Lecture 6

Induction

The basics of induction

The most obvious interesting thing about the natural numbers is that it’s natural to start listing them, one after the other: N={0,1,2,3,4,}.\mathbb{N}= \left\{ 0, 1, 2, 3, 4, \ldots \right\}. This, of course, is how counting works.

It turns out that this way of thinking about the integers gives us a very powerful tool for proving things one integer at a time: the principle of mathematical induction, usually known to mathematicians simply as induction.

Informally, I like to think of the following example:

If I can reach the bottom (rung number zero?) of a ladder,
and if I’m on any rung I can reach the next rung up,
then I can reach any rung on the ladder.

Why, for example, can I reach the fourth rung? One can imagine a detailed proof of this, as follows:

The connection with counting is obvious: our proof visibly counts up to four.

Writing that out was okay, but you are probably glad I didn’t write out a proof that we could reach the hundred and seventy-eighth rung. I suppose that we could do so, writing “and so on” at some point: but that’s a little vague (what about situations where it isn’t obvious what “and so on” means)?

It’s helpful to have a way which isn’t vague.

So here’s a formal version:
Definition: [Induction] Let P(n)P(n) be a statement that depends on a natural number nn. Then, if

  1. the statement P(0)P(0) is true, and

  2. for all kNk\in\mathbb{N}, if P(k)P(k) is true, then P(k+1)P(k+1) is true,

then the statement P(n)P(n) is true for all nNn\in\mathbb{N}.

Here are some useful words:

We call part () the base case, and part () the induction step. These words agree quite well with our mental picture of a ladder!

When we are trying to prove the induction step P(k)P(k+1)P(k)\Rightarrow P(k+1) we refer to P(k)P(k) as the induction hypothesis.

Example of induction

We’ll prove many things by induction in this course, but this is one:

Proposition

For any natural number nn, we have the following formula for the sum of the first nn positive integers: 1+2++(n1)+n=n(n+1)2.1+2+\cdots+(n-1)+n = \frac{n(n+1)}{2}.

Let P(n)P(n) be the statement above for some particular nn.

So P(3)P(3) is the statement that says 1+2+3=3×4/21+2+3 = 3\times 4/2, and P(10)P(10) is the statement that 1+2+3+4+5+6+7+8+9+10=(10×11)/2.1+2+3+4+5+6+7+8+9+10 = (10\times 11)/2. Notice that P(n)P(n) is not a number, it’s a statement.

Proof

We will prove P(n)P(n), which says that 1+2++(n1)+n=n(n+1)21+2+\cdots+(n-1)+n = \frac{n(n+1)}{2} for all nn by induction.

For our base case, P(0)P(0) says that the sum of no integers at all is 0×1/20\times 1/2, which is true, as the sum of no integers is zero.

Now we will do our induction step, proving P(k)P(k+1)P(k)\Rightarrow P(k+1) for all kk. Suppose P(k)P(k) is true: we need to show that P(k+1)P(k+1) is true.

The statement P(k)P(k) tells us that 1+2++(k1)+k=k(k+1)2.1+2+\cdots+(k-1)+k = \frac{k(k+1)}{2}. We need to prove P(k+1)P(k+1), which would say that 1+2++(k1)+k+(k+1)=(k+1)(k+2)2.1+2+\cdots+(k-1)+k+(k+1) = \frac{(k+1)(k+2)}{2}. Now note that 1+2++(k1)+k+(k+1)=(1+2++(k1)+k)+(k+1)=k(k+1)2+(k+1)(by the induction hypothesis)=k(k+1)+2(k+1)2=(k+1)(k+2)2.\begin{aligned} & 1+2+\cdots+(k-1)+k+(k+1)\\ =& \left(1+2+\cdots+(k-1)+k\right)+(k+1)\\ =& \frac{k(k+1)}{2}+(k+1)\quad\text{(by the induction hypothesis)}\\ =& \frac{k(k+1)+2(k+1)}{2}\\ =& \frac{(k+1)(k+2)}{2}.\end{aligned} This is exactly the statement P(k+1)P(k+1), which is what we needed for the induction step, and that completes the proof.

