This preview shows pages 1–3. Sign up to view the full content.
Handout #26
CS103
April 23, 2010
Robert Plummer
Introduction to Mathematical Induction
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 rung, 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.
..).
What we would like
to show is that all of the following propositions are true:
P
1
: I can reach rung 1.
P
2
: I can reach rung 2.
P
3
: I can reach rung 3.
P
4
: I can reach rung 4.
P
5
: I can reach rung 5.
P
6
: I can reach rung 6.
P
7
: I can reach rung 7
....
P
n
: I can reach rung n.
(for any n)
...
If you think about it, that's a fairly tall order, because we are trying to prove infinitely many
propositions!
It's
Mathematical Induction that will come to our rescue.
Rephrasing the ladderclimbing problem using this notation, we are trying to prove that P
n
is true for all
positive n.
The two assertions we need to prove in order to show this are:
1) P
1
i
s
t
r
u
e
2) for any positive k, if P
k
is true, then P
k+1
is true
If we can prove both of these statements, then P
n
holds for all positive integers n, just as you could climb to
an arbitrary rung of the ladder.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2
The foundation for arguments of this type is the
Principle of Mathematical Induction
, which can be
used as a proof technique on statements that have a particular form.
We can state it this way:
A proof by
mathematical induction
that a proposition P
n
is true for every positive integer n
consists of two steps:
BASE CASE: Show that the proposition P
1
is true.
INDUCTIVE STEP: Assume that P
k
is true for an arbitrarily chosen positive integer k, and show that
under that assumption, P
k+1
must be true.
From these two steps we conclude (by the principle of mathematical induction) that for all positive
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '09

Click to edit the document details