Page 308 - HOW TO PROVE IT: A Structured Approach, Second Edition

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294 Mathematical Induction
An important property of the natural numbers that is related to mathematical
induction is the fact that every nonempty set of natural numbers has a smallest
element. This is sometimes called the well-ordering principle, and we can prove
it using strong induction.

Theorem 6.4.4. (Well-ordering principle) Every nonempty set of natural num-
bers has a smallest element.
Scratch work

Our goal is ∀S ⊆ N(S = ∅ → S has a smallest element). After letting S
be an arbitrary subset of N, we’ll prove the contrapositive of the conditional
statement. In other words, we will assume that S has no smallest element and
prove that S = ∅. The way induction comes into it is that, for a set S ⊆ N,to
say that S = ∅ is the same as saying that ∀n ∈ N(n /∈ S). We’ll prove this last
statement by strong induction.
Proof. Suppose S ⊆ N, and S does not have a smallest element. We will prove
that ∀n ∈ N(n /∈ S), so S = ∅. Thus, if S = ∅ then S must have a smallest
element.
To prove that ∀n ∈ N(n /∈ S), we use strong induction. Suppose that n ∈ N
and ∀k < n(k /∈ S). Clearly if n ∈ S then n would be the smallest element of
S, and this would contradict the assumption that S has no smallest element.
Therefore n /∈ S.

Sometimes, proofs that could be done by induction are written instead as
applications of the well-ordering principle. As an example of the use of the
√
well-ordering principle in a proof, we present a proof that 2 is irrational. See
exercise 2 for an alternative approach to this proof using strong induction. See
exercise 16 for another application of the well-ordering principle.

√
Theorem 6.4.5. 2 is irrational.
Scratch work

Because irrational means “not rational,” our goal is a negative statement, so
√
proof by contradiction is a logical method to use. Thus, we assume 2is
√
rational and try to reach a contradiction. The assumption that 2 is rational
√ √
means that there exist integers p and q such that p/q = 2, and since 2is
positive, we may as well restrict our attention to positive p and q. Because
this is an existential statement, our next step should probably be to choose
√
positive integers p and q such that p/q = 2. As you will see in the proof,
√
simple algebraic manipulations with the equation p/q = 2 do not lead to any