First Logic Puzzle

Implication Rules

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.

Teachers: Read the following before, besides or after the two logic puzzles in this chapter, as you like.

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.

The First Puzzle

A One-Way Implication Rule

To help you think and possibly cook more carefully, we look at a very simple puzzle. The puzzle consists of a rule and five questions. The questions test your ability to think carefully and to read exactly what is written. Once you have understood the answers and why they are true, your ability to think carefully and clearly will have advanced. The rule for the puzzle is as follows:

When Aunt Jane visits her nephew Tom's home,
Tom goes out to play.

Try to answer the five questions below. Be careful. The questions may trip you. Answers follow. See if you agree with them.

Five Questions

Answers are given twice

in popup boxes, and
in text below (as in the printed version)

See if you agree with them.²

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]

Hint: The rule provides no information and no reason explaining why Tom goes out to play whenever his Aunt Jane visits. The rule only describes what happens when Aunt Jane visits. We cannot say if he goes out to play to avoid Aunt Jane. We cannot say if he looks forward to her visits. The answers to the above questions only depend on the wording of the question and the given information or rule(s). So control your imagination. Do not assume or imagine too much.

Suggestion: Discuss the questions with your family and friends. Some people will get correct answers immediately. Others require persuasion. Still others will not understand. Talking with people about the questions shows how well they think.

The First Answer

The first question is

When the rule is obeyed, what can you say happens for sure when Aunt Jane visits her nephew's home?

Its answer is easy: Tom goes out to play.

The Second Answer

The second question is

When the rule is not disobeyed, what can you say happens for sure about Aunt Jane when Tom is out playing?

The answer is nothing. The rule only tells what happens when Aunt Jane visits. It does not say what must happen when Tom goes out to play. Tom could go out to play without Aunt Jane visiting. The rule does not say, nor does it suggest that Tom may only play outside when Aunt Jane visits. The rule does not say Aunt Jane must visit when Tom goes out to play.

When the rule is not disobeyed, we cannot say much for sure or certain about Aunt Jane when Tom goes out to play. All we can say for sure is that she may be visiting or she may not be visiting. When she is not visiting, the rule cannot be disobeyed. When she is visiting, the rule is obeyed and so not disobeyed. In either case, the rule is not disobeyed.

The above rule is one way. It says what should happen when Aunt Jane visits without saying that she must be visiting when Tom goes out to play. When Tom goes out to play, the rule is not disobeyed when Aunt Jane is not visiting. It gives no information on her whereabouts. An example of a two-way rule is given later. See the second puzzle.

The Third Answer

The answer to the third question

When the rule is not disobeyed, what can you say happens for sure about Tom when Aunt Jane is not visiting?

is like that of the second. When Aunt Jane is not visiting, the rule is not disobeyed if Tom goes out, and the rule is not disobeyed if Tom does not go out. When the rule is not disobeyed we can say nothing for certain about Tom when Aunt Jane is not visiting. The rule does not say that the only time Tom can go out to play is when his Aunt Jane visits. Again, the rule is only one-way. When Aunt Jane is not visiting, no information can be extracted from the rule. It says nothing about Tom.

The Fourth Answer

The fourth question is

What must happen for the given rule to be disobeyed?

The rule is disobeyed if Aunt Jane visits and Tom does not go out to play. That is, the situation where Aunt Jane visits and Tom does not go out to play must happen for the rule to be disobeyed.

The Fifth Answer

The fifth question is

When the rule is not disobeyed, what can you say happens for sure about Aunt Jane when Tom does not go out to play?

The rule will be disobeyed when Aunt Jane visits and Tom does not go out to play. To avoid the rule being disobeyed when Tom does not go out to play, Aunt Jane must not be visiting. The fifth answer is Aunt Jane is not visiting.

The contrapositive way of writing the above rule is When Tom not go out to play, Aunt Jane not visit. For this contrapositive rule to be never disobeyed, what can you say for sure when Aunt Jane visits? Answer: Not (Tom Not go out to play), that is, Tom goes out to play. The contrapositive of the contrapositive is the original rule. Later chapters on logic give more information, just a little more, about the contrapositive.

Some Vocabulary

The above rule is called a one-way implication rule. The first aim of this chapter is to show you the difference between one- and two-way implication rules. The meaning and use of the word implication will be talked about later. The five questions should help you see the difference between a one-way and a two-way implication rule. Seeing this difference signals that you can interpret precisely what a rule means.

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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Chapter 2.