|
YOU are better than YOU think. Show yourself how:
|
-/[]\- 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;
|
-/[]\- 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 28
|
| row | A | NOT (A) |
| 1 | occurs | occurs not |
| 2 | occurs not | occurs |
The following table
| row | A | NOT A | NOT (NOT A) |
| 1 | occurs | occurs not | occurs |
| 2 | occurs not | occurs | occurs not |
| row | situation A | situation B | A and B |
| 1 | occurs | occurs | occurs |
| 2 | occurs | occurs not | occurs not |
| 3 | occurs not | occurs | occurs not |
| 4 | occurs not | occurs not | occurs not |
The situation A and B occurs provided
rows 2, 3 and 4 in the above never occur.
In each row, a possible combination of the occurrence or nonoccurrence of the situations A and B is shown in the middle two columns. In the last column, we put a note to say whether or not, the situation A and B occurs or occurs not.
* Language Note. The phrase A and B is also labelled (called) the conjunction of the situations A and B. The situation A and B is said to be true when and only when both the situations A and B occur (= are true).
In everyday speech when you use the word or in a phrase like John or Andrew will go to the store, the usual expectation is that only one will go, not both. But there is another use of the word or favored in logic. The word or is employed in the at least one sense (as is done in logic and mathematics). With this sense or usage, the previous phrase is understood in the inclusive sense: John or Andrew, or both, will go to the store. We now proceed and we will use the word or in the at least one sense.
The situation (A or B ) is said to occur if at least one of the two situations A and B occurs. Otherwise, it is said not to occur. This is summarized in the following table.
| row | situation A | situation B | A or B |
| 1 | occurs | occurs | occurs |
| 2 | occurs | occurs not | occurs |
| 3 | occurs not | occurs | occurs |
| 4 | occurs not | occurs not | occurs not |
The situation A or B can be said to occur
provided the situation in row 4 does not occur.
We have to be careful with the word or. Its meaning depends on the speaker and possibly the listener. That is, confusion and ambiguity results when two people in question use the same words but give them different meanings. To eliminate this ambiguity in everyday speech, write and say one of the following:
Any rule which can be stated in the form if a first situation A occurs, then a second situation B occurs, in brief, if A then B or A implies B, is called a one-way implication.
A one-way implication which is never disobeyed is said to hold and to be (always) true. For a one-way implication rule if A then B, we recall the following:
| row | situation A | situation B | if A then B |
| 1 | occurs | occurs | obeyed |
| 2 | occurs | occurs not | disobeyed |
| 3 | occurs not | occurs | not disobeyed |
| 4 | occurs not | occurs not | not disobeyed |
In each row, a possible combination of the occurrence or nonoccurrence of the situations A and B is shown in the middle two columns. In the last column, we put a note to say whether or not the if-then rule is obeyed, disobeyed, or not disobeyed.
Row 2 represents the situation in which A occurs but B does not. Observe that in this situation, the rule is disobeyed. In the situations represented by the other three rows, the rule is not disobeyed. A one-way implication rule if A then B is said
to be always true,
to always hold
when it is never disobeyed. The one-way implication if A then B is always true when the situation described in row 2 in the above table never occurs.
Remark. If situation A never occurs, the implication rule if A then B is never disobeyed amd it is said to be vacuously true.
A rule which can be stated, or restated, in the form
The first situation A occurs when and only when the second
situation B occurs
or in the form
The first situation A occurs if and only if the second situation B occurs
is called a two-way implication rule. For each two-way implication rule
note that:
The rule is obeyed when both situations occur.
The rule is disobeyed when the first situation A occurs without the second situation B occurring.
The rule is disobeyed when the second situation B occurs without the first situation A.
The rule is not disobeyed when both situations do not occur.
In brief, the two situations in a two-way implication rule must both occur or both must not occur, for the rule to be not disobeyed.
The next table summarizes the above remarks for any two-way implication rule A if and only if B.
| row | situation A | situation B | A if and only if B |
| 1 | occurs | occurs | obeyed |
| 2 | occurs | occurs not | disobeyed |
| 3 | occurs not | occurs | disobeyed |
| 4 | occurs not | occurs not | not disobeyed |
As said before, a two-way implication rule is said to be always true when it is
never disobeyed. This requires that the situations in rows 2 and 3 of the above
table do not occur. That is, the above two-way implication rule A iff B is
true (never disobeyed) provided neither of the situations A and B occurs without
the other.
The converse to the implication rule if A then B is the rule if B then A. Note that interchanging the first and second situation A and B yields the converse to a rule. From this definition or perspective, we see that the converse of the converse is the original rule. Check this.
When we know a rule if A then B is never disobeyed, we have no guarantee that the converse rule if B then A is never disobeyed. The reason for this is as follows. The rule if A then B is true if the situation A never occurs without the situation B. The converse rule if B then A is true if the situation B cannot occur without the situation A.
Reminder. Now we can easily answer the following question: What can we say for sure about the event A when (i) the rule if A then B is never disobeyed, and (ii) the event B occurs? Your answer should be not much, or nothing, without further information.
Next: Chapter 29, Contrapositive Form of Implication and Conditionals IF A THEN B
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
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
|
www.whyslopes.com
|