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Appetizers and Lessons for Mathematics and Reason
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 20 pages in French: Algèbre  
 Définition d'une variable  
La raison basée sur les  règles et modelés

www.whyslopes.com > Volume 2,  Three Skills for Algebra  1995  >   28 Occurrence Tables     [ Back ] [ Next ]

Chapter 28
Occurrence Tables

Previous Chapter:  27 Pronouns and Symbols  in Logic

Note: Online Book Pattern Based Reason includes this chapters and more on logic and reason.

1  The Special Usage of Three Words

Given a situation A, we can talk about the negative situation not A. Given a situation A and another situation B, we may talk about two further situations

  1. A and B (conjunction), and
  2. A or B (inclusive or).
The meanings of the terms or phrases are explained next.

NOT A and NOT (NOT A)

Given a single situation A, we can speak of another situation NOT A. The situation NOT A is said to occur when the situation A does not occur. Further, the situation NOT A is said not to occur when the situation A occurs. This is summarized in the following table.

row A NOT (A)
1 occurs occurs not
2 occurs not occurs

Language note: a situation A is said to be true when it occurs and not true (false) when it does not occur.

The following table

row A NOT A NOT (NOT A)
1 occurs occurs not occurs
2 occurs not occurs occurs not

shows that the situation NOT (NOT A) occurs when A occurs and that the situation NOT (NOT A) does not occur when A does not occur. This suggests that the situation A is equivalent to the situation NOT (NOT A).

The word AND

The situation A and B is said to occur if both situations A and B occur. Otherwise, it is said not to occur. See the table below.

row situation A situation B A and B
1 occurs occurs occurs
2 occurs occurs not occurs not
3 occurs not occurs occurs not
4 occurs not occurs not occurs not

The situation A and B occurs provided
rows 2, 3 and 4 in the above never occur.

In each row, a possible combination of the occurrence or nonoccurrence of the situations A and B is shown in the middle two columns. In the last column, we put a note to say whether or not, the situation A and B occurs or occurs not.

*  Language Note. The phrase A and B is also labelled (called) the conjunction of the situations A and B. The situation A and B is said to be true when and only when both the situations A and B occur (= are true).

The At-Least-One-Usage of the word OR

In everyday speech when you use the word or in a phrase like John or Andrew will go to the store, the usual expectation is that only one will go, not both. But there is another use of the word or favored in logic. The word or is employed in the at least one sense (as is done in logic and mathematics). With this sense or usage, the previous phrase is understood in the inclusive sense: John or Andrew, or both, will go to the store. We now proceed and we will use the word or in the at least one sense.

The situation (A or B ) is said to occur if at least one of the two situations A and B occurs. Otherwise, it is said not to occur. This is summarized in the following table.

row situation A situation B A or B
1 occurs occurs occurs
2 occurs occurs not occurs
3 occurs not occurs occurs
4 occurs not occurs not occurs not

The situation A or B can be said to occur
provided the situation in row 4 does not occur.
 

Remember the at least one usage differs from the exactly one usage of A or B which means either A or B occurs, but not both. In contrast, in the at least one usage, A or B means either A or B occurs, or both.

We have to be careful with the word or. Its meaning depends on the speaker and possibly the listener. That is, confusion and ambiguity results when two people in question use the same words but give them different meanings. To eliminate this ambiguity in everyday speech, write and say one of the following:

  • A or B, or both,
  • A and/or B
  • A or B, but not both.
When listening, you will have to ask what is meant. Legal texts use the phrase A and/or B to signal that at least one of the two cases A and B can occur.

2  One-Way Implications

Any rule which can be stated in the form if a first situation A occurs, then a second situation B occurs, in brief, if A then B or A implies B, is called a one-way implication.

