CS103A
HO# 47
Induction III, Fibonacci
2/27/08
1
CI
→
WOP
Well Ordering Property: If A is a non-empty set of positive integers,
then A has a least element.
CI
→
WOP
Well Ordering Property: If A is a non-empty 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 non-empty 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 non-empty 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.