Set Theory and Logic


An Origin of Transitivity and Other Useful Relation’s Properties

Authors: Misha Mikhaylov

Usually itemizing relations’ properties those of them are always pointed out – some appearance of reflexivity, symmetry and transitivity. Also it is not so clear whether they are introduced artificially – i.e. axiomatically, rather for the sake of convenience or it may be done due to inartificial reasons. At the same time an origin of them is not so clear – whether they appear chaotically and independently on each other or there should be rigorous association between them. It is shown here that request of relation’s reversibility leads to these properties’ presence or absence. Often symmetry appearances are defined by using of ambiguous way. In fact, anti-symmetry is not direct negation for symmetry – there is also something that may be called as asymmetry or it may be something else. To avoid it here there was found the unified method of their definition. The same thing may be told about reflexivity and it was shown that just the only intransitivity may be represented as direct negation of transitivity.

Comments: 11 Pages.

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Submission history

[v1] 2014-09-08 16:49:27
[v2] 2014-09-14 04:03:45
[v3] 2016-02-23 12:30:09

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