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

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

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.2: Direct Methods For Solving Linear Systems
Problem 47EQ
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Let us build a geometry, S, using the three axioms of incidence geometry with one additional axiom added: Incidence Axiom 1: for every point P and every point Q (P and Q not equal), there exists a unique line, 1, incident with P and Q Incidence Axiom 2: for every line 1 there exist at least two distinct points incident with 1. Incidence Axiom 3: there exist (at least) three distinct points with the property that no line is incident with all three of them Incidence Axiom 4: there exist at most four points 1) The elliptic parallel property is true, false or independent of the axioms of S. Which one is it? Prove your answer. (the elliptic parallel property says: Given a line 1 and a point P not on 1, there exists no line through P parallel to 1.) 2) The Euclidean parallel property is true, false, or independent of the axioms of S. Which one is it? Prove your answer. (The Euclidean parallel property says: given a line 1 and a point p not on 1, there exists exactly one line through P parallel to 1.)
Transcribed Image Text:Let us build a geometry, S, using the three axioms of incidence geometry with one additional axiom added: Incidence Axiom 1: for every point P and every point Q (P and Q not equal), there exists a unique line, 1, incident with P and Q Incidence Axiom 2: for every line 1 there exist at least two distinct points incident with 1. Incidence Axiom 3: there exist (at least) three distinct points with the property that no line is incident with all three of them Incidence Axiom 4: there exist at most four points 1) The elliptic parallel property is true, false or independent of the axioms of S. Which one is it? Prove your answer. (the elliptic parallel property says: Given a line 1 and a point P not on 1, there exists no line through P parallel to 1.) 2) The Euclidean parallel property is true, false, or independent of the axioms of S. Which one is it? Prove your answer. (The Euclidean parallel property says: given a line 1 and a point p not on 1, there exists exactly one line through P parallel to 1.)
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