A one-way implication which is never disobeyed is said to hold and to be (always) true. For a one-way implication rule if A then B, we recall the following:

  1. The rule is obeyed when both situations occur.
  2. The rule is not disobeyed when the first situation A does not occur but the second B occurs.
  3. The rule is not disobeyed when the first situation A does not occur and also the second situation B does not occur.
  4. The rule is disobeyed if the first situation A occurs but the second situation B does not.
The last two items 3 and 4 can be summarized by saying that disobeying a one-way implication rule is impossible when the first situation A does not occur. When not disobeyed, the rule is said to be obeyed by default. The following table, an occurrence table for the one-way implication rule if A then B, summarizes what has been said.

 

row situation A situation B if A then B
1 occurs occurs obeyed
2 occurs occurs not disobeyed
3 occurs not occurs not
disobeyed
4 occurs not occurs not not
disobeyed

In each row, a possible combination of the occurrence or nonoccurrence of the situations A and B is shown in the middle two columns. In the last column, we put a note to say whether or not the if-then rule is obeyed, disobeyed, or not disobeyed.

Row 2 represents the situation in which A occurs but B does not. Observe that in this situation, the rule is disobeyed. In the situations represented by the other three rows, the rule is not disobeyed. A one-way implication rule if A then B is said

  1. to be always true,

  2. to always hold

when it is never disobeyed. The one-way implication if A then B is always true when the situation described in row 2 in the above table never occurs.

Remark. If situation A never occurs, the implication rule if A then B is never disobeyed amd it is said to be vacuously true.

3  Two-Way Implication Rules

A rule which can be stated, or restated, in the form
  The first situation A occurs when and only when the second situation B occurs
or in the form
The first situation A occurs if and only if the second situation B occurs
  is called a two-way implication rule. For each two-way implication rule note that:

  1. The rule is obeyed when both situations occur.

  2. The rule is disobeyed when the first situation A occurs without the second situation B occurring.

  3. The rule is disobeyed when the second situation B occurs without the first situation A.

    4.  

  4. The rule is not disobeyed when both situations do not occur.

In brief, the two situations in a two-way implication rule must both occur or both must not occur, for the rule to be not disobeyed.

The next table summarizes the above remarks for any two-way implication rule A if and only if B.

row situation A situation B A if and only if B
1 occurs occurs obeyed
2 occurs occurs not disobeyed
3 occurs not occurs disobeyed
4 occurs not occurs not not disobeyed


As said before, a two-way implication rule is said to be always true when it is never disobeyed. This requires that the situations in rows 2 and 3 of the above table do not occur. That is, the above two-way implication rule A iff B is true (never disobeyed) provided neither of the situations A and B occurs without the other.

4  Converses to One-Way Implications

The converse to the implication rule if A then B is the rule if B then A. Note that interchanging the first and second situation A and B yields the converse to a rule. From this definition or perspective, we see that the converse of the converse is the original rule. Check this.

When we know a rule if A then B is never disobeyed, we have no guarantee that the converse rule if B then A is never disobeyed. The reason for this is as follows. The rule if A then B is true if the situation A never occurs without the situation B. The converse rule if B then A is true if the situation B cannot occur without the situation A.

Reminder. Now we can easily answer the following question: What can we say for sure about the event A when (i) the rule if A then B is never disobeyed, and (ii) the event B occurs? Your answer should be not much, or nothing, without further information.

Next:  Chapter 29, Contrapositive Form of Implication and Conditionals IF A THEN B

 

 

Three Skills
For 
Algebra
 Volume 2
Printed in Canada
ISBN 0-9697564-2-9
Read slowly,  this work may enrich your skills & knowledge. Take the risk.

Chapters and Appendices

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

Book Entrance

 

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

For Senior High School  & Calculus Students

  <| (o)   (o)   |> 
 \     | |      / 
\___ _/
||
 -/[]\- 
||
   / \_ 

Words  to clearly introduce algebra and variables have been missing in course design. For people who cannot do algebra, 
the missing words may explain or ease their difficulties.  Volume 2 ,Three Skills for Algebra,  in Chapters 8 to 14 & 18 etc, puts words before symbols to providing the missing words in a way that enrich the comprehension of all.  Those words form the middle part of a algebra (and logic) lessons aimed at helping or improving all of  high school mathematics and also calculus course design & delivery. 

