Talk:Affine plane (incidence geometry)
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Equal or disjoint
editPer the question in a recent reversion, according to Parallel (geometry)#Reflexive variant, the definition of parallel lines as "equal or disjoint", as in this article, is needed because Euclidean parallelism is not a reflexive relation and hence not an equivalence relation. As it turns out, for a symmetric relation, transitivity implies reflexivity by identifying first and third variables in the transitive axiom. — Rgdboer (talk) 02:16, 13 March 2018 (UTC)
- Your second statement is false. It is a somewhat standard exercise in an introduction to proofs class to find the error in that argument.--Bill Cherowitzo (talk) 04:01, 13 March 2018 (UTC)
- I agree that transitivity does not implies reflexivity. For example the empty relation is transitive but not reflexive. Also, you say that "Euclidean parallelism is not an equivalence relation". This depends how parallelism is defined (including or not equality), and so, such an assertion may not be claimed to be true or false without referring to an explicit definition (on Euclid times parallelism were not reflexive, but presently, most authors prefer to include equality into parallelism, for avoiding considering special cases in statements and proofs).
- However, the problem of your edit is not there. The fact is that wikilinks are aimed for explaining to the readers the terms that they may not understand or for which they may want more explanation. Here you have linked "equal or disjoint" to the assertion that parallelism is a transitive relation. This is true that parallelism is "equal or disjoint" is transitive, but this assertion is misplaced in the lead, and wiki linking to an assertion without stating it is highly confusing for most readers. If a link would be needed, this would be to disjoint. D.Lazard (talk) 08:15, 13 March 2018 (UTC)
Just amplifying the claim that in contemporary mathematics, it has been preferred to make parallelism an equivalence relation. At least one can refer to Artin's use in Geometric Algebra. Also Hartshorne's Euclid and beyond page 68, basically following the same program. Rschwieb (talk) 13:56, 13 March 2018 (UTC)