1) Consider the following geometry called S: Undefined Terms: point, line, incidence Axioms: I) Each pair of lines in S has precisely one point in common. II) Each point in S is incident with precisely two lines. III) There exists precisely four distinct lines in S. i) prove the theorem: there exist precisely six points in S. ii) prove the theorem: there exist precisely three points on each line. iii) Is this system categorical? Justify your answer. iv) can is be proved that precisely one of the following properties hold in S: a) The elliptic parallel property b) The Euclidean parallel property c) The hyperbolic parallel property If so, which one? Prove your answer.
1) Consider the following geometry called S: Undefined Terms: point, line, incidence Axioms: I) Each pair of lines in S has precisely one point in common. II) Each point in S is incident with precisely two lines. III) There exists precisely four distinct lines in S. i) prove the theorem: there exist precisely six points in S. ii) prove the theorem: there exist precisely three points on each line. iii) Is this system categorical? Justify your answer. iv) can is be proved that precisely one of the following properties hold in S: a) The elliptic parallel property b) The Euclidean parallel property c) The hyperbolic parallel property If so, which one? Prove your answer.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1) Consider the following geometry called S:
Undefined Terms: point, line, incidence
Axioms:
I) Each pair of lines in S has precisely one point in common.
II) Each point in S is incident with precisely two lines.
III) There exists precisely four distinct lines in S.
i) prove the theorem: there exist precisely six points in S.
ii) prove the theorem: there exist precisely three points on each line.
iii) Is this system categorical? Justify your answer.
iv) can is be proved that precisely one of the following properties hold in S:
a) The elliptic parallel property
b) The Euclidean parallel property
c) The hyperbolic parallel property
If so, which one? Prove your answer.
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