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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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Chapter 31
Direct and Indirect Reason
Previous Chapter: Proofs and Logic- 30 Truth Tables
Note: Online Book Pattern
Based Reason includes this chapters and more on logic
and reason.
To prove a statement is to show that it must hold. To refute a statement is
to show that it cannot hold. In between proof and refutation1
lies uncertainty or We don`t know.
1 Proof and Refutation Methods
In a given situation:
- what evidence or proof do we need to say a rule if A then B is
never disobeyed?
- what evidence or proof do we need to say the rule is false?
False here means sometimes disobeyed. The proof or disproof techniques described
next are usually employed in a context where some implication rules are assumed
to be never disobeyed. The first question (1) then becomes what further
implications rules are never disobeyed. The techniques described next provide
answers in some but not all circumstances.
Refutation: Proving Falseness
To show that a rule if A then B is false is simple, see if we can find
(or show there exists or must be) a situation in which A occurs and B does
not. Then the implication rule A implies B is false. In other words, it
is not always obeyed or it is sometimes disobeyed.
A Direct Approach: Applying Implication Rules
You may show that the rule If A then B is always true by showing that
when situation A occurs so must B. Such a proof could employ a
chain of reason (deduction) using trusted implication rules: rules that are not
disobeyed in the situation at hand. Such a proof shows that when A occurs,
so must B. This says and shows that the situation A and NOT B never
occurs. See the chapter Chains of Reason for an illustration of this
approach.
The Contrapositive Method (Indirect)
Alternatively we can show the (contrapositive) rule If NOT B then NOT A is
always true by showing that when situation NOT B occurs then so must NOT
A. Such a proof again shows the situation A and not B never occurs.
That is what we need. (See the previous chapter The Contrapositive.)
The Contradiction Method (Indirect)
The aim here is to find a situation C with the following properties:
- the situation C does not occur (is obviously false) and
- (A and NOT B) implies situation C.
These properties tell us that the situation (A and NOT B) cannot occur -
why? So that the implication rule A implies B is never disobeyed. Next
are a few words to explain why:
The contrapositive form of the implication rule if (A and NOT B) then C is
what we need. It says if NOT C then NOT (A and NOT B). But the situation NOT
C, by chance or discovery, occurs. Thus the situation (A and not B) can
never occur. And that is what we mean when we say that the rule A implies B always
holds.
* Remark (a-looking we will go). This proof by
contradiction method can be applied without knowing in advance what the
situation C will be. We search for it. That is, for the sake of finding
such a situation C, we assume the situation A and not B occurs.
After this we follow whatever chains of reason we can to reach a conclusion that
an absurd (or obviously false) situation C occurs.
Footnotes:
1Proof and
Refutation was the title of a work by the philosopher Latakos, Cambridge
University Press, 1976 with subsequent corrections and reprinting.
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www.whyslopes.com
Volume 2, Three Skills for Algebra -
Preview, starter & further lessons for logic and algebra
to (i) improve work & study skills; (ii) to to ease or avoid
algebra (math) fears & difficulties; and (iii) to fill gaps in the
exposition of mathematics.
Foreword, Chapters and Appendices follow.
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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