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YOU are better than YOU think. Show
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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
Do not leave here without it - Logic
mastery will develops critical thinking, improve reading and
writing, and give a firmer base for work and studies at many levels.
Good luck.
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Caution: Site advice
is approximately correct, for some circumstances, not all.
Site How-TOs are
logically developed, but not tried and tested. That leaves
room for thought and refinement.. |
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After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside
site area on solving
linear2007 Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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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Explore collaborative whiteboards
from groupboard,
twiddla or
scriblink.
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1 Why Set Theory
Set membership, union, intersection and complements form a language for a
precise description of mathematical ideas.1 Set theory is emphasized in mathematics since
the axiomatic method of analysis, more precisely the arithmetic properties of
integers, real numbers and functions can be logically codified and described in
it. Outside calculus, you may see the role of set concepts in some presentations
of analytic geometry. Lines, planes, surfaces and sometimes solid objects are
regarded as set of points. Sets and set membership also have a role in
combinatorics and counting, and in the combinatorial based parts of probability.
Combinatorics counts or is concerned with the number of ways objects in sets can
be grouped or placed together.
2 Set and Rule Viewpoints Reconciled
When two different views of an idea are presented, they need to be
reconciled. In the first instance, we could regard a function as a well-defined
computational rule on a domain. The rule could be given by a formula, and the
domain might be defined to the maximal set of real (or complex) numbers for
which division by zero and other undefined operations are avoided. A set of
ordered pairs which satisfies the vertical line property could be viewed as just
another (and then a more general way) of representing or defining a function.
Here
- a formula y = f(x) (or rule for computing y given
some x in a domain) can be used to define a set of order pairs with the
vertical line property, and
- a set with the vertical line property can be used to define a rule for
computing y when x lies in the domain of the set.
Graphing and sketching functions gives the opportunity to show the equivalence
between sets of ordered pairs with both the vertical and horizontal line
properties and one-to-one rules for computation. With this equivalence, graphs
and their transposes can define computation rules for functions or their
inverses.
The foregoing links together and treats equally the set-theoretic definition
of functions and the rule-based definition. The issue of which rule is better is
not important except in the set theoretic foundations of mathematics. Both
definitions should be viewed as interchangeable.
3 Relations
The use of the word relation in set theory requires some explanation. An
explanation is given here since, as suggested earlier, definitions and theories
without examples to illustrate them hold vacuously. Now requiring that an
equation be satisfied by (x,y) when giving the value of x
restricts the value of y, and vice-versa. The satisfaction of an equation
by two quantities (x,y) defines what is called a relationship
between them.
From equations to sets. For instance observe graphing and
sketching the solutions of an equation, for example y2+2x2
= 1, defines sets of ordered pairs (a solution set) in the plane. Requiring that
an equation be satisfied by the ordered pair (x,y) is equivalent
to requiring that this ordered pair belongs to the solution set of the equation.
In set theory, a set S of ordered pairs is said to be a relation: The
knowledge that a point (x,y) belongs to a set S links
(relates) the values of x and y. In particular, for a set S
of points in the plane, giving the value of x restricts the value of y
to the intersection of the set S with a vertical line through the point (x,0)
while specifying the value of y restricts the value of x to the
horizontal line through the point (0,y). This is exactly what happens
when S is a solution set for an equation.
Back to an Equation. Note given a set S of points,
we define the characteristic function XS of the set by
putting XS(x,y) = 1 whenever (x,y)
belong to the set S, and putting XS(x,y)
= 0 otherwise. Now the set S is the solution set of the artificial
equation XS(x,y) = 1. So we can go back
to the equation viewpoint of relations.
Footnotes:
1But it is not
always employed in mathematics instruction. In college-level calculus texts, the
set theoretic representation of functions may be mentioned or emphasized for
function of single variable, but it is frequently omitted in the description of
functions of several variables to science and engineering student for whom
specialization in math is no longer of interest.
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[ Back ] [ Next ]
Three Skills for Algebra
www.whyslopes.com
Foreword, Chapters
& Appendices
Foreword
1. Introduction
2. Implication Rules
3. Chains of Reason
4. Romeo and Juliet
4. Induction Mathematical
5 Knowledge Islands
6 Old Language
7 Arith Skill Check
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable
9. Algebra Talk
10 Two More Skills
11 Why Shorthand
12 Shorthand Usage
13 What's Next
14 Compound Interest
15 Linear Equations
PS I. Distributive Law
PS II. Polynomials
16 Painless Proofs
17 Pythagoras
18 Rules of Algebra
19 Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums
23 Summation Notation
24 Your Money
25 Induction & Recursion
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
A. Advice For Learning
Words Before Symbols:
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice
Complex number: starter lesson
Solving Linear Equations:
A. Letters and Lengths
B. & C. Solving Linear Eq'ns
with stick diagrams.
(i) x + 20 = 29
(ii) 2x + 5 = 20
(iii) 3x + 10 = 32
(iv) 5a + 16 = 3a+ 24
(v) (½)x + 8 = 24½
(vI) (¾)a + 16 = (¼)a+ 24
(vii) (¾)q + 17 = 32
(viii) 13 =[2/3]x +7 twice
(x) Animated Examples
(i) Integral Coefficients (A)
(ii) Integral Coefficients (B)
(iii) Fractional Coefficients
(iv) With
Parameters
Problem Solving with Linear
Equations in one or many
unknowns, and in essentially
one unknown - Symbols before
words.
C. Solving Linear Eq'ns
without
Stick Diagrams
D.
Problems in
essentially one unknown
E: 2D Systems - Sub Methods.
F. Larger Systems
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