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 parallel 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.
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 parallel 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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