4. Implication Rules [10]

Previous Chapter: Chapter 3, Elements of Reason Next Chapter: [5. Deception]
Implication Rules
Chapter 4

Introduction

In this chapter, you will meet two puzzles. They show the difference between one- and two-way implication rules. Mastering the difference is a simple, first step, in rule and pattern-based thought. This first step is needed to precisely read rules, definitions and statements in all disciplines, including mathematics.

Are you a careful thinker? Can you understand exactly the meaning of a rule or pattern? Instructions for building or creating provide rules and patterns which say and suggest that when this is done, that should happen. Every cook and dressmaker knows the importance of following instructions carefully. Instructions and suggestions which are not repeatable and results which are not reproducible are not of interest to a cook or dressmaker.

To read rules carefully, do not imagine too much. To decide or choose among opinions and actions, you must understand the exact meaning of written and spoken words. You need this skill to understand, to follow, to write and to change rules, guidelines, instructions and laws, etc.

Use your imagination in language courses. Use your imagination when you are reading novels (and newspaper opinion columns). When reading newspapers or listening to radio and television ask: Is the story presented in a one-sided way? Headlines may suggest conclusions which are not in the stories or the text. Look at the details. Here imagination allows you to guess what the full story might be. But imagination provides only suggestion, not proof. Confidence in suggestions must come after proof is given, not before.

Also use your imagination for poorly written rules to guess their meanings. Guesses and speculations give possible meanings. These may or may not be correct. Proof and evidence, or tests, may decide which among various possibilities, if any, are correct.

Each of us needs to understand fully or as much as is possible, whatever we might be doing or learning. In reasoning, some rules and patterns are reliable. Others are guidelines. Each of us needs to know which is which.

Next Subsection: First Puzzle

Teachers: Read the following before, besides or after the two logic puzzles in this chapter, as you like. (Material not in printed version follows)

Explaining the difference between the meaning of If A then B and B if and only A is the purpose of the following two logic puzzles. The questions in the puzzle below are intended to introduce and emphasize the difference.

Suppose the following:

the local store sells a newspaper if John enters.
the same store sells a newspaper if Jeremy enters.

Then we cannot state

the local store sells a newspaper if and only if John enters it (the local store). .

since the local store also sells a newspaper if Jeremy enters.

So there is a difference in meaning between the two suggestions or statements

the local store sells a newspaper if John enters.
the local store sells a newspaper if and only if John enters it (the local store). .

Seeing the difference in meaning in this simple example is the key to precision reading and writing.

More generally, there is a difference in meaning between the two suggestions or statements

situation B arises if and only if situation A arises

situation B arises if situation A arises.

Here we may say occurs or happens instead of arise, or omit the word arise altogether. That be said and done, we make the convention that the two statements

Situation B if situation A (B if A form)

If situation A then situation B (if A then B)

have the same meaning.

Here we assume that the two following two statements have the same meaning:

B if and only if A,

A if and only B

mean the same.

Understanding there is a difference in meanings is the key to greater precision and exactness in work and study. If you see the difference, you will we hope make an effort to respect the difference and use the difference in following and writing rules or instruction.

Postscript: October 2006. We may know the following:

Situation A occurs if situation B occurs

We may not know if there are furthers situations C such that

Situation A occurs if situation C occurs.

When further such situations C are possible. we cannot use the two-way implication rule

Situation A occurs if and only if situation B occurs.

The foregoing observation provides an alternative starting point

Next Subsection: First Puzzle

A. More Arithmetic a must for algebra etc

D. Logic In Mathematics

G. Algebra with Take Home Value

I. Vectors & Functions

Decimal Lesson - Reference
Counting & Addition   (8 lessons)
Comparison to Subtraction (9 lessons)
Multiplication ( 11 lessons)
Long Division (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites
-Primes Factorization
-Greatest Common Divisors & Multiples.
-Prime Factorization Aids (Learn how to find factors quickly)
-Prime Factorization Examples
-Counting & Generating. Factors.
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts
Ratios & Fractions (3 lessons): Similarities & Differences
Units in calculations Fractions with Units

B. Basic Algebra

Solving Linear Equations
- in one unknown. Intro with stick diagrams?
the normal way & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of word problems.
- still in more unknowns: Gaussian Elimination via substitution, by equality or comparison, by operations on equations.

C. More Algebra

Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra
What is a Variable. The answer here is a simple prequel to the modern mathematics viewpoint.

First, every rule & pattern U meet in math, logic & science will be used forwards and backwards. Get a head start with this theme by reading Chapter 14 in Three Skills for Algebra. Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial backward use of the proportionality formula.

Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.

If you can not read or write precisely, you will have difficulty in following instructions. One wordy remedy is given by chapters 2 to 5 in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in Euclidean-Geometry In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.

E. More Geometry

The Pythagorean Theorem. Chapter 17 from in Three Skills for Algebra uses algebra and geometry   to show why the Pythagorean equation for right triangles holds. Its forward and backward use is common exercise.. At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be fractional multiples of a unit length. That geometrically implies a need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with Maps and Plans drawn to scale give coordinate-dependent base for senior high school development of similarity, trig, vectors and straight lines.
Complex Numbers: This lesson on Complex Numbers draws on Euclidean and Analytic geometry. Sbortcuts simplifiy trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products.

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry, personal finance, and calculus. To learn more, if you study calculus, see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters

identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry: - Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one. I may combine them later. -In my school days, slopes appeared year after year.   This Why Slopes calculus preview on graphs of functions y = f(x) explains why. Enjoy.
Quadratics and Polynomials: - Operations on Polynomials: Meet a light and ultraquick geometric introduction to multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials. Compare Fractions with Units. Enrichment: A Plus: The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic.
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials high school mathematics has technical value as part of the senior high school mathematics preparation for calculus. This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials, their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.

Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms. Vectors and their properties appear in physics, its mathematical description and formulation.
- Functions - Forwards & Backwards. Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics. comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic.

J. Pre-Calculus Skill Check

- Arithmetic Skill Check. In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.
- All the skills and concepts in Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with your other duties. Good luck.

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Previous Chapter: Chapter 3, Elements of Reason Next Chapter: [5. Deception] Implication Rules Chapter 4

Introduction

Previous Chapter: Chapter 3, Elements of Reason Next Chapter: [5. Deception]
Implication Rules
Chapter 4