Friday, February 7 Scoring the Kennesaw State University High School Mathematics Competition

1. Friday, February 7 Scoring the Kennesaw State University High School Mathematics Competition 10:00 – 3:00 Math and Stats Building, room 246

2. Warm-up: In the diagram, there are exactly seven points A, B, C, D, E, F, G and seven lines (including the dotted one) each incident with 3 points. Is this a model of incidence geometry? I-1: For every point P and every point Q not equal to P, there exists a unique line lincident with P and Q. I-2: For every line lthere exist at least two distinct points incident withl. I-3: There exist three distinct points with the property that no line is incident withall three of them. C F A G D E B

3. Last time we looked at three different parallel properties The Euclidean Parallel Postulate For every line l and for every point P that does not lie on l, there exists a unique line mthrough P that is parallel to l. The Elliptic Parallel Property All lines meet (there are no parallel lines) The Hyperbolic Parallel Property For every line l and for every point P that does not lie on l, there exist at least two lines through P parallel to l.

4. Warm-up: In the diagram, there are exactly seven points A, B, C, D, E, F, G and seven lines (including the dotted one) each incident with 3 points. Is this a model of incidence geometry? Which parallel postulate/property holds? I-1: For every point P and every point Q not equal to P, there exists a unique line lincident with P and Q. I-2: For every line lthere exist at least two distinct points incident withl. I-3: There exist three distinct points with the property that no line is incident withall three of them. C F A G D E B

5. I-1 I-2 I-3 I-1 I-2 I-3 I-2 I-3 Which parallel postulate/property holds?

6. I-1 I-2 I-3 I-1 I-1 I-2 I-1 I-3 I-3 I-2 Which parallel postulate/property holds?

8. I-1 I-2 I-3 of positive 4-tuples (a, b, c, d) where a < b < c < d. So for example (1 , the points (1,), (1, 5), (1, 9), (, (1, 9). I-1 I-1 I-2 I-1 I-3 I-3 I-2

9. Another example: Consider the number 2310 = (2)(3)(5)(7)(11) Interpret points as the prime factors of 2310. Interpret lines as all products of pairs of prime factors of 2310. Points: {2, 3, 5, 7, 11} Lines: {6, 10, 14, 15, 21, 22, 33, 35, 55, 77} Interpret “incidence” as divisibility. (e.g. line 10 contains points 2 and 5). Is this a model of Incidence Geometry?

10. Review exercises (yellow handout) C C N N “3 is an odd number and 9 is odd.”

11. C C N N C

12. C C N N C  (Isosceles s), base s are  N

13. C C N N C N N  (a that is not isosceles)

14. C C N N C N N N

15. C C N N C N N N N contrapositive is also

16. C C N N C N N N Proposition 2.5 For every point P, there exist at least two distinct lines through P. N N

17. C C The statement holds in Euclidean Geometry N N C N N m N N n l N C To show independence, find a model in which the property holds and another model in which it doesn’t.

18. Exercise 1 • What is the negation of (P V Q)?  (P V Q)  P &  Q • What is the negation of (P &  Q)?  (P &  Q)  P V Q • Use Logic Rules 3, 4, and 5 to show that • P  Q is the same as  P V Q.

19. Exercise 1 •  (P V Q)  P &  Q •  (P &  Q)  P V Q • Use Logic Rules 3, 4, and 5 to show that • P  Q is the same as  P V Q.

20. Exercise 1 •  (P V Q)  P &  Q •  (P &  Q)  P V Q • Use Logic Rules 3, 4, and 5 to show that • P  Q is the same as  P V Q. • By L.R. 3, P  Q [ (P Q)] Logic Rule 3: The statement (S) means the same as S. Logic Rule 4: The statement [HC] means the same as H&C. Logic Rule 5: The statement [S1&S2] means the same as S1 S2.

21. Exercise 1 •  (P V Q)  P &  Q •  (P &  Q)  P V Q • Use Logic Rules 3, 4, and 5 to show that • P  Q is the same as  P V Q. • By L.R. 3, P  Q [ (P Q)] • BY L.R. 4,  (P  Q) P &  Q Logic Rule 3: The statement (S) means the same as S. Logic Rule 4: The statement [HC] means the same as H&C. Logic Rule 5: The statement [S1&S2] means the same as S1 S2.

