5.3 (a) • Start from A,B,C be non-collinear. (why do you have that?) Can you look at line LAB, IBC, ICA (why these give you lines?), and find one more new point D, E, F on each of them? • Now can you find another point on AE, say point G? 5.3(b) • Can you first recall a projective plane of 7 points that you have seen in tutorial? (the Fano plane) • Can you show that every projective plane of 7 points is isomorphic to the Fano plane? (We will discuss more on isomorphism in detail this Thursday, although you probably have already heard of this from the tutorial.) 5.3(c) We have listed many examples when verifying the axioms of incidence during lecture, you might want to review that part, to be honest, many of the examples there still work =) (for sure I vill help you review the independence on Thursday)

5.3 (a) • Start from A,B,C be non-collinear. (why do you have that?) Can you look at line LAB, IBC, ICA (why these give you lines?), and find one more new point D, E, F on each of them? • Now can you find another point on AE, say point G? 5.3(b) • Can you first recall a projective plane of 7 points that you have seen in tutorial? (the Fano plane) • Can you show that every projective plane of 7 points is isomorphic to the Fano plane? (We will discuss more on isomorphism in detail this Thursday, although you probably have already heard of this from the tutorial.) 5.3(c) We have listed many examples when verifying the axioms of incidence during lecture, you might want to review that part, to be honest, many of the examples there still work =) (for sure I vill help you review the independence on Thursday)

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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[Geometry] How do you solve this? The second photo has hints

I1: Given any two discrete points there exists a unique line containing the,

I2: Given any line there exists at least two distinct points lying on it

I3: There exists three non-collinear points

6.3 (a)
• Start from A,B,C be non-collinear. (why do you have that?)
• Can you look at line LAB, IBC, ICA (why these give you lines?), and find one more new point D, E, F on each of them?
• Now can you find another point on LAE, say point G?
6.3(b)
• Can you first recall a projective plane of 7 points that you have seen in tutorial? (the Fano plane)
• Can you show that every projective plane of 7 points is isomorphic to the Fano plane? (We will discuss more on isomorphism in detail this Thursday, although you probably have already
heard of this from the tutorial.)
6.3(c) We have listed many examples when verifying the axioms of incidence during lecture, you might want to review that part, to be honest, many of the examples there still work =) (for sure I
will help you review the independence on Thursday)

6.3 A projective plane is a set of points and subsets called lines that satisfy the following
four axioms:
Pl. Any two distinct points lie on a unique line.
P2. Any two lines meet in at least one point.
P3. Every line contains at least three points.
P4. There exist three noncollinear points.
Note that these axioms imply (11)-(13), so that any projective plane is also an inci-
dence geometry. Show the following:
(a) Every projective plane has at least seven points, and there exists a model of a
projective plane having exactly seven points.
(b) The projective plane of seven points is unique up to isomorphism.
(c) The axioms (P1), (P2), (P3), (P4) are independent.

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