Appetizers and Lessons for Mathematics and Reason 
New Visitors: Visit site entrance www.whyslopes.com to see what  1000+ pages offer. 

29 Contrapositive
Back ] Book Entrance ] Next ]



Three Skills
For 
Algebra
Volume 2

Vol 2, Three Skills for Algebra covers many  topics in algebra and logic that students starting calculus should have mastered or will have to master. Also includes arithmetic review problems to catch common mistakes of students starting calculus.  A fourth skill in this misnamed volume gives a unifying theme for high school maths.

Chapters and Appendices

Book Entrance

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

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

Chapter 29
The Contrapositive

Previous Chapter: Proofs and Logic- 28 Occurrence Tables

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

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

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

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.

3  Vacuously True

The one-way implication rule If A then B is said to be vacuously true if and only if the situation A never occurs.

The contrapositive If NOT B then NOT A is vacuously true if and only if the situation NOT B never occurs, that is if and only if the situation B always occurs. Therefore an implication rule and its contrapositive are vacuously true in different circumstances.

Finally, an innovation perhaps, the two-way implication rule A if and only if B is said to be vacuously true in the situation where A and B are both always true or both always false.

An implication rule says that when a first situation A occurs then so must a second situation B. The associated contrapositive implication rule says that when the second situation B does not occur, then the situation A cannot occur. The previous part of this chapter explains why an implication rule is never disobeyed if and only if its contrapositive is never disobeyed. In consequence, a chain of reasoning which shows the contrapositive form of an implication rule is never disobeyed also shows the implication rule is never disobeyed.

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.)

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.


Next: Chapter 30 Truth Tables, Revisited

 
Schools/Colleges:  Hire the site author, as an online instructor, a technical support for teachers, or advisor for curriculum review.    Site Reviews may serve as references.  See how online whiteboards with  voice and real-time writing make online help possible - board content printable.  Text or written work scanned or saved to a  pdf  may be  uploaded  for discussion in the whiteboard.  

www.whyslopes.com

Parents: Help your Child/Teen Learn

Online Volumes
 
(orders)
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

Math How-TOs etc  2008
1. Arithmetic
2. Algebra 
3. More Algebra 
4. Geometry  
5. More Geometry
6. Calculus

Site Description/Reviews  by 3rd parties

Site  Math Lessons
1. Arithmetic Flash Videos  11-2008
2.  Algebra Videos (to appear)
3. Fractions and More 
4.. 
Solving Linear Equations  04-2005
5. Euclidean-Geometry To Complex No.s 
6.  Analytic Geometry/Functions 2006
7.  Number Theory. 2006-7
8.
  Exponents, Radicals & logs. 2008
9 Calculus  2005
10..Real  Analysis 1995
11 Electric Circuits Etc  2007
12. .Algebra, Odds & Ends, HS level-2001
13.Maps, Plans,  Similarity &Trig, with
Complex   Numbers
, 12-2009. 

For Math Instructors/Tutors/
Curriculum Committees


1. K0-11Applied Math Program Outline  
2. Mathematics education  essays 
3. LAMP - an earlier applied math program.
4.
(150 pages)

www.whyslopes.com/search

Would you like to show yourself or others how to be an  algebra power users?

 Back ] Up ] Next ] [Top of this Page]  

Road Safety Message  Do not walk on a road with your back to the traffic - rule of thumb
Please report by
email,  errors in mathematics or grammar or terminology to site author
If a mathematics topic you need is not covered in site pages,  report that as well. Topics in most demand
will be covered first in site growth.  

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,  
Mathematics Consultant/Tutor/Instructor, All Rights Reserved.