2) Consider the following geometry called T: Undefined Terms: point, line, incidence Axioms: I) There exist precisely three distinct points incident with every line II) Each point in T is incident with precisely two distinct lines III) There exists at least one point in T. i) show that there exist at least two non-isomorphic models for T. [Hint: try to construct a 4-line model and a 6-line model] ii) show that the Euclidean property is independent of the axioms of T iii) prove the theorem: there exist at least two distinct lines in T
2) Consider the following geometry called T: Undefined Terms: point, line, incidence Axioms: I) There exist precisely three distinct points incident with every line II) Each point in T is incident with precisely two distinct lines III) There exists at least one point in T. i) show that there exist at least two non-isomorphic models for T. [Hint: try to construct a 4-line model and a 6-line model] ii) show that the Euclidean property is independent of the axioms of T iii) prove the theorem: there exist at least two distinct lines in T
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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![2) Consider the following geometry called T:
Undefined Terms: point, line, incidence
Axioms:
I) There exist precisely three distinct points incident with every line
II) Each point in T is incident with precisely two distinct lines
III) There exists at least one point in T.
i) show that there exist at least two non-isomorphic models for T. [Hint: try to construct a 4-line
model and a 6-line model]
ii) show that the Euclidean property is independent of the axioms of T
iii) prove the theorem: there exist at least two distinct lines in T
iv)) Consider the statement A: For every point P and every point Q, not equal to P, there exists a
unique line incident with P and Q
*Is statement A a theorem in T? If so, prove it.
*State the negation of A (call it B). Is B a theorem in T? If so, prove it.
*Is A independent of the axioms of T? If so, prove this.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F31367789-01e4-4129-b30b-49084de51bdb%2F6780bf1d-73e9-486c-8415-cd88290ca80d%2F6x82wth_processed.png&w=3840&q=75)
Transcribed Image Text:2) Consider the following geometry called T:
Undefined Terms: point, line, incidence
Axioms:
I) There exist precisely three distinct points incident with every line
II) Each point in T is incident with precisely two distinct lines
III) There exists at least one point in T.
i) show that there exist at least two non-isomorphic models for T. [Hint: try to construct a 4-line
model and a 6-line model]
ii) show that the Euclidean property is independent of the axioms of T
iii) prove the theorem: there exist at least two distinct lines in T
iv)) Consider the statement A: For every point P and every point Q, not equal to P, there exists a
unique line incident with P and Q
*Is statement A a theorem in T? If so, prove it.
*State the negation of A (call it B). Is B a theorem in T? If so, prove it.
*Is A independent of the axioms of T? If so, prove this.
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