Show that the following systems are not models for the Incidence Axioms, that is, in each case determine which of the axioms are violated. (Note, if only one axiom is violated, then you have shown the independence of that axiom.) (a) Points: {1,2,3, 4, 5} Lines: {1,2, 3}, {1, 4}, {1,5}, {2,4}, {2, 5}, {3, 4}, {3,5}, {4, 5} Planes: {1,2, 3, 4}, {1, 2, 3, 5}, {1, 4, 5}, {2, 3, 4, 5} (b) Points: {1,2, 3, 4, 5} Lines: {1,2, 3}, {1, 4}, {1, 5}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5} Planes: {1,2,3, 4}, {1, 2, 3, 5}, {1, 2, 3}, {1,4, 5}, {2, 4, 5}, {3, 4, 5} (c) Points: {1,2,3, 4} Lines: {1,2, 4}, {1, 3}, {2, 3}, {3, 4} Planes: {1,2,3}, {1,3, 4}, {2, 3, 4}

Show that the following systems are not models for the Incidence Axioms, that is, in each case determine which of the axioms are violated. (Note, if only one axiom is violated, then you have shown the independence of that axiom.) (a) Points: {1,2,3, 4, 5} Lines: {1,2, 3}, {1, 4}, {1,5}, {2,4}, {2, 5}, {3, 4}, {3,5}, {4, 5} Planes: {1,2, 3, 4}, {1, 2, 3, 5}, {1, 4, 5}, {2, 3, 4, 5} (b) Points: {1,2, 3, 4, 5} Lines: {1,2, 3}, {1, 4}, {1, 5}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5} Planes: {1,2,3, 4}, {1, 2, 3, 5}, {1, 2, 3}, {1,4, 5}, {2, 4, 5}, {3, 4, 5} (c) Points: {1,2,3, 4} Lines: {1,2, 4}, {1, 3}, {2, 3}, {3, 4} Planes: {1,2,3}, {1,3, 4}, {2, 3, 4}

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

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Show that the following systems are not models for the Incidence
Axioms, that is, in each case determine which of the axioms are violated.
(Note, if only one axiom is violated, then you have shown the
independence of that axiom.)
(a) Points: {1,2,3, 4, 5}
Lines: {1,2, 3}, {1, 4}, {1,5}, {2,4}, {2, 5}, {3, 4}, {3,5}, {4, 5}
Planes: {1,2, 3, 4}, {1, 2, 3, 5}, {1, 4, 5}, {2, 3, 4, 5}
(b) Points: {1,2, 3, 4, 5}
Lines: {1,2, 3}, {1, 4}, {1, 5}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}
Planes: {1,2,3, 4}, {1, 2, 3, 5}, {1, 2, 3}, {1,4, 5}, {2, 4, 5}, {3, 4, 5}
(c) Points: {1,2,3, 4}
Lines: {1,2, 4}, {1, 3}, {2, 3}, {3, 4}
Planes: {1,2,3}, {1,3, 4}, {2, 3, 4}

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