Mathematical Induction
Often times, we would like to make a statement about the
natural numbers (0,1,2,.
..) such as: for all natural numbers n,
the sum of 1+2+3+.
..+n = n(n+1)/2.
However, proving such a statement may be difficult if we are
not very creative or proficient with algebra. In particular, it
would be nice if we could prove a given statement without
plugging in every possible value of n (which would clearly take
forever.
..) or using terribly clever mathematics.
Induction makes this possible. The induction principle is as
follows:
Let A
⊆
N. (N is the set of nonnegative integers.) where the
following two properties hold:
1)
0
∈
A
2)
k
∈
A
⇒
k+1
∈
A
Then we have that A = N.
In layman’s terms, the way we will use this principle to prove
statements is the following way.
Given an open statement s(n), where n is an arbitrary non
negative integer, we must show these two following things to
prove the statement for all nonnegative integers n:
1) s(0)
2)
s(k)
⇒
s(k+1) for all nonnegative integers k.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentNow, the first step is relatively simple. Since s(0) is a simple
statement, you must simply plug in 0 into the open statement
and assess the validity of the statement.
The second step must be broken down into two steps. Notice
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 Mathematical Induction, Natural number, open statement

Click to edit the document details