Three Skills
For
Algebra
Volume 2
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Chapters and Appendices
Book Entrance
First Logic Puzzle
Second Logic Puzzle
One-versus Two-Way Implications
Implications versus Suggestions
Foreword
1. Introduction
2. Implication Rules [4]
3. Chains of Reason [3]
4. Induction Mathematical
4. Romeo and Juliet
6 Old Language
5 Knowledge Islands [2]
7 Arith Skill Check [4 X 2]
Arith Webvideos
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable [8]
9. Algebra Talk [7]
10 Two More Skills[5]
11 Why Shorthand
12 Shorthand Usage [10]
13 What's Next
PS: The 4-th Skill For Algebra
14 Compound Interest [6]
15 Linear Equations [5]
16 Painless Proofs
17 Pythagoras
PS I. Distributive Law
PS II. Polynomials
18 Rules of Algebra [20]
19 Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums [2]
23 Summation Notation
24 Your Money [3]
25 Induction & Recursion [4]
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
Pathways for Learning
Would you like to show yourself or others how to be algebra
power users?
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
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Chapter 2,
Implication Rules
The Second Puzzle
Previous Section: The First
Puzzle
A Two-Way Implication Rule
Try answering the five questions again, using this two-way (implication) rule
Tom goes out to play when and only when Aunt Jane visits his home.
instead of the original rule. How will the answers change? Rather, which
answers change? This second rule can be restated as follows.
Tom goes out to play when Aunt Jane visits his home.
and also
Tom goes out to play only when Aunt Jane visits his home.
The first when part of this rule is disobeyed in the situation where
Aunt Jane visits and Tom does not go out to play. The only when part of this
rule is disobeyed in the situation when Tom goes out to play without his Aunt
Jane visiting. Here are the five questions again.
- When the rule is obeyed, what can you say happens for sure when Aunt Jane
visits her nephew's home? This is easy. [Answer]
- When the rule is not disobeyed, what can you say happens for sure about
Aunt Jane when Tom is out playing? Be careful. [Answer]
- When the rule is not disobeyed, what can you say happens for sure about
Tom when Aunt Jane is not visiting? Be careful, again. [Answer]
- What must happen for the given rule to be disobeyed? This is another
easy question. [Answer]
- When the rule is not disobeyed, what can you say for sure about Aunt Jane
when Tom does not go out to play? See the answer to the fourth question. [Answer]
Answers are given twice
- in popup boxes, and
- in text below. (as in
the printed version)
See if you agree with them.2 |
The two-way implication rule for the second puzzle is:
Tom goes out to play when and only when Aunt Jane visits his home
instead of the original rule. How will the answers change? Rather, which answers
change? This second rule can be restated as follows.
Tom goes out to play when Aunt Jane visits his home
and also
Tom goes out to play only when Aunt Jane visits his home.
The first when part of this rule is disobeyed in the
situation where Aunt Jane visits and Tom does not go out to play. The only
when part of this rule is disobeyed in the situation when Tom goes out to
play without his Aunt Jane visiting. The questions and answers follow.
- When the rule is obeyed, what can you say happens for sure when Aunt Jane
visits her nephew's home? Answer: Tom must be out playing (no change).
- When the rule is not disobeyed, what can you say happens for sure about
Aunt Jane when Tom is out playing? Answer: Aunt Jane must be visiting (the
answer has changed).
- When the rule is not disobeyed, what can you say happens for sure about
Tom when Aunt Jane is not visiting? Answer: Tom is not outside playing (the
answer has changed).
- What must happen for the given rule to be disobeyed? Answer: Either Aunt
Jane must be visiting and Tom does not go out to play or Tom must be out
playing without Aunt Jane visiting (the answer has changed).
- When the rule is not disobeyed, what can you say happens for sure about
Aunt Jane when Tom does not go out to play? Answer: Aunt Jane is not
visiting (no change).
Chapter Subsections: [ First Logic Puzzle ] [ Second Logic Puzzle ] [ One-versus Two-Way Implications ] [ Implications versus Suggestions ]
Next: One- Versus
Two-Way Implications
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