Projective Plane: Axiomatic Approach A projective plane II is a set of points and subsets called lines that satisfy the following four axioms: • P1 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 exists three non-collinear points. A projective plane is called finite, if there are only finitely many points on the plane. 1. Show that for any line 1 of II, there exists a point A s.t. A does not lie on 1. 2. Show that for any two distinct points A, B of II, there exits a line / which does not pass through neither of them. 3. To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line. 4. Show that for any given finite projective plane, there exists some n ≥ 2 s.t. every point in II lie on n + 1 lines and every line in II contains n+1 points. 5. Prove the duality principle we explained in lecture. 6. Use the duality principle to claim the dual statements of subquestions 1-4.
Projective Plane: Axiomatic Approach A projective plane II is a set of points and subsets called lines that satisfy the following four axioms: • P1 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 exists three non-collinear points. A projective plane is called finite, if there are only finitely many points on the plane. 1. Show that for any line 1 of II, there exists a point A s.t. A does not lie on 1. 2. Show that for any two distinct points A, B of II, there exits a line / which does not pass through neither of them. 3. To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line. 4. Show that for any given finite projective plane, there exists some n ≥ 2 s.t. every point in II lie on n + 1 lines and every line in II contains n+1 points. 5. Prove the duality principle we explained in lecture. 6. Use the duality principle to claim the dual statements of subquestions 1-4.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Transcribed Image Text:Projective Plane: Axiomatic Approach A projective plane II is a set of
points and subsets called lines that satisfy the following four axioms:
• P1 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 exists three non-collinear points.
A projective plane is called finite, if there are only finitely many points on the
plane.
1. Show that for any line 1 of II, there exists a point A s.t. A does not lie on
1.
2. Show that for any two distinct points A, B of II, there exits a line 1 which
does not pass through neither of them.
3. To a given infinite straight line, from a given point which is not on it, to
draw a perpendicular straight line.
4. Show that for any given finite projective plane, there exists some n ≥ 2
s.t. every point in II lie on n + 1 lines and every line in II contains n + 1
points.
5. Prove the duality principle we explained in lecture.
6. Use the duality principle to claim the dual statements of subquestions 1-4.
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