Question

3) Let us build a geometry, S, using the three axioms of incidence geometry with one additional axiom added: Incidence Axiom l: For every point P and every point Q (P and Q not equal), there exists a unique line, I, incident with P and Q. Incidence Axiom 2: For every line / there exist at least two distinct points incident with Incidence Axiom3: There exist (at least) three distinct points with the property that no line is incident with all three of them. Axiom 4: There exist at most four points. (i) The elliptic parallel property is true, false or independent of the axioms of S Which one is it? Prove your answer.. [Recall that the elliptic parallel property says: Given a line and a point P not on 1, there exists no line through P parallel to I.] (ii) The Euclidean parallel property is true, false or independent of the axioms of S. Which one is it? Prove your answer. [Recall that the Euclidean parallel property says: Given a line I and a point P not on l, there exists exactly one line through P parallel to L.]

Answer 1

Elliptic parallel property is false because their can be a unique line which passes through the point P & the line is not l here. Ie, ex: - Let point be P(x, y) & line be L be a line & line k be passing through the point (x, y) Euclidean parallel property is Free because their can be only one line passed through P(x, y) such that it is parallel to line L. you can observe in the above example that no other line can be parallel to L other than K.