Verify by showing that they follow the three incidence axioms:(I1) Given any two discrete points there exists a unique line containing them(I2) Given any line there exists at least two distinct points lying on it(I3) There exists three non-collinear pointsWe will use Hilbert’s axiom system that is constructed with the three primitive terms point, line and plane (lie on). 2. Rp²lowing:Consider the RP2 Geometry, whose interpretation is the fol-point: equivalent classes of points on the sphere, obtained by identifyanti-polar pairs, i.e.,RP² = {[x, y, z]|(x, y, z) = x² + y² + z² = 1,(x, y, z) ~ (-x, -y, -z)}.line The great circles of the unit sphere.● lie on: A point p = [x, y, z] € RP² lies on a line 7 if (x, y, z) ≤ l.EDiscuss if it is a model of geometry, by verifying the 3 axioms one by one.

As per the question we are given the Hilbert's axion system defined on the real projective plane ℝℙ2…

Answered: Discuss if it is a model of geometry,…

Math

Advanced Math

Discuss if it is a model of geometry, by verifying the 3 axioms one by one.

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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Question

Verify by showing that they follow the three incidence axioms:

(I1) Given any two discrete points there exists a unique line containing them
(I2) Given any line there exists at least two distinct points lying on it
(I3) There exists three non-collinear points

We will use Hilbert’s axiom system that is constructed with the three primitive terms point, line and plane (lie on).

2. Rp²
lowing:
Consider the RP2 Geometry, whose interpretation is the fol-
point: equivalent classes of points on the sphere, obtained by identify
anti-polar pairs, i.e.,
RP² = {[x, y, z]|(x, y, z) = x² + y² + z² = 1,
(x, y, z) ~ (-x, -y, -z)}.
line The great circles of the unit sphere.
● lie on: A point p = [x, y, z] € RP² lies on a line 7 if (x, y, z) ≤ l.
E
Discuss if it is a model of geometry, by verifying the 3 axioms one by one.

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