B 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.
B 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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Transcribed Image Text: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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