22. Exercise 1 •  (P V Q)  P &  Q •  (P &  Q)  P V Q • Use Logic Rules 3, 4, and 5 to show that • P  Q is the same as  P V Q. • By L.R. 3, P  Q [ (P Q)] • BY L.R. 4,  (P  Q) P &  Q • Combining these two steps, P  Q  (P &  Q) • By L.R. 5,  (P &  Q)  P V Q (or by part b above) • Therefore, P  Q  P V Q Logic Rule 3: The statement (S) means the same as S. Logic Rule 4: The statement [HC] means the same as H&C. Logic Rule 5: The statement [S1&S2] means the same as S1 S2.

23. The negation of Euclid’s fourth postulate is “There exist (at least) two right angles that are not congruent.” 3. The negation of Euclid’s parallel postulate is “There exist a line l and a point P such that either every line through P meets l or there exists more than one line parallel to l through P.” 6. Completed in class last time

24. For each pair of axioms of incidence geometry, invent an interpretation in which those two axioms are satisfied but the third axiom is not. a. Axioms 1 and 2, but not 3. b. Axioms 1 and 3, but not 2. c. Axioms 2 and 3, but not 1.         I-1: For every point P and every point Q not equal to P, there exists a unique line lincident with P and Q. I-2: For every line lthere exist at least two distinct points incident withl. I-3: There exist three distinct points with the property that no line is incident withall three of them.

25. 9 a. Axiom 1 fails (skew lines). None of the parallel properties hold. I-1: For every point P and every point Q not equal to P, there exists a unique line lincident with P and Q. I-2: For every line lthere exist at least two distinct points incident withl. I-3: There exist three distinct points with the property that no line is incident withall three of them.

26. 9 b. All three axioms hold. Elliptic parallel property holds. I-1: For every point P and every point Q not equal to P, there exists a unique line lincident with P and Q. I-2: For every line lthere exist at least two distinct points incident withl. I-3: There exist three distinct points with the property that no line is incident withall three of them.

27. 9 c. All three axioms hold. Hyperbolic parallel property holds. I-1: For every point P and every point Q not equal to P, there exists a unique line lincident with P and Q. I-2: For every line lthere exist at least two distinct points incident withl. I-3: There exist three distinct points with the property that no line is incident withall three of them.

28. 9 d. All three axioms hold. Elliptic parallel property holds. I-1: For every point P and every point Q not equal to P, there exists a unique line lincident with P and Q. I-2: For every line lthere exist at least two distinct points incident withl. I-3: There exist three distinct points with the property that no line is incident withall three of them.

29. P  s Q   S  Q  9 d. All three axioms hold. Elliptic parallel property holds. P I-1: For every point P and every point Q not equal to P, there exists a unique line lincident with P and Q. I-2: For every line lthere exist at least two distinct points incident withl. I-3: There exist three distinct points with the property that no line is incident withall three of them.

30. We’ll look at question 10 later.

31. Isomorphism of models of incidence geometry Two models of incidence geometry are isomorphic if there exists a one-to-one correspondence P P between the points of the models and a one-to-one correspondence ll between the lines of the models such that P Il if and only if P Il.

32. 5 points, 10 lines 3 Consider the number 2310 2310 = (2)(3)(5)(7)(11) Points: 2, 3, 5, 7, 11 Lines: 6, 10, 14, 15, 21, 22, 33, 35, 55, 77 (i.e. all products of pairs of prime factors) 7 2 5 Interpret “incidence” as divisibility. (e.g. line 10 contains points 2 and 5). 22 11 Are these two models isomorphic?