You may know other ways of proving that. (I can think of a few.) But I hope you’re impressed with this as a strong potential method for proving identities.

Nonexamples of induction

Let’s now try proving some completely false statements using induction. The plan is (of course) not to succeed, but to understand where we need to be careful.

Clearly this statement is complete and utter rubbish.

If you believe that induction is a reliable method of proof (and I do, and I hope you do too), then it had better be the case that we’re not using induction correctly.

Anyway, here’s an induction “proof”. Suppose that k=k+83k = k+83 for some kk. We’ll prove that (k+1)=(k+1)+83(k+1) = (k+1)+83. But we have k+1=(k+83)+1(by assumption)=(k+1)+83(by rearrangement).\begin{aligned} {}k+1 &= (k+83)+1\qquad\text{(by assumption)}\\ {}&= (k+1)+83\qquad\text{(by rearrangement)}.\end{aligned} This completes the proof.

What’s the problem with the argument above?

There’s no base case.

If you don’t have a base case, such as P(0)P(0), then it’s of no use to prove that P(k)P(k+1)P(k)\Rightarrow P(k+1) for all kk. It’s no use to be able to climb a ladder if the bottom of the ladder is unreachable.

Here’s another, more subtle example:

Again, we find ourselves hoping very strongly that there’s a mistake in the use of induction in what follows. I’ll write it out and we can see if we can spot it.

In order to do this, we’ll let P(n)P(n) be the statement “Given any nn horses, all of them have the same colour”. We’ll prove P(n)P(n) for all nn by induction: that will give us what we want, because we can take nn to be the number of horses in the world.

We’ll take P(1)P(1) as the base case of the induction. This is the statement “Given any one horse, all of them have the same colour”: this is obviously true.

Now we’ll prove the induction step. We will assume that P(k)P(k) is true (“given any kk horses, all of them have the same colour”): our job is to prove that P(k+1)P(k+1) is true (“given any (k+1)(k+1) horses, all of them have the same colour”).

So suppose we have (k+1)(k+1) horses. Name two of them Alice and Zebedee.

image

Excluding Alice, there are kk horses, which all have the same colour, by the induction hypothesis. So all the horses except Alice have the same colour as Zebedee.

Also, excluding Zebedee, there are kk horses, which all have the same colour, again by the induction hypothesis. So all the horses except Zebedee have the same colour as Alice.

Hence all the horses except Alice and Zebedee have the same colour as both Alice and Zebedee, which says that all the horses have the same colour. That ends the proof.

What’s wrong with this?

The particular case P(1)P(2)P(1)\Rightarrow P(2) doesn’t work.

I find this surprisingly subtle.

In fact, it’s a parody of a valid style of argument. If it is the case that any two things are the same, then we could prove using exactly this method that they’re all the same. In fact, this is something you already know, since “all are alike” and “no two differ” are synonymous phrases.

Variants

There are other techniques which use the same idea of induction, but not quite the same formal principle as I’ve written out above.

We can start with a base case which isn’t P(0)P(0). For example, if

  1. P(15)P(15) is true, and

  2. P(k)P(k) implies P(k+1)P(k+1) for all k15k\geq 15,

then P(n)P(n) is true for all n15n\geq 15.

Perhaps you want to think of that as saying “if have a door which leads to the fifteenth rung of a ladder, and you know how to climb ladders, then you can get to every rung above the fifteenth”.

Actually, this is really just ordinary induction in disguise.

Indeed, if we define Q(n)Q(n) to be P(n+15)P(n+15), then proving Q(n)Q(n) for all nNn\in\mathbb{N} by induction is the same as proving P(n)P(n) for all integers n15n\geq 15.

Perhaps you want to think of that as “ignoring all the bits of the ladder below the fifteenth rung, imagining the ladder starts outside your door, and starting counting rungs from there”. Or perhaps you’re bored of the ladder analogy now.

Actually, you should have been prepared for this variant: my induction proof that “all horses have the same colour” started with 11, not 00. (Okay, that proof was wrong. But there was nothing wrong with that bit of the proof: there’s nothing wrong with induction starting from 11. It was something else that was wrong).