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

(A =⇒ B) ⇐⇒ (not B =⇒ not A).

Remember, the arrow ⇐⇒ means that the implication is true in both directions. This means that we
need to prove two things:

1. Whenever the statement A implies B is true, then the statement not B implies not A is
also true. We can write this as:

AimpliesB =⇒ notB impliesnot A.

2. Whenever the statement not B implies not A is true, then the statement A implies B is also
true, which we can write as

notB impliesnot A=⇒ AimpliesB.

To prove 1 we start by assuming that A implies B, which means that whenever A is true then B is also

true. So if B is not true, then A cannot be true, because every time A is true, B follows. Therefore every
time B is not true, A is also not true, which means not B implies not A.

Now we need to prove 2, so we start by assuming the statement “not B implies not A” is true. This means
that whenever B is not true, then A is not true. So if A is true, then B must also be true, because every
time B is not true, A is also not true.

♥ 5

Traditionally the end of a proof is signified by , but in this text readers are encouraged to think outside
the , so we’ll end our proofs with ♥. The words converse and contrapositive may sound similar at first,

but they have very different meanings. The words themselves can help us remember their meanings.

• Converse rhymes with reverse, and the converse means that the implication (the direction of
the arrow) is reversed.

• The contrapositive is like a double negative: the result is positive. First, both A and B are
negated to (not A) and (not B). Next, the direction of the arrow of implication is reversed,
so the implication goes from (not B) towards (not A). This double negative:

(not B) =⇒ (not A)

is equivalent to the positive statement: A =⇒ B.

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