33. Isomorphism of models of incidence geometry Two models of incidence geometry are isomorphic if there exists a one-to-one correspondence P P between the points of the models and a one-to-one correspondence ll between the lines of the models such that P Il if and only if P Il. M Points B, A, L, D Lines {B, A} , {B, L} , {B, D} , {A, L} , {A, D} , {L, D} H A T

34. Isomorphism of models of incidence geometry Two models of incidence geometry are isomorphic if there exists a one-to-one correspondence P P between the points of the models and a one-to-one correspondence ll between the lines of the models such that P Il if and only if P Il. M L L D D H A B A B A T

35. If two models are isomorphic, then all results in one are valid in the other. If the Euclidean parallel postulate is valid in one model, then in an isomorphic model, the Euclidean parallel postulate must also hold.

36. Is there a four-point model of incidence Geometry that is NOT isomorphic to the one below, or are all-four point models isomorphic? M H A T

37. Is there a four-point model of incidence Geometry that is NOT isomorphic to the one below, or are all-four point models isomorphic? O  M    H C L D A 4 points and 4 lines (CO, OL, CL, OD) T

38. Isomorphism in Algebra Addition mod 3

39. Isomorphism in Algebra 2 = 3 = 1  = Addition mod 3 1 2  2 1  1 2  Can the numbers 1, 2, 3 and the numbers 1, . 2be paired in such a way that the sums and products correspond? 1 , 2 , 3

40. A system of axioms is said to be consistent if no contradiction can be proved from the axioms. For example, the incidence axioms are consistent. No contradiction can be reached using them alone.

41. A system of axioms is said to be consistent if no contradiction can be proved from the axioms. In an inconsistent system, every statement is provable.

42. A system of axioms is said to be consistent if no contradiction can be proved from the axioms. In an inconsistent system, every statement is provable. Let’s develop an example…

43. Notice that we have seen one model of incidence geometry where all lines have the same number of points. , and earlier this class period we had a model where all lines but one had the same number of points. Both models have the elliptic parallel property. Notice that this model has one line with exactly three points while all others have exactly two points. O  C F A    C L D G 4 points and 4 lines (CO, OL, CL, AD) D E 7 points and 7 lines 2 2 3 2 Each line has 3 points B

44. What if we alter the situation somewhat; i.e. what if one line had exactly two points and all the rest had exactly three points?

45. Consider the following hypothetical Axioms:

46. Consider the following hypothetical Axioms: K-1 There exist at least 3 distinct non-collinear points. I-1: For every point P and every point Q not equal to P, there exists a unique line lincident with P and Q. I-2: For every line lthere exist at least two distinct points incident withl. I-3: There exist three distinct points with the property that no line is incident withall three of them.

47. Consider the following hypothetical Axioms: K-1 There exist at least 3 distinct non-collinear points. I-3 I-1: For every point P and every point Q not equal to P, there exists a unique line lincident with P and Q. I-2: For every line lthere exist at least two distinct points incident withl. I-3: There exist three distinct points with the property that no line is incident withall three of them.

48. Consider the following hypothetical Axioms: K-1 There exist at least 3 distinct non-collinear points. I-3 K-2 For every pair of distinct points P and Q, there exists a unique line incident with both. I-1: For every point P and every point Q not equal to P, there exists a unique line lincident with P and Q. I-2: For every line lthere exist at least two distinct points incident withl. I-3: There exist three distinct points with the property that no line is incident withall three of them.

49. Consider the following hypothetical Axioms: K-1 There exist at least 3 distinct non-collinear points. I-3 K-2 For every pair of distinct points P and Q, there exists a unique line incident with both. I-1 I-1: For every point P and every point Q not equal to P, there exists a unique line lincident with P and Q. I-2: For every line lthere exist at least two distinct points incident withl. I-3: There exist three distinct points with the property that no line is incident withall three of them.

50. Consider the following hypothetical Axioms: K-1 There exist at least 3 distinct non-collinear points. I-3 K-2 For every pair of distinct points P and Q, there exists a unique line incident with both. I-1 K-3 There exists exactly one line l containing exactly 2 points K-4 Every line not equal to l contains exactly 3 points. I-1: For every point P and every point Q not equal to P, there exists a unique line lincident with P and Q. I-2: For every line lthere exist at least two distinct points incident withl. I-3: There exist three distinct points with the property that no line is incident withall three of them.