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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 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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-/[]\- 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. |
Chapter 30
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| row | A | not(A) |
| 1 | T | F |
| 2 | F | T |
The statement A is always true when statement NOT A is never true.
The statement NOT A is always true when statement A is never true. Here instead of saying never true, we may say always false.
The truth (T) or falseness (F) of the statement A and B depends on the respective truth or falseness of the statements A and B. This situation is summarized in the following table.
| row | statement A | statement B | A and B |
| 1 | T | T | T |
| 2 | T | F | F |
| 3 | F | T | F |
| 4 | F | F | F |
The statement A and B is said to be always true (to always hold) if the situations in rows 2, 3 and 4 of the above table never occur.
The statement A or B is said to be (mathematical usage) when and only when at least one of the statements A and B is true. The following table summarizes this situation. It shows when the statement A or B is true and when it is false.
| row | statement A | statement B | A or B |
| 1 | T | T | T |
| 2 | T | F | T |
| 3 | F | T | T |
| 4 | F | F | F |
With this usage, the statement A or B is guaranteed to be true provided the situation in row 4 of the above table never occurs.
| row | statement A | statement B | if A
then B |
| 1 | T | T | T |
| 2 | T | F | F |
| 3 | F | T | T |
| 4 | F | F | T |
The implication if A then B is said to be vacuously true when statement A is always false.
The following truth table if for the two-way implication A if and only if B. We observe the two-way implication is always true if the situations in rows 2 and 3 never occur.
| row | statement A | statement B | A if and
only if B |
| 1 | T | T | T |
| 2 | T | F | F |
| 3 | F | T | F |
| 4 | F | F | T |
Remember the letter F signals false, and corresponds to the idea of rule being disobeyed. Also remember that the letter T signals true and corresponds to the ideas of a rule being obeyed, or not disobeyed.
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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
Real Player Videos
Perfect arithmetic skills with whole numbers & fractions after or besides chapters 1 to 14.
Arithmetic Videos Summary Addition with Decimals Subtraction with Decimals Multiplication with Decimals Fraction Arithmetic Recognizing Primes Long Division for Decimals Square Root Simplification Greatest Common Divisors Least Common Multiples 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 lessonSolving 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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