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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

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
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.
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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