CS103A
HO# 47
Induction III, Fibonacci
2/27/08
1
CI
→
WOP
Well Ordering Property: If A is a nonempty set of positive integers,
then A has a least element.
CI
→
WOP
Well Ordering Property: If A is a nonempty set of positive integers,
then A has a least element.
Proof. Suppose that A is a set of positive integers without a least element,
and let P(n) be the proposition that n
∉
A.
We will show that
∀
n P(n) by
complete mathematical induction, i.e., that A must be empty.
BASE CASE: Since 1 is the smallest positive integer, 1
∉
A, because if so, 1
would be the least element of A. So P(1) is true.
INDUCTIVE STEP:
Assume:
for some positive integer k, P(i) for 1
≤
i
≤
k.
Show P(k + 1).
CI
→
WOP
Well Ordering Property: If A is a nonempty set of positive integers,
then A has a least element.
Proof. Suppose that A is a set of positive integers without a least element,
and let P(n) be the proposition that n
∉
A.
We will show that
∀
n P(n) by
complete mathematical induction, i.e., that A must be empty.
BASE CASE: Since 1 is the smallest positive integer, 1
∉
A, because if so, 1
would be the least element of A. So P(1) is true.
INDUCTIVE STEP:
Assume:
for some positive integer k, P(i) for 1
≤
i
≤
k.
Show P(k + 1).
If k + 1
∈
A, k + 1 would be the least element in A, since no integer less
than k + 1 is in A by the inductive hypothesis.
Thus k + 1
∉
A, and P(k+1) is true.
Thus by C.I.,
∀
n P(n).
Since A is a set of positive integers to which no integer
belongs, we have shown that if A has no least element, it must be empty.
CI
→
WOP
Well Ordering Property: If A is a nonempty set of positive integers,
then A has a least element.
Proof.
We will show that if A is a set of positive integers without a
least element, then A is empty.
Let P(n) be the proposition that n
∉
A.
We will show that
∀
n P(n) by complete mathematical induction.
BASE CASE: Since 1 is the smallest positive integer, 1
∉
A, because if so, 1
would be the least element of A. So P(1) is true.
INDUCTIVE STEP:
Assume:
for some positive integer k, P(i) for 1
≤
i
≤
k.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '07
 Plummer,R
 Computer Science, Natural number, Prime number, inductive hypothesis, Fn1 + Fn2

Click to edit the document details