Talk:Equivalence relation
|
|
| Contents |
Duplication of definition?
Should the three properties really be defined here or is it enough to refer to binary relation? Or should these properties have their own pages a la Eric Weisstein? -- Jan Hidders
- I don't think the small amount of duplication is a problem, and it aids understanding.
- PS: Hopefully, we can soon get up to the level of depth Weisstein had (unsigned comment by Gareth Owen 09:25, Jul 13, 2001 — Paul August ☎ 17:43, May 23, 2005 (UTC))
Ok. I would agree with that. But do you think that Wikipedia should define the same terms, i.e., if Eric has a definition of a certain mathematical concept then so should Wikipedia? Or should Wikipedia be more "course grained"? -- Jan Hidders
- Duplication, in addition to aiding "understanding", also helps to make Wikipedia more robust, in that an error in one place can be corrected by looking somewhere else. Paul August ☎ 17:44, May 23, 2005 (UTC)
Content move
I moved this from the main page:
- Given an arbitrary relation R, we define the equivalence relation ~ as follows:
- (Reflexivity) Define x~x.
- (Symmetry) Define x~y whenever R(x,y) *or* R(y,x) is true.
- (Transitivity) Define x~y whenever there exists z such that R(x,z) and R(z,y) is true.
This is incorrect; the equivalence relation generated by R is a bit more complicated. In the third step, one has to allow for a whole sequences z1,...,zn of intermediate elements, and for each i one has to allow for R(zi,zi+1) *or* R(zi+1,zi). AxelBoldt 22:10 Nov 23, 2002 (UTC)
"in thermal equilibrium with"?
On what set is "in thermal equilibrium with" an equivalence relation? (unsigned comment by anon 69.202.74.207 — Paul August ☎ 17:43, May 23, 2005 (UTC))
Asymptotical equivalence
I did not (yet) find anything about the notion of asymptotical equivalence
(x_n) ~ (y_n) iff (x_n-y_n) = o(y_n)
and the "crude" equivalence x=Theta(y) <=> x=O(y) and y=O(x). — MFH: Talk 17:14, 23 May 2005 (UTC)
- PS: found the first one (slightly different and not completely equivalent definition) at Asymptotic... see Talk:Asymptotic analysis... — MFH: Talk