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Blast into Math! Pure mathematics: the proof of the pudding is in the eating

In the converse, the direction of the arrow, which is the direction of implication, is reversed.

Let’s do some examples!

1. Statement: if 2+ x =4, then x =2. It is helpful to first put the statement in the form A
implies B and to determine A and B. As a general rule whatever immediately follows the
word “if” is the hypothesis which we call A, and whatever immediately follows the word
“then” is the conclusion which we call B. So in this example, “A” is “2+ x =4,” and “B” is
“x =2.” We can write the statement as

2+ x =4 =⇒ x =2.

To form the converse, we simply reverse the arrow of implication. So, the converse is

x =2 =⇒ 2+ x =4,

which means: if x =2, then 2+ x =4.

2. Statement: if you have the flu, then you are ill. This is no longer a mathematical statement,
but we can nonetheless form its converse because it is a statement of the form “A implies B.”
What immediately follows the word if is: you have the flu. So, A is in this case: you have the

flu. What immediately follows the word then is: you are ill. So B is: you are ill. We can write
this succinctly as:

youhavethe flu =⇒ youare ill.

To form the converse we simply reverse the arrow, so the converse is

youare ill =⇒ youhavethe flu,

which means: if you are ill, then you have the flu.

In the last example the original statement is true because the flu is a type of illness, so if you have the flu,
then you are ill. Is the converse true? Well, if you’re ill (which I sincerely hope is not the case), then you
might have the flu or you might have something else, like a cold or Appendicitis. In general a statement
and its converse have different meanings, and if a statement is true then its converse need not be true.
However, it can happen that both a statement and its converse are true.

In the first example, the statements 2+ x =4 and x =2 are equivalent This means that both

2+ x =4 =⇒ x =2

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