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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 29
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| row | A | B | if A then B |
NOT B | NOT A | if NOT B then NOT A |
| 1 | occurs | occurs | obeyed | occurs
not |
occurs
not |
not
disobeyed |
| 2 | occurs | occurs not |
disobeyed | occurs not |
occurs | disobeyed |
| 3 | occurs not |
occurs | not disobeyed |
occurs not |
occurs | not disobeyed |
| 4 | occurs not |
occurs not |
not disobeyed |
occurs | occurs | obeyed |
In the fourth column, headed by the rule if A then B for each combination of occurrences of A and B, we note if the rule is obeyed, disobeyed or not disobeyed.
Next, in the fifth and sixth columns headed by situations NOT B and NOT A, for each of the four combinations we note if these situations occur or not.
In the last column, we finally note if the rule if NOT B then NOT A is obeyed, disobeyed or not disobeyed. The entries in the last column depend on those in the fifth and sixth columns. The entries in the latter two in turn depend on those in the previous columns.
Now we can answer the questions: when are the two one-way implication rules (if A then B) and (if NOT B then NOT A) true? Remember we say these implication rules are true if each is never disobeyed. Both implications are true, that is, never disobeyed, when the situation row 2, A and NOT B, never occurs. Both implications are false when the situation in row 2, namely (A and NOT B), occurs. So we conclude from the table that the two rules are equivalent: each implies the other.1
The one-way implication rule If A then B is said to be vacuously true if and only if the situation A never occurs.
The contrapositive If NOT B then NOT A is vacuously true if and only if the situation NOT B never occurs, that is if and only if the situation B always occurs. Therefore an implication rule and its contrapositive are vacuously true in different circumstances.
Finally, an innovation perhaps, the two-way implication rule A if and only if B is said to be vacuously true in the situation where A and B are both always true or both always false.
An implication rule says that when a first situation A occurs then so must a second situation B. The associated contrapositive implication rule says that when the second situation B does not occur, then the situation A cannot occur. The previous part of this chapter explains why an implication rule is never disobeyed if and only if its contrapositive is never disobeyed. In consequence, a chain of reasoning which shows the contrapositive form of an implication rule is never disobeyed also shows the implication rule is never disobeyed.
Note that a hint or preview of the contrapositive was provided by the discussion of the first logic puzzle (questions 4 and 5) in the chapter Implication Rules. (You might wish to revisit that puzzle.)
1The rule if NOT B then NOT A is disobeyed if the NOT B occurs but NOT A does not. That is, it is disobeyed precisely when B does not occur, while A does. But the rule if A then B is disobeyed precisely in this situation where A occurs and B does not. This tells us that both rules are not disobeyed provided the situation where A occurs and B does not never occurs. So if one rule is true (never disobeyed), then so is the other.
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2. Three Skills for AlgebraForeword, 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
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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