Book contents
- Frontmatter
- Contents
- Preface
- I Study skills for mathematicians
- II How to think logically
- III Definitions, theorems and proofs
- IV Techniques of proof
- 20 Techniques of proof I: Direct method
- 21 Some common mistakes
- 22 Techniques of proof II: Proof by cases
- 23 Techniques of proof III: Contradiction
- 24 Techniques of proof IV: Induction
- 25 More sophisticated induction techniques
- 26 Techniques of proof V: Contrapositive method
- V Mathematics that all good mathematicians need
- VI Closing remarks
- Appendices
- Index
26 - Techniques of proof V: Contrapositive method
from IV - Techniques of proof
- Frontmatter
- Contents
- Preface
- I Study skills for mathematicians
- II How to think logically
- III Definitions, theorems and proofs
- IV Techniques of proof
- 20 Techniques of proof I: Direct method
- 21 Some common mistakes
- 22 Techniques of proof II: Proof by cases
- 23 Techniques of proof III: Contradiction
- 24 Techniques of proof IV: Induction
- 25 More sophisticated induction techniques
- 26 Techniques of proof V: Contrapositive method
- V Mathematics that all good mathematicians need
- VI Closing remarks
- Appendices
- Index
Summary
If it isn't hurting, it isn't working.
Prime Minister John Major's comment after being told that his government's policies were hurting the countryThe contrapositive method is very confusing to grasp initially – however, one student told me that she liked using it as it made her feel that she had done something clever. I think this feeling comes from the fact that the method is very indirect.
We saw in Chapter 8 that the statement ‘A ⇒ B’ is equivalent to ‘not B ⇒ not A’. The contrapositive method is the use of this equivalence. The indirectness and feeling of being clever is that in using it to prove ‘A ⇒ B’ we start with not B (and proceed to show that not A is true).
Since this idea causes a lot of confusion for beginners we shall start with some revision of Chapter 8 and show, via a different proof to the one in that chapter, that a statement and its contrapositive are equivalent. We shall then see the contrapositive in action.
Revision of the contrapositive
Definition 26.1
The contrapositive of the statement ‘A ⇒ B’ is
‘not(B) ⇒ not(A)’:
Examples 26.2
(i) ‘I am Winston Churchill implies I am English’ has contrapositive ‘I am not English implies I am not Winston Churchill.’
(ii) ‘If I am Jane, then I am a woman’ has contrapositive ‘If I am not a woman, then I am not Jane.’
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- Information
- How to Think Like a MathematicianA Companion to Undergraduate Mathematics, pp. 180 - 184Publisher: Cambridge University PressPrint publication year: 2009