Maggie Johnson
Handout #29
CS103A
Introduction to Induction
Key topics:
* Introduction and Definitions
* Examples of Weak Induction
* Proper Proof Form
* Examples of Strong Induction
* A Faulty Proof and Some Interesting Proofs
* Some Useful Formulas
One of the most important tasks in mathematics is to discover and characterize regular patterns or
sequences. The main mathematical tool we use to prove statements about sequences is
induction
.
Induction is a very important tool in computer science for several reasons, one of which is the fact
that a characteristic of most programs is repetition of a sequence of statements.
To illustrate how induction works, imagine that you are climbing an infinitely high ladder. How do
you know whether you will be able to reach an arbitrarily high rung? Suppose you make the
following two assertions about your climbing abilities:
1) I can definitely reach the first rung.
2) Once I get to any rung, I can always climb to the next one up.
If both statements are true, then by statement 1 you can get to the first one, and by statement 2,
you can get to the second. By statement 2 again, you can get to the third, and fourth, etc.
Therefore, you can climb as high as you wish. Notice that both of these assertions are necessary
for you to get anywhere on the ladder. If only statement 1 is true, you have no guarantee of getting
beyond the first rung. If only statement 2 is true, you may never be able to get started.
Assume that the rungs of the ladder are numbered with the positive integers (1,2,3.
..). Now think
of a specific property that a number might have. Instead of "reaching an arbitrarily high rung", we
can talk about an arbitrary positive integer having that property. We will use the shorthand P(n) to
denote the positive integer n having property P. How can we use the ladderclimbing technique to
prove that P(n) is true for all positive n? The two assertions we need to prove are:
1) P(1) is true
2) for any positive k, if P(k) is true, then P(k+1) is true
Assertion 1 means we must show the property is true for 1; assertion 2 means that if any number
has property P then so does the next number. If we can prove both of these statements, then P(n)
holds for all positive integers, just as you could climb to an arbitrary rung of the ladder.
The foundation for arguments of this type is the
Principle of Mathematical Induction
. The
Principle of Mathematical Induction can be used as a proof technique on statements that have a
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View Full Documentparticular form. First of all, we must be working with positive integers or numbers of objects, and
we must be talking about some kind of sequence or pattern. The Principle tells us that if we can
prove the assertion is true for 1 (or some small starting value), and then prove that it is also true for
k+1 (assuming it is true for k), we have proven the assertion for all positive integers, because if it is
true for 1 and k+1, then it is true for k. When it is true for all three, it is true for all positive
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 Fall '07
 Plummer,R
 Computer Science, Mathematical Induction, Natural number, induction hypothesis

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