Wikipedia:Reference desk/Archives/Mathematics/2015 July 17

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July 17

Partial equivalence relations

Given a set X, a partial equivalence relation R on X, and an element x of X, why is it true that if x is not R-related to itself, then no element of X is ever R-related to x? GeoffreyT2000 (talk) 00:19, 17 July 2015 (UTC)

(We have an article, Partial equivalence relation, btw.) If x R y then symmetry implies y R x and transitivity would then imply x R x. Apparently (I'm just getting this from the article), partial equivalence relations arise as kernels of partial functions the same way equivalence relations arise as kernels of ordinary functions. So you can also think of it as: If f(x)=f(y) then f(x) must exist, so f(x)=f(x). --RDBury (talk) 00:39, 17 July 2015 (UTC)