For Avid Readers in School & Out - Online Books 
   1.  Elements of Reason. 1996 
1A. Pattern Based Reason  1995 
1B. Math Curriculum Notes 1996 
2. Three Skills for Algebra  1995 
3.Why Slopes & More.Math 1995
Tour their  forewords.   

Calculus Prep or Help: See Volumes 2 & 3, and this bigger Calculus Guide.  If your  calculus   questions is not answered here, submit it. Over time, that may complete the site development of calculus. 

For Parents: Speaking Skills, Reading & Writing,  Preparing for Science,  ends, values and methods for work and study,  parent- friendly maths skill development booklets for ages 4-14.

Mostly For High School

Intro to Solving Linear Equations 
- a different paths for junior and even senior high school students. Question for Tutors: When do you use and when you skip the stick diagram method here?

Fraction Skills,  thought-based  development, Ages 10 to 14 may need a tutor.  Students who have to understand in order to do may like the development in all or part. 

For Senior High School Mathematics & Calculus

5 wordy Logic Chapters
4 curious Algebra Chapters
Words before & besides symbols. A Key Algebra forward & backwards Chapter    

First Calculus Preview (1st intro)
Four Calculus Chapters  (2nd intro)
Intro to Complex Numbers (long)
Intro to Mathematical Induction (romantic & wordy at first)

Tutors & Instructors: These lessons introduce skills differently Would you recommend them? 

More Topics 

 1. Decimal Arithmetic  Reference!
2. Integers - Intro to Signed No.s
3.  Fractions - fully explained.
4.  Fractions  with Units  
5.   Number Theory, 
6.    Solving Linear Equations  
7  Formulas for- & backwards -  
8.  Proportionality, Back- & For-wards.   
9. Logic Chapters:   
10.  Euclidean-Geometry  
11.  Slopes & Equations of Straight Lines.  (Take I. See take II below)
12.  Why Study Slopes. 
13. Maps, Plans,  Similarity & Trig,  
  (Take II included here)
14.  Quadratics: Starter lessons
15.  Polynomials: Starter lessons 
16 Why Factor Polynomials:  
17   Functions - Forwards & Backwards.  
18.  Exponents, Radicals & logs.  
19. Complex Numbers before trig (new advance/ starter lesson)
20.  DC Electric Circuits Etc 
21. Real  Analysis 
22. The Olde Complex No, Trig
& Vector Section.
23. More Calculus Stuff
- written after Volumes 2 and 3.

Level I Material: New Stuff
Time and Date Matters
 Level I Arithmetic. 
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
 Logic Chapters (leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of Maps, Plans,  Similarity & Trig,  to appear here).

For Instructors
- Education Essays   (opinions, possibilities, references) 
- Free Advice and Directions for teaching primary & high school maths will be given in online meeting place with voice & whiteboard.   
- Math & Logic  How-TOs 
1. Arithmetic
2. Algebra
3. More Algebra
4.  Beginner Geometry
5.  More Geometry
6. Calculus 
7. Show Work or Logic 
 These may be too dense for students.

Offering ideas to change education makes this site different.  Nothing ventured, nothing gained.  Site material is mathematically  correct, and where not, please report errors. The two level program POMME in the site entrance implies multiple paths for instruction. Supporting those paths in turn implies a clear destination  for site development and perhaps a new name.

 

www.whyslopes.com > Volume 2,  Three Skills for Algebra  1995  >   28 Occurrence Tables     [ Back ] [ Next ]

Road Safety Message   Walk on a side walk. If that is not possible, try  not to  walk on a road with your back to the traffic.
Try to see what  trucks, cars, buses or bicycles are coming, so that you may step out of their way.  Put safety first. .

Support for Technical Mathematics from Number Theory to Calculus Prep
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
  1. Geometric Sums Etc,
  2. Notation For Sums,
  3. Personal Money Maths and
  4. Some Finite Mathematics
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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