D. Sets & Sequences

Appendix D
Sets and Sequences

Greatest and Least Bounds

A real number M is said to be an upper bound for a set S of real numbers if M ³ s whenever s belongs to the set S. Similarly, a real number K is said to be a lower bound for a set S of real numbers if K £ s whenever s belongs to the set S. The following theorem asserts the existence of a least upper bound B = sup(S) and also a greatest lower bound A = inf(S) for each set S contained in a finite interval [a,b].

Theorem D.1 [On Greatest and Least Bounds] If a set S is contained in a finite interval [a,b] then there is a subinterval [A,B] containing S such that among all intervals of the form [p,q] containing S, the interval [A,B] is the smallest. The number A in this theorem is called the greatest lower bound (The name most positive lower bound might be more appropriate) of the set S. It can be obtained to k decimal places by modifying the proof of the Bolzano-Weierstrass theorem, so that the I_k is the leftmost interval of length 10^-k with the property that there are no points of S to the left of it. The left-end point of I_k then yields A to k decimal places. The decimal expansion defines a number A = inf(S) (read the infimum of S) with the property that no elements of S is to the left of it. In other words, the number A = inf(S) is the greatest number with the property that no numbers in S are lower than it. It is called the greatest lower bound. That is, the number A = inf(S) is lower or £ all numbers in the set S, and it is the largest number with this property. Moreover, by construction, in every interval of the form [A, A+10^-k] where k is a whole number, there must be at least one element of S. Thus, if A is not an element of S, then the set S must be infinite and A is the leftmost limit point of S. On the other hand, if A belongs to S, it is the leftmost or least element of S and S could be finite or infinite. (Exercise: What can be said about the membership of A in the set S when the set S is finite?)

Similarly, the supremum of S is the number sup(S) = B. It is also called the least upper bound (The name least positive upper bound might be more appropriate) of the set S. It can be obtained to k decimal places by modifying the second (or first proof) so that I_k is the rightmost interval of length 10^-k with the property that there are no points of S to the right of it. The left-end point of I_k then yields B to k decimal places. The number B = sup(S) is the leftmost point or number with the property that no element of S is to the right of it. In other words, the number B is the least number with the property that no numbers in S are above it. Here B is above or ³ all numbers in the set S. Moreover, by construction, in every interval of the form [B-10^-k,B] where k is a whole number, there must be at least one element of S. Thus if B is not an element of S, then the set S must be infinite and B is the leftmost limit point of S. On the other hand, if B belongs to S, it is the rightmost or greatest element of S. Here S may be finite or infinite.

Question. What can be said about the existence of the least and greatest lower bounds of a set S when the latter is contained in a semi-infinite interval of the form [a,¥) or (-¥,b]?

Two Exercises

What can be said about the membership of B in the set S if the set S is finite?

Show if S = U ÈV Ì [a,b] then

sup

(S) =

max

(

sup

(U),

sup

(V))

and

inf

(S) =

min

(

inf

(U),

inf

(V))

A. More Arithmetic a must for algebra etc

D. Logic In Mathematics

G. Algebra with Take Home Value

I. Vectors & Functions

Decimal Lesson - Reference
Counting & Addition   (8 lessons)
Comparison to Subtraction (9 lessons)
Multiplication ( 11 lessons)
Long Division (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites
-Primes Factorization
-Greatest Common Divisors & Multiples.
-Prime Factorization Aids (Learn how to find factors quickly)
-Prime Factorization Examples
-Counting & Generating. Factors.
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts
Ratios & Fractions (3 lessons): Similarities & Differences
Units in calculations Fractions with Units

B. Basic Algebra

Solving Linear Equations
- in one unknown. Intro with stick diagrams?
the normal way & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of word problems.
- still in more unknowns: Gaussian Elimination via substitution, by equality or comparison, by operations on equations.

C. More Algebra

Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra
What is a Variable. The answer here is a simple prequel to the modern mathematics viewpoint.

First, every rule & pattern U meet in math, logic & science will be used forwards and backwards. Get a head start with this theme by reading Chapter 14 in Three Skills for Algebra. Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial backward use of the proportionality formula.

Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.

If you can not read or write precisely, you will have difficulty in following instructions. One wordy remedy is given by chapters 2 to 5 in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in Euclidean-Geometry In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.

E. More Geometry

The Pythagorean Theorem. Chapter 17 from in Three Skills for Algebra uses algebra and geometry   to show why the Pythagorean equation for right triangles holds. Its forward and backward use is common exercise.. At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be fractional multiples of a unit length. That geometrically implies a need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with Maps and Plans drawn to scale give coordinate-dependent base for senior high school development of similarity, trig, vectors and straight lines.
Complex Numbers: This lesson on Complex Numbers draws on Euclidean and Analytic geometry. Sbortcuts simplifiy trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products.

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry, personal finance, and calculus. To learn more, if you study calculus, see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters

identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry: - Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one. I may combine them later. -In my school days, slopes appeared year after year.   This Why Slopes calculus preview on graphs of functions y = f(x) explains why. Enjoy.
Quadratics and Polynomials: - Operations on Polynomials: Meet a light and ultraquick geometric introduction to multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials. Compare Fractions with Units. Enrichment: A Plus: The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic.
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials high school mathematics has technical value as part of the senior high school mathematics preparation for calculus. This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials, their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.

Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms. Vectors and their properties appear in physics, its mathematical description and formulation.
- Functions - Forwards & Backwards. Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics. comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic.

J. Pre-Calculus Skill Check

- Arithmetic Skill Check. In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.
- All the skills and concepts in Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with your other duties. Good luck.

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The Rest © 1995 onward by site author, Alan Selby, All Rights Reserved.

Appendix D
Sets and Sequences

Greatest and Least Bounds

the Real Analysis appendices of

Why Slopes
and
More Math
Volume 3

Printed in Canada
ISBN 0-9697564-3-7

Appendix D Sets and Sequences

Greatest and Least Bounds

the Real Analysis appendices of

Why Slopes and More Math Volume 3

Printed in Canada ISBN 0-9697564-3-7

Appendix D
Sets and Sequences

Why Slopes
and
More Math
Volume 3

Printed in Canada
ISBN 0-9697564-3-7