Help Haiti.    Help flood victims in Pakistan

Appetizers and Lessons for Mathematics and Reason
by A. Selby, Ph. D.   Feedback & Questions

 20 pages in French: Algèbre  
 Définition d'une variable  
La raison basée sur les  règles et modelés

www.whyslopes.com > Volume 1A, Pattern Based Reason  >   22. Contrapositive     [ Back ] [ Next ]

Chapter 22
The Contrapositive

Previous: Chapter 21, Occurrence Tables for Implications

  • PS: Use of the contrapositive form of an implication B IF A provides one form of indirect reason.
  • PS: The occurrence table in the earlier chapter 21 for B IF A will be used to explain or provide a justification for truth tables for material implications B IF A (or equivalently, IF A THEN B).

1  Introduction

In the chapter Implication Rules, we asked the following question: What can you say for sure about Aunt Jane when Tom does not go out to play and the following rule is never-disobeyed:

Each time Aunt Jane visits her nephew Tom's house, Tom goes out to play.

The answer was: NOT Aunt Jane visit. That is, when the previous rule holds, the following rule also holds

Each time her nephew Tom does not go out to play,
Aunt Jane does not visits Tom's house.
 

This is a contrapositive way or form of writing the original rule.

With the foregoing in mind, we can define the contrapositive way of writing other implication rules. The contrapositive form of writing the implication (or conditional statement) if A then B is if NOT B then NOT A. For example, the contrapositive way of writing
 

if Aunt Jane visits her nephew Tom's house
then Tom goes out to play
is

if NOT (Tom go out to play) then 
NOT (Aunt Jane visits her nephew Tom's house).
 

Language (or grammar) courses would prefer us to write

if (Tom does not go out to play) then 
(Aunt Jane does not visit her nephew Tom's house).

2  Equivalence of a one-way implication with its contrapositive

Note that a hint or preview of the contrapositive was provided by the discussion of the first logic puzzle (questions 4 and 5) in the chapter Implication Rules. (You might wish to revisit that puzzle.)

The occurrence table below is intended to show you that if an implication rule if A then B is true (never disobeyed) then the contrapositive rule if NOT B then NOT A is true (never disobeyed), and vice versa.

row A B if A 
then B		 NOT B NOT A if NOT B
then 
NOT A
1 occurs occurs obeyed occurs

not

 occurs

not

 not

disobeyed
2 occurs occurs 
not		 disobeyed occurs
 not		 occurs disobeyed
3 occurs
 not		 occurs not 
disobeyed		 occurs 
not		 occurs not 
disobeyed
4 occurs
 not		 occurs
 not  		 not
  disobeyed		 occurs occurs obeyed
Table for the contrapositive assertion:
(A implies B)
if and only if
(NOT B implies NOT A).

Filling The Table

First we look at the four combinations of the occurrences of the situations A and B. When A occurs we have two possibilities for B. When A does not occur, we have two possibilities for B as well. This gives a total of four cases or rows and fills in the first three columns.

In the fourth column, headed by the rule if A then B for each combination of occurrences of A and B, we note if the rule is obeyed, disobeyed or not disobeyed.

Next, in the fifth and sixth columns headed by situations NOT B and NOT A, for each of the four combinations we note if these situations occur or not.

In the last column, we finally note if the rule if NOT B then NOT A is obeyed, disobeyed or not disobeyed. The entries in the last column depend on those in the fifth and sixth columns. The entries in the latter two in turn depend on those in the previous columns.

Answers to Two Questions

Now we can answer the questions: when are the two one-way implication rules (if A then B) and (if NOT B then NOT A) true? Remember we say these implication rules are true if each is never disobeyed. Both implications are true, that is, never disobeyed, when the situation row 2, A and NOT B, never occurs. Both implications are false when the situation in row 2, namely (A and NOT B), occurs. So we conclude from the table that the two rules are equivalent: each implies the other.1

1The rule if NOT B then NOT A is disobeyed if the NOT B occurs but NOT A does not. That is, it is disobeyed precisely when B does not occur, while A does. But the rule if A then B is disobeyed precisely in this situation where A occurs and B does not. This tells us that both rules are not disobeyed provided the situation where A occurs and B does not never occurs. So if one rule is true (never disobeyed), then so is the other.

Question

Recall that the rule if NOT B then NOT A is called the contrapositive way of saying if A then B. What is the contrapositive of the contrapositive? The answer is essentially the original implication: why? Hint: Replace NOT (NOT A) by A in the statement of the contrapositive of the contrapositive.

Next: Chapter 22, Part II, Vacuously True Implication Rules

 

Pattern
Based
Reason
 Volume 1A
Printed in Canada
ISBN 0-9697564-5-3

Volume 1 = 1A+1B
bounded together


Table of Contents
Foreword
PS. Three Remark
1. What is reason
2. Inductive Ed Principles
2. Communication
3. Elements of Reason
4. Implication Rules [10]
5. Hype & Deception
5. Hype & Ethics
6. Chains of Reason [4]
7.  Longer Chains of Reason
7. Mathematical Induction
8. Language Change [2]
9. Next Chapters, About.
10. Limits to Freedom [2]
11. Accidental Patterns
12. Two Analogies
12.  Knowledge Islands
13. Euclidean Model
13. Euclidean Reason
14 Math: Deductive/Empirical [6]
15. Objectivity
15. Objectivity, More
16 Rules-Patterns Origins [10]
Knowledge & Story Telling
17. Objective Ways
17. Trial & Error Discovery
18. Conciousness
19. Symbols & Logic
20. Pronouns & Symbols
21. Truth Tables I. [3]
22. Contrapositive
22. Vacuously True
24. Indirect Reason More
24PS. Excluded Middle Law
24PS.  Proof by Absurdity
PS. Reality vs Imagination
PS. Ahistorical Logic
Links Elsewhere - Go GoGo

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
+History Lost or Missing

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 1A, Pattern Based Reason  >>  22. Contrapositive     [ 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.

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby,  All Rights Reserved.