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YOU are better than YOU think. Show
yourself how:
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Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately
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What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
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A. Functions and Relations - Set Viewpoint
People and how they think are shaped by their language,
written or verbal, and vice-versa. Set-based concepts and operations provide a
language and framework for pure mathematics and some but not yet all,
applications.
An introduction or set-based view and codification of functions
(how one variable may be determined by another) and of relations (how
one variables values may be related or limited by that of another)
follows. The explanations are as simple as possible but no simpler.
The explanations in the following pages not only do they
introduce concepts, they also provide a mathematical context for them.
Yet the language here is artificial and technical. This set based view of
mathematics shapes and provides a mould for further topics in pure and
applied mathematics. It has some advantages, and it required
for advanced studies in mathematics.
Set Existence and Construction
(technical starting point)
A set is a collection of objects. In mathematics, the objects are required to
be of a numerical kind - constructed from numbers. Sets of objects which
are not number may be used for illustration of set related ideas and concepts,
but those sets are not part of pure mathematics. In mathematics, the
existence of a set IR of real number (or sets that lead to the
existence of the latter) is assumed. In pure mathematics, there are three ways
to define or construct sets from existing sets.
- Set Builder, Subset Definition: If A is a set and P is
condition that may be satisfied or not by elements of A, then there
is proper or improper set
{ a in A | condition P holds}
Read the latter line as the set of p in A such that the condition P
holds for a in the set A. Read the vertical bar | as such that or for
which. Variations on this notation are possible and permissible - at
this level of instruction.
- Product Set Formation: If A and B are sets, then there exist a set
of ordered pairs
A x B = { (a, b) | a in A and b in B}
exists and consists of all order pairs (a,b) such that the first element
(abscissa) a belongs to the first set A and the second element
(ordinate) b belongs to the second set B. The case A = B
leads to the set
A2= A x A = { (a, b) | both a and b are in A}
For example,
IR2= IR x IR = { (a, b) | both a and b are in
IR}
Letters x and y may be used in place of a and b.
- Power Set Formation (optional): If A is a set, then there exists a
set of all proper and improper subsets of A, namely
P(A) = { B | B is a subset, proper or not, of A}
Pure mathematics essentially assumes the existence of the set of Natural
numbers (Peano's axioms or axiom of infinity) and then defines or construct all
further numbers using set-based definitions. That being said, secondary
mathematics is and cannot be pure. The decimal view of real numbers and diagrams
are needed or should be used in a mixed or applied mathematics manner to
introduce and develop arithmetic, algebraic and geometric reasoning skills. The
designers of the modern mathematics curricula in the 1950's onward adhered too
closely to pure mathematics.
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www.whyslopes.com
Analytic Geometry
Area Entrance
Entrance + Pages Below this page
Pages at Current LevelFNs & Dependency
FN With Finite Sets
FN Vertical Line Rule
FN Infinite Domains
FN Sets-Theory
FN Interval Notation
FN: Sets - Continued
(FN) Sets & Relations I
(FN) Relations & Sets
FN Source Target Domain Range
(FN) Injective or Not
(FN) Sign & Zero Analysis
(FN) Increasing/Decreasing
(FN) Extrema
FN Numerical Exercises
FN Step Sawtooth Abs.Value
FN Horizontal Line Rule
FN Inverse Functions
FN Many Ways to Define
Area Entrance
(C) Complex Numbers
(FN) What are Functions?
(FN) Functions - More
SZM: Sign Analysis
(L) Lines Summary
(P) Polynomials (*,+,-)
(Q) Quadratics
(D) Simplify Square Roots
(T) Unit Circle Trig
Conic Sections
Links
More Links
Pages at Above this Page
Extras: Not all perfect.
Equal Sign Use/Abuse
Real Numbers
Say More Positive
Linear Inequalities
Triangle Inequality
Absolute Value |x|
|x| Eq'ns & Inequalities
Rectangular Coords
Shortest Path
Distance Formulas
Add & Multiply Points
Polar Coordinates
Radians
(A) Vectors
(A) Coordinate Arithmetic
(A) Navigation on Maps
(A) Addition Geometrically
(A) Rotation
(PT) Translations
(PT) Dilatations
PT: Rotations
Links to Site Pages outside this site area follow - co- and pre-
requisites.
Road
Safety Message
Easy Consequences of this (newest) Complex
Number. Starter Lesson follow below to provide an alternate
development of HS or college maths.
Vec & Cmplx No Applet
B2 C. Conjugates
B3 Pythagoras
B4 Distance
B5 Rt Triangle Similarity
B6 Trig., Functions
B7 Dot & Cross Products
B8 Cosine Law
B9 Exponential & cis fns
B10 Easy Trig Identities
B11 Set Viewpoint
Arithmetic Videos - Real Player Format
Decimal Addition Methods
Decimal
Subtraction Methods
Decimal
Multiplication Methods
Decimal Division Methods
Fractions
Primes
Greatest Common Divisor
Divisors
Least Common Multiples
Square Root
Simplification
Using formulas forwards
& Backwards - A unifying theme for algebra skill development - the 4th
skill in Volume 2, Three
Skills for Algebra!
What
is a Variable?
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