THE WELL ORDERING PROPERTY

1
THE WELL ORDERING PROPERTY

• Definition Let B be a set of integers. An
  integer m is called a least element of B if m is
  an element of B, and for every x in B, m?x.
• Example 3 is a least element of the set
  4,3,5,11.
• Example Let A be the set of all positive odd
  integers. Then 1 is a least element of A.
• Example Let U be the set of all odd
  integers. Then U has no least element. The proof
  is left as an informal exercise.

2
WELL ORDERING

• The Well Ordering Property Let B be a non
  empty set of integers such that there exists an
  integer b such that every element x of B
  satisfies
• b lt x. Then there exists a least element of
  B.
• Therefore, in particular If B is a non empty
  set of non negative integers then there exists a
  least element in B.

3
WELL ORDERING

• An Application of the Well Ordering Property
  A Proof of the Existence Part of the Division
  Algorithm
• For every integer a and positive integer d
  there exist integers q and r such that 0?rltd and
  a qd r.

4
WELL ORDERING

• Proof Let A be the set of all values a kd for
  integers k, such that a kd is non negative. We
  show that A is not empty If a is non negative,
  let k0. a-kda, and a is non negative. If a is
• negative, let ka. a-kda-ada(1-d). (1-d) is
  not
• positive (since d is positive), and a is
  negative,
• so the product a(1-d) is non negative.
• By the well ordering property A has a least
  element. Let that least element be r a qd. So
  a qd r and 0?r. We claim that rltd. This claim
  can be proved by contradiction.

5
WELL ORDERING

• Suppose that d?r. Note that
• a qd r qd d (r-d) (q1)d r
• where r (r-d) ? 0. Also r r-d lt r,
  because d gt 0. So
• a (q1)d r lt r a qd, and
• a (q1)d r ? 0. So a (q1)d is in A,
  and a (q1)d lt a qd contradicting the fact
  that a qd is the least element of A.