Let $A,B$ be sets and let $f: A \to B$. Then we say that $A,B$ are isomorphic under $f$ if $f$ is a linear function that maps $A$ onto $B$ in a one-to-one manner; that $A,B$ are homeomorphic under $f$ if $f$ is a continuous function that maps $A$ onto $B$ in a one-to-one manner; that $A,B$ are ($C^{r}$)-diffeomorphic under $f$ if $f$ is a $C^{r}$ function with $C^{r}$ inverse that maps $A$ onto $B$ in a one-to-one manner; and so on.

I think it may be convenient to have a name for a pair of sets between which there is a "simple", need-not-have-additional-property bijection, for if not then one may have to use many words to articulate.

I am not aware if there is already one such name, so I would like to know it if any exists and, if no such name exists, I would like to solicit some ideas for coining a new terminology, say simply calling such a pair of sets morphic under the bijection.

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    $\begingroup$ Is it not just "they have the same cardinality" $\endgroup$ – Elliot G Aug 30 '15 at 4:16
  • $\begingroup$ @ElliotG: Yes. This question aims to go after a short terminology. Constantly referring to having-the-same-card is still somewhat not that ideal. $\endgroup$ – Megadeth Aug 30 '15 at 4:18
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    $\begingroup$ @ElliotG: I edited my question to make my point more clear (I hope so!). My point here is to emphasize the relation not only between the pair of sets but the function involved. $\endgroup$ – Megadeth Aug 30 '15 at 4:29
  • $\begingroup$ I believe you used the word: "bijection". $\endgroup$ – Eric Towers Aug 30 '15 at 5:08

Probably the most standard term of the sort you're looking for is "equipotent", though it isn't used particularly often (more often, people will just say two sets "have the same cardinality" or "are in bijection"). People who think about categories a lot sometimes say "isomorphic as sets".

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    $\begingroup$ Also equipollent. $\endgroup$ – Asaf Karagila Aug 30 '15 at 4:39
  • $\begingroup$ Thank you. I wonder if there is something like "$A$ is equipotent to $B$"? $\endgroup$ – Megadeth Aug 30 '15 at 4:45
  • $\begingroup$ @GudsonChou: Yes, you can say that, though it would be more common to say "$A$ is in bijection with $B$" or "$A$ has the same cardinality as $B$". $\endgroup$ – Eric Wofsey Aug 30 '15 at 4:51
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    $\begingroup$ Some possibly interesting numbers: "is equipotent to" has 13,300 hits on Google, "has the same cardinality as" has 14,800, "is in bijection with" has 32,200, "is equinumerous with" has 2,300, and "is equipollent to" has 5,800. $\endgroup$ – Eric Wofsey Aug 30 '15 at 4:58
  • $\begingroup$ Is "$A$ is in injection with $B$" also in use? $\endgroup$ – Megadeth Aug 30 '15 at 6:23

You can call them equinumerous.

  • $\begingroup$ Thank you. First-time-ever heard of it. Do you know if it is conventional to say something like $A$ and $B$ are equinumerous under $f$, given that $f$ is a bijection between $A$ and $B$? $\endgroup$ – Megadeth Aug 30 '15 at 4:30
  • $\begingroup$ @GudsonChou: It is more common to say "in bijection via $f$" (or some variant with a different preposition). $\endgroup$ – Eric Wofsey Aug 30 '15 at 4:34
  • $\begingroup$ @GudsonChou People would know what you mean but the way you phrase it sounds a little stilted. Maybe something like "$A$ and $B$ are equinumerous, as seen via a bijection $f$" might be better. $\endgroup$ – user223391 Aug 30 '15 at 4:36
  • $\begingroup$ @avid19: Thank you. (I am not a native English speaker, so .... :)) $\endgroup$ – Megadeth Aug 30 '15 at 4:39

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