
chapter 10.

chapter I Preview

Continue Previous Readings, chapter I, 1
c01_s1_r01:
Equality of Sets
c01_s1_r02:
Elements of Sets
c01_s1_r03:
An Example Postponed
c01_s1_r04:
Subsets
c01_s1_r05:
Set Equality in Terms of Subsets
c01_s1_r06:
Reading Subset Symbols
c01_s1_r07:
The Empty Set
c01_s1_r08:
Unions and Intersections of Sets

Continue Previous Readings, chapter I, 2
c01_s2_r01:
Complement of a Set
c01_s2_r02:
Examples from the Number Line: Union & Complement
c01_s2_r03:
The Intersection of the Complements = The Complement of the Union
c01_s2_r04:
Difference of Sets
c01_s2_r05:
Indexing Families of Sets
c01_s2_r06:
Ordered Pairs
c01_s2_r07:
The Cartesian Product of Two Sets

Continue Previous Readings, chapter I, 3
c01_s3_r01:
Description of Functions
c01_s3_r02
Notation for Functions and More Examples Postponed
c01_s3_r03
Definition:   f : XY
c01_s3_r04
Exercise:     Y Y     XY XY
c01_s3_r05
General Definition of a Function
c01_s3_r06
Composition of Functions
c01_s3_r07
Functions that are One-One, Onto, or Both
c01_s3_r08
The Identity Function
c01_s3_r09
Inverse Functions
c01_s3_r10
The Image of a Set Under a Function
c01_s3_r11
The Inverse Image of a Set Under a Function

Continue Previous Readings, chapter I, 4
c01_s4_r01:
Shall We Postpone this Section?
c01_s4_r02:
Definitions for Finite and Infinite Sets
c01_s4_r03:
Proper Subsets of Infinite Sets
c01_s4_r04:
The two parts of :    if and only if
c01_s4_r05:
Proper Subsets of Infinite Sets (continued)
c01_s4_r06:
Sequences

chapter II Preview

chapter II, 1_preview
c02_s1_f03_and_f04:
F3 and F4: Unique Solutions to   x+a=b   and   xa=b
c02_s1_f05:
F5: Zero times any real number is zero.
c02_s1_f06:
F6: The additive inverse of the additive inverse is the original number.
c02_s1_f07:
F7: The multiplicative inverse of the multiplicative inverse is the original number.
c02_s1_f08:
F8: The additive inverse of a sum is the sum of the additive inverses.
c02_s1_f09:
F9: The multiplicative inverse of a product is the product of the multiplicative inverses.
c02_s1_f10:
F10: The additive inverse of any real number is equal to   -1   times the real number.

chapter II, 2_preview
c02_s2_O1:
O1: Trichotomy
c02_s2_O2:
O2: Transitivity
c02_s2_O3:
O3: The sum of the greaters is greater.
c02_s2_O4:
O4: The product of the greater positives is greater.
c02_s2_O5:
O5: The rules of sign in addition and multiplication.
c02_s2_O6:
O6: The squares of the real numbers are greater than or equal to zero.
c02_s2_O7:
O7: The multiplicative inverses of the positive real numbers are greater than zero.
c02_s2_O8:
O8: The multiplicative inverse of the positive greater is less.
c02_s2_O9:
O9: The Rules of Arithmetic

chapter II, 3_preview
c02_s3_LUB1:
LUB1:     x,     n{...,-3,-2,-1,0,1,2,3,...}     nx
Click the link AND the topic index to see the meaning of the symbols.
c02_s3_LUB2:
LUB2:   Existence of arbitrarily small positive rational numbers
c02_s3_LUB3:
LUB3:   Nearest integers to any real number
c02_s3_LUB4:
LUB4:   Rationals with a given denominator near a given real
c02_s3_LUB5:
LUB5:   Arbitrarily accurate approximation of reals by rationals
decimals:
Decimal representation of real numbers

chapter II, 4
The existence of square roots

TO BE ... CONTINUED - from cultural starting points with the aid of collaborators.
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