|
YOU are better than YOU think. Show
yourself how:
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
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;
|
// _ _ \\
/\ /\
<| (o) (o) |>
| |
| |
\
/
\ = /
|
Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
-/[]\-
||
_ / \
||||||||||||||||||||||||||||
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.
| |
Previous Chapter: 25 Inductive Proofs and
Recursive Defintion/Calculation
Note: Online Book Pattern
Based Reason includes these chapters and more on
logic and reason.
Logos is a Greek word for thought. Previous algebra and symbol
free chapters on reason showed how implication rules can be directly used or
chained together to arrive at conclusions. In daily life with the exception
perhaps of detective stories, the direct use of rules and patterns is usually
sufficient (enough).
Yet in mathematics, direct and indirect chains of reasoning appear. The study
of logic, that is, methods or laws for rule- and pattern-based thought, has been
motivated by the need in mathematics to reach conclusions. In particular, proofs
based on (1) mathematical induction, (2) proof by contradiction, and (3) the
contrapositive all stem or originate from the conclusion-reaching needs of
mathematics. The chapter Direct and Indirect Reason below, will describe
methods (2) and (3). Suggestion: try to read this last chapter to see how much
can be immediately understood.
The subject of logic as it is studied within mathematics courses, is often
presented as an algebraic (or symbolic) perspective of the methods of reason.
The next chapters present the algebraic perspective. They with the earlier
algebra-free discussion of implication rules and chains of reason give some
preparation for the description of the indirect methods (2) and (3) for in the
last chapter Direct and Indirect Reason
The algebraic description of logic also has a role in the
design and simplification of electrical controls and computing circuits.
The algebraic description of logic further allows algebraic methods for
arriving at conclusions, in particular mathematical induction, to be applied
to the drawing conclusions about rule-based reason and logic. The algebraic
description of logic provides models of mathematical logic. Conclusions drawn
about the models then reflect on the limitations and reach of logical or
rule-based thought in mathematics.
1 About the Next Chapters
The next five chapters
- Shorthand or Pronouns in Logic
- Occurrence Tables,
- The Contrapositive
- Truth Tables and
- Direct and Indirect Reason
continue the description of logic.
The occurrence (or obedience) tables invented and introduced below identify
those situations in which implication rules are obeyed, disobeyed or not
disobeyed. The latter notions are intended to simplify the explanation of truth
tables. An implication rule is said to be true in the case when it is obeyed or
it is at least not disobeyed. An implication rule is said to be false or not
true when it is disobeyed.
The language previously used to explain and justify the
entries of truth tables overuses the word true. The introduction of the three
notions of an implication rule if A then B being obeyed, disobeyed or
not disobeyed aims to avoid this situation. Such implication rule is
said to be false in situations where it is disobeyed, and it is said to hold
(or be true) in those situations where it is obeyed or at least not disobeyed.
Finally, the implication rule is said to be always true in the circumstances
of interest provided it is never disobeyed in those circumstance. See the text
for further explanation.
The chapter The Contrapositive shows the
equivalence of an implication rule with its contrapositive formulation. The
analysis is based on the three notions of a rule being obeyed, disobeyed or not
disobeyed.
The chapter Direct and Indirect Reason describes and
explains direct and indirect methods for reaching or proving conclusions. Among
the indirect methods, this chapter describes in particular, how an implication
rule can be shown to always hold by (a) showing its contrapositive form always
hold, or by (b) looking for absurdities that would occur if the implication rule
did not hold. The second method (b) is more indirect than the first method (a).
Next: Chapter 27, Pronouns in Logic
| |
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
|