Book contents
- Frontmatter
- Contents
- Preface
- I Study skills for mathematicians
- II How to think logically
- 6 Making a statement
- 7 Implications
- 8 Finer points concerning implications
- 9 Converse and equivalence
- 10 Quantifiers – For all and There exists
- 11 Complexity and negation of quantifiers
- 12 Examples and counterexamples
- 13 Summary of logic
- III Definitions, theorems and proofs
- IV Techniques of proof
- V Mathematics that all good mathematicians need
- VI Closing remarks
- Appendices
- Index
9 - Converse and equivalence
from II - How to think logically
- Frontmatter
- Contents
- Preface
- I Study skills for mathematicians
- II How to think logically
- 6 Making a statement
- 7 Implications
- 8 Finer points concerning implications
- 9 Converse and equivalence
- 10 Quantifiers – For all and There exists
- 11 Complexity and negation of quantifiers
- 12 Examples and counterexamples
- 13 Summary of logic
- III Definitions, theorems and proofs
- IV Techniques of proof
- V Mathematics that all good mathematicians need
- VI Closing remarks
- Appendices
- Index
Summary
But the fact that some geniuses were laughed at does not imply that all who are laughed at are geniuses. They laughed at Columbus, they laughed at Fulton [steamboat inventor], they laughed at the Wright brothers. But they also laughed at Bozo the Clown.
Carl Sagan, Broca's Brain, 1979Statements of the form A ⇒ B are at the heart of mathematics. We have seen that for an implication A ⇒ B we can take its inverse (not(A) ⇒ not(B)) and its contrapositive (not(B) ⇒ not(A)). In this chapter we will look at another implication: B ⇒ A; this is called the converse of A ⇒ B. We shall see that a statement and its converse are not the same. One may be true and the other false, both may be true or both may be false.
If A ⇒ B and B ⇒ A are both true, then we say that A and B are equivalent statements. Mathematicians really like equivalent statements, particularly if the A and B seem to have no obvious connection.
The converse
Definition 9.1
The converse of the statement ‘A ⇒ B’ is ‘B ⇒ A’.
The converse of
‘If I am Winston Churchill, then I am English’
is
‘If I am English, then I am Winston Churchill.’
This simple example shows that, even if a particular statement is true, its converse need not true. It may be true or it may not be true. Investigation is required before we can say.
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- Information
- How to Think Like a MathematicianA Companion to Undergraduate Mathematics, pp. 75 - 79Publisher: Cambridge University PressPrint publication year: 2009