Class 3

Well Ordering

Transcript

1. cs2102: Discrete Mathematics Class 3: Well-Ordering Principle David Evans & Mohammad Mahmoody University of Virginia
3. There are lots of office hours! (‘no Fridays’ is intentional) Don’t wait to get help.
4. Plan - Well-Ordering Principle - Closer look: ordering / well ordering - General Well-Ordering Theorem? - Using Well-Ordering Principle
5. Well-Ordering Principle Every nonempty set of non-negative integers has a smallest element.
6. Well-Ordering Principle Every nonempty set of non-negative integers has a smallest element. …is the crux of mathematical induction, which is the basis for many of the most important proofs in CS.
7. What is an Ordered Set? Definition. An ordered set : (1) If ≠ then either < or < (2) < and < then <
8. Examples of Ordered Sets
11. Well Ordered Sets Definition. An ordered set (with respect to an ordering `relation’ like < ) is well-ordered if any of its non-empty subsets ⊆ has a minimum.
12. Well Ordering Principle (again) Well Ordering Principle. The set of natural numbers under the order “<“ is well ordered. Every nonempty set of non-negative integers has a smallest element. Why called “principle” ?
13. Examples of Well Ordered Sets
14. Well-Ordered? Set: the integers Comparator: a < b
15. Well-Ordered? Set: the integers Comparator: |a| < |b|
16. Well-Ordered? Set: the non-negative reals Comparator: <
17. Well-Ordered? Set: the non-negative rationals Comparator: < Definition. A number is rational if it can be written as a ratio of two integers.
18. Set: the non-negative rationals Comparator: ? Is there some comparator that makes the non-negative rationals well-ordered?
19. General Well-Ordering Theorem For every set S, there is some ordering on that makes well-ordered. Is this true?
20. Slack break… For every set S, there is some ordering on that makes well-ordered.
21. Gyula Kőnig There is no comparator that results in a well- ordering of the real numbers.
22. Gyula Kőnig Claimed counter-proof: There is no comparator that results in a well- ordering of the real numbers. Proof withdrawn as incorrect
23. Georg Cantor Well-ordering theorem is a fundamental law of thought. What should we (mathematics) do when there is something that seems like it should be true, but no one can prove?
24. “General Well Ordering” Axiom? Is it really that obvious? Is it `equivalent’ to something else that is obvious?
25. Axiom of Choice For any collection of non-empty sets, there is a choice function that picks exactly one element from each set.
28. Bertrand Russell (1872-1970) To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed.
29. For every set S, there is some ordering on that makes well-ordered. For any collection of non-empty sets, there is a choice function that picks one element from each set. implies Well-Ordered Theorem Axiom of Choice
30. For every set S, there is some ordering on that makes well-ordered. For any collection of non-empty sets, there is a choice function that picks one element from each set. Well-Ordered Theorem Axiom of Choice implies
31. Foundations of Math ZF Axioms: Zermelo–Fraenkel Set Theory ZFC Axioms: ZF + Axiom of Choice
32. Axiom of Choice: Still Controversial… General Well Ordering Theorem on “larger sets”… Banach-Tasrski Paradox Well Ordering Principle (as stated) not controversial
33. How to Use Well Ordering Principle ?
34. Can I bet any dollar amount using just $2 and $5 chips?
35. Betable Numbers A number is betable if it can be produced using some combination of $2 and $5 chips.
36. Betable Numbers A number is betable if it can be produced using some combination of $2 and $5 chips. Are all integers greater than $3 betable?
37. Betable Numbers A number is betable if it can be produced using some combination of $2 and $5 chips. Are all integers greater than $3 betable?
38. Proof: Using Well-Ordering Principle A number is betable if it can be produced using some combination of $2 and $5 chips. All integers > 3 are betable.
39. Charge • If you need to enroll in the class, bring us a course action form to sign (now, or later) • Due Friday (6:29pm): PS1 • Read MCS Ch 2, 3 (at least through 3.5) before Thursday’s class