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

In mathematics when we say that a sentence is true, this means that it is always true in every possible case.
If a sentence is only sometimes true, then it is not true in the mathematical sense, so in the mathematical
sense it is false. A mathematical sentence is either true or false. To prove that a mathematical sentence is
true, we must demonstrate that it is true for every possible case. Let’s do an example. After a long drive
through the Swedish countryside we make the conclusion that every house in Sweden is red, because
every house we have seen after driving a few hundred kilometers was red.However, we have not proven

the statement “every Swedish house is red,” because we have not proven that the statement is true in
every possible case. To disprove a statement we just need to find one case (one house) for which the
statement is false, but to prove the statement we would need to visit every house in Sweden! We would

also need precise definitions of “house” and “red.” In mathematics we can prove statements because we
have precise definitions; these are mathematical words. However, we must always proceed carefully and
thoroughly because a proof must hold for every case (for every house).

Using negation, we can form the contrapositive of the statement A implies B.

Definition 2.3.5. The contrapositive of the statement A implies B is the statement not B implies not A.

The contrapositive is particularly useful because A implies B is equivalent to its contrapositive; let’s
prove this!

Proposition 2.3.6. (Contrapositive Proposition) The statement A =⇒ B is equivalent to its
contrapositive.

Proof: We must prove that A implies B means the same thing as not B implies not A. We can write this as

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