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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. [Hi model and a 6-line model]

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. [Hi model and a 6-line model]

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter5: Similar Triangles
Section5.2: Similar Polygons
Problem 11E: a Does the similarity relationship have a reflexive property for triangles and polygons in general?...
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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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