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YOU are better than YOU think. Show yourself how:
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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
Do not leave here without it - Logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
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Caution: Site advice is approximately
correct, for some circumstances, not all. Site How-TOs are logically
developed, but not tried and tested. That leaves room for thought and
refinement.. |
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After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
linear2007 Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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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Explore collaborative whiteboards from
groupboard, twiddla or
scriblink.
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Mathematical Induction
Chapter 7, part II
Previous: Chapter 7,Part I
The principle of mathematical induction stated below describes the Romeo and
Juliet ladder idea, see Chapter 7,part I , in the
algebraic shorthand notation favored in mathematics. The last part of this
chapter will not make sense to you if you are not familiar with this shorthand
notation. If this is the case, you may skip this description of mathematical
induction.
We assume that when or if we have counted to any number n, we can count to
the next one as well. Just add one to the count n. This gives the next number
in our count which is written n+1. This offers a way to begin counting all the
whole numbers 1, 2, 3, 4 and so on.
Suppose or imagine for each whole number n, there is a situation An.
This gives a step on the ladder. Now the next whole number after a whole number
n is given by adding 1, that is n+1. So the next step after An. is
written as An+1. The principle of mathematical induction says the
following:
If
- for each whole number n, there is a situation An.
- every time the situation An. occurs, the next situation Am.
with m = n + 1 must also occur; and
- the first A1. situation occurs,
then all the situations An. (where n is a whole number)
occur.
The word occurs can be replaced by the expression can be reached.
The principle of mathematical induction is quite simple. It requires the
following: (1) there is a ladder; (2) on the ladder, from each step we can reach
the next; and (3) the first step is reachable. When these three requirements are
met, the principle of mathematical induction says: all the steps can be
climbed or reached. That is all there is to this inductive principle.
Question. What can be said about the reach-ability of An
where n > 4, if we find a ladder for which requirements (1) and (2)
are met, and we somehow know A4. is reachable? Hint: Imagine a
ladder where the first three steps are broken, but the fourth is somehow
climbable. Is the ladder climbable?
Next: Chapter 8, A Language
Change
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www.whyslopes.com
Volume 1A, Pattern Based Reason
Chapters 1 to 24
FOREWORD
Three Remarks
1 Introduction
2 Communication
3. Elements of Reason
4 Implication Rules
5. Deception
6 Chains of Reason
7 Longer Chains
For & From Consistency
8. Language Change
9 Next Chapters
10 Responsibility
11 Accidental Patterns
12 Knowledge Islands
13 Euclidean Logic
14 Deductive
& Empirical
Views of Mathematics
15 Objectivity
16 Origin of Rules
and Patterns
17 Objective Ways
18. Waking up
19. Symbols & Logic
20. Pronouns or Symbols
21. Truth Tables I.
22. Truth Tables II
22. Biconditional
22. Contrapositive
23. IF-THEN table
24. Indirect Reason Again
To reason often means to persuade someone of
the need for an idea or action. That someone could be yourself. So be
careful.
1A Logic Postscripts
- online only
+Proof by
Absurdity alias proof by contradiction
+How the demand
for consistency supports the law of the excluded middle
+Reality versus or with the aid of Imagination
+Links for reason, logic and crtical thinking
+Three Remarks
+History
Lost or Missing
There is a difference between
knowing how to spend money,
and having money to spend.
There is likewise a difference
between mastering a skill
and having meeting a situation in which it applies.
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