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from II - How to think logically
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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