Class 3: Well-Ordering

cs2102: Discrete Mathematics
University of Virginia, Fall 2016

See course site for notes:
https://uvacs2102.github.io

Transcript

1. cs2102: Discrete Mathematics Class 3: Well-Ordering David Evans University of Virginia
2. Plan Well-Ordering Principle Well-Ordered Sets (and Comparators) Well-Ordered Theorem? Well-Ordered Data Types Well-Ordering Principle Proofs The Well-Ordering Principle is the crux of mathematical induction, which is the basis for many of the most important proofs in CS.
4. There are lots of office hours! Don’t wait to get help.
5. Well Ordering Principle
6. Well-Ordering Principle Every nonempty set of non-negative integers has a smallest element.
7. Ordering Sets Definition. A set is well-ordered with respect to an ordering function (e.g., <), if all of its non-empty subsets has a minimum element. Why does this mean the whole set is well-ordered?
8. Definition. A set is well-ordered with respect to an ordering function (e.g., <), if all of its non-empty subsets has a minimum element.
9. Sensible Ordering Functions Definition. A set is well-ordered with respect to an ordering function (e.g., <), if all of its non-empty subsets has a minimum element.
10. Java Transgression
11. Java Transgression
12. Well-Ordered Sets? Set: the integers Comparator: a < b
13. Well-Ordered Sets? Set: the integers Comparator: |a| < |b|
14. Well-Ordered Sets? Set: the integers Comparator: if | a | = | b |: a < b else: |a| < |b|
15. Well-Ordered Sets? Set: the non-negative rationals Comparator: < Definition. A number is rational if it can be written as a ratio of two integers.
16. Set: the non-negative rationals Comparator: ? Is there some comparator that makes the non-negative rationals well-ordered?
17. Well-Ordered Theorem For every set, there is some comparator (that is transitive and trichotomic) that makes the set well-ordered. Is this true?
18. Slack break… For every set, there is some comparator that is transitive and trichotomic that makes the set well-ordered.
19. Georg Cantor Well-ordered theorem is a fundamental law of thought. (1883)
20. Gyula Kőnig There is no comparator that results in a well- ordering of the real numbers.
21. Gyula Kőnig Claimed counter-proof: There is no comparator that results in a well- ordering of the real numbers. Proof withdrawn as incorrect
22. Georg Cantor Well-ordered theorem is a fundamental law of thought. (1883) What should we (mathematics) do when there is something that seems like it should be true, but no one can prove?
23. Axiom of Choice For any collection of non-empty sets, there is a choice function that picks exactly one element from each set.
24. 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.
27. For every set, there is some comparator that is transitive and trichotomic that makes the set well-ordered. For any collection of non-empty sets, there is a choice function that picks exactly one element from each set. implies Well-Ordered Theorem Axiom of Choice
28. For every set, there is some comparator that is transitive and trichotomic that makes the set well-ordered. For any collection of non-empty sets, there is a choice function that picks exactly one element from each set. implies Well-Ordered Theorem Axiom of Choice
29. Still Controversial… ZF Axioms: Zermelo–Fraenkel Set Theory ZFC Axioms: ZF + Axiom of Choice
30. Well-Ordering in Programming Is a + 1 > a always true?
31. “glitch in control unit causes generators to shut down if left powered on for 248 days”
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. Are all integers greater than $3 betable?
36. Betable Numbers A number is betable if it can be produced using some combination of $2 and $5 chips.
37. Charge • Maintain Well Orderliness! • If you need to enroll in the class, bring me 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