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6.6 In an incidence geometry, consider the relationship of parallelism, "l is parallel to m," on the set of lines. (a) Give an example to show that this need not be an equivalence relation. (b) If we assume the parallel axiom (P), then parallelism is an equivalence relation. (c) Conversely, if parallelism is an equivalence relation in a given incidence geom- etry, then (P) must hold in that geometry.

6.6 In an incidence geometry, consider the relationship of parallelism, "l is parallel to m," on the set of lines. (a) Give an example to show that this need not be an equivalence relation. (b) If we assume the parallel axiom (P), then parallelism is an equivalence relation. (c) Conversely, if parallelism is an equivalence relation in a given incidence geom- etry, then (P) must hold in that geometry.

Advanced Engineering Mathematics
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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6.6 In an incidence geometry, consider the relationship of parallelism, "7 is parallel to m," on the set of lines. (a) Give an example to show that this need not be an equivalence relation. (b) If we assume the parallel axiom (P), then parallelism is an equivalence relation. (c) Conversely, if parallelism is an equivalence relation in a given incidence geom- etry, then (P) must hold in that geometry.
Transcribed Image Text:6.6 In an incidence geometry, consider the relationship of parallelism, "7 is parallel to m," on the set of lines. (a) Give an example to show that this need not be an equivalence relation. (b) If we assume the parallel axiom (P), then parallelism is an equivalence relation. (c) Conversely, if parallelism is an equivalence relation in a given incidence geom- etry, then (P) must hold in that geometry.
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