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Handout 10: Induction

February 26, 2002

Now, we look at induction. Mathematical induction is a method of proof which we can often use when we want to establish the truth of an infinite list of propositions. The method is a natural one to use in a variety of situations in computer science as well. Some applications are quite mathematical in flavor, such as verifying that a certain formula holds for all positive integers. Another frequent use of the method is to show that a computer program or algorithm with loops performs as expected.

Induction?

Consider a video game which begins with a spaceship in the middle of the screen. In 5 seconds, an alien appears. 5 seconds later, the alien splits into two, and then five seconds, later, each of these aliens splits into two, and so on. Every 5 seconds, the number of aliens doubles. The player’s task is to eliminate the aliens before they fill the screen. Now after 30 seconds, assuming that the play is not very skillful and kills no aliens, how many aliens will there be 30 seconds after the game starts? 1 2 4 8 16 32 – 32 aliens. What about after 5 minutes? We know that . And it seems reasonable to guess that after 5 minutes.

The method of mathematical induction applies to just such situations, ones in which

1. We know the answer in the beginning,

2. We know how to determine the answer at one stage from the answer at the previous stage, and

3. We have a guess at the general answer.

Principle of Mathematical Induction

Consider a list , , , …, of propositions indexed by P. All the propositions are true provided

Basis (B) is true;

Induction (I) is true whenever is true.

We will refer to the fact that is true, as the basis for induction and we will refer to is true whenever is true as the inductive step (or the induction hypothesis). In the notation of the propositional calculus, the inductive step is equivalent to:

The implication is true for all n in P.

Note: The principle is not itself a proof that is true for all n, but it tells us that if we can somehow show (B) and (I) then all ’s are true. There is no free lunch. The work goes into showing the basis and the induction, which must be verified before the Principle of Mathematical Induction can be applied. In practice, the basis will usually be easy to check.

So, remember that there are two basic ingredients for a valid induction proof: the basis and the inductive step. In addition, if there is any possible doubt, it should be made clear that one is giving a proof by induction.

Steps

(1) Verify the statement for . This is the basis for induction, i.e., show that is true.

(2) Write down the induction step (hypothesis) (the statement for n = k) in the form “Now suppose that …” and be explicit about what is being assumed.

(3) Write down what is to be proven (the statement for n = k+1) in the form “We must show that … “ again being very explicit about what is to be shown; and finally (after waiting a second);

(4) Give a convincing argument as to why the statement for n = k + 1 is true (and make sure this argument uses the induction hypothesis).

(5) Conclude with “By the Principle of Mathematical Induction, all of the propositions are true.”

Based on (Goodaire and Parmenter 2002, and Ross and Wright 1988).