CSCE 235

Handout 16: Common Mistakes in Proof by Induction

February 27, 2004

There are many types of problems that we have talked about in this class: the summation of a series of numbers, proving that some numbers are divisible by k, recursive definitions, and so on. Common mistakes include not setting up the problem correctly.

I will repeat the strategy here to supplement Handout 15 (February 25 2004). First, the strategy consists of three steps:

(1) You want to establish the basis. Show that it is true by inspection. Why can we simply show it by inspection? This is because usually the basis is very simple to prove: by plugging in the numbers.

(2) You want to establish the induction step. Show that it is true. Show that the induction hypothesis holds. How do you do that? You have to first define what the nth proposition is, and then given that proposition, you show that the (n+1)th proposition is also true.

(3) Finally, you want to conclude by saying “By the Principle of Mathematical Induction, we …”.

Prove that the following is true using the Principle of Mathematical Induction:

for all .

First, we want to establish the basis. For ,

So, the basis is true by inspection.

Next, we want to establish the induction step. [Now, we need to say what the induction hypothesis is, or what we know is true.] We know that for some , holds. [Why? Because we have just shown in the Basis that the hypothesis does hold for !] We want to show that for , the following is also true:

.

[But wait a minute! How do we come up with the above? Why do we use ? If we go back to what has been taught in the class early in our Induction topic, we know that we want to do this: given the nth proposition, you want to show that the (n+1)th proposition is also true. All this is saying is: given a proposition or hypothesis for a current case that is true, you want to show that the next proposition is also true. The current case in our problem is when and the next case is when !!! Don’t confuse the n in our discussion with the n in the equation! These are simply variables in different contexts. Now, let’s look at the second issue. Some students had trouble identifying what the nth proposition is, and what the (n+1)th proposition is. Given the summation type, we can actually rewrite it this way:

That is, we consider the summation of a series of numbers up to (and including) the mth term where . That is the mth proposition. Note the notation change. We cannot use n here because it has been used in the equation! So, for the (m+1)th proposition, we consider the summation up to the (m+1)th term! So we have

This same strategy applies to other types of Induction problem. So, do not get fixated at this particular type of Induction problem. You must understand why this is done in order to deal with other types of Induction problem.]

To show that the proposition for is true,

.

[Now, do you understand why we can write the following equation in the above:

?

Some of you may not fully understand this and thus make some grave mistakes in the proof. What is ? The summation can be re-written as:

Now, so what? Well, if you look carefully, the summation of all terms up to the mth term is something that we already know! That is:

.

So, that means ! This is why we can write what we write. You must understand this. Once you understand this, you can apply this idea to other types of Induction problem! Note also that this is what drives the induction: you use what is known to be true for the current proposition to prove the next proposition! This is the core of induction, the core that gives induction its power! Conceptually, it says that we can break down a problem into two parts: one consists of a new term, and the other consists of a term that we know the answer for. So, now, we can substitute the answer for that term into the mix, incorporating it with the new term, driving towards the goal state (i.e., showing the next proposition holds).]

Thus, we have shown that the induction step holds: is true when is true.

Therefore, by the Principle of Mathematical Induction, we have shown that is true for all .