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I sometimes hear people saying something like

Let $f$ be a function from a set $A$ into another set $B$.

But, of course, by saying this, they do not want to exclude the possibility of $A$ and $B$ being equal, they just mean that $B$ can be different than $A$.

My question is: Is it, technically speaking, correct to use the word "another" in this case? Also, independent of the question whether "another" is correct or not, I wonder if the word "another" could confuse readers oder listeners, that is, if the word "another" is good from an expository point of view.

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    $\begingroup$ That is not a "technical" sentence. One speaker may be using "another" to indicate that $A \ne B$, while a different speaker might not care to make that distinction, and is only using the word as a bit of verbal filler. If one wanted to be clear, would write "$f : A \to B$ where $A \ne B$". $\endgroup$
    – Xander Henderson
    Commented Sep 18, 2018 at 18:53
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    $\begingroup$ It's generally best to be as clear as possible. Why use "another" when you could just say "into a set $B$" and avoid any possible confusion? $\endgroup$ Commented Sep 18, 2018 at 18:55
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    $\begingroup$ This is a question better suited for ELU.se than it is for math.se $\endgroup$
    – amWhy
    Commented Sep 18, 2018 at 18:57
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    $\begingroup$ @amWhy While I agree that the question is about English language usage, and not strictly math, the question is so fundamentally rooted in conventional mathematical jargon that I suspect the community there would suggest that the question be asked here in order to better reach an audience who can actually help. $\endgroup$
    – Arthur
    Commented Sep 18, 2018 at 19:10
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    $\begingroup$ I disagree that this question should be closed as "primarily opinion based". Yes, the interpretation of the sentence is primarily opinion based - but then the OP (and those who use similar statements) should realise this. Therefore, the question admits a non-opinion based answer: the statement is ambiguous and should be rewritten. $\endgroup$
    – user1729
    Commented Sep 19, 2018 at 10:55

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This is common usage but it really isn't a good thing. If one means that the set is distinct just say a "distinct set B." If B is possibly A then just say "from A to B" without any further comment. To some extent "another" in this context is a verbal "filler" that doesn't do much.

For that matter, it seems unlikely that there are going to be many contexts where one has a function from A to B and it is going to matter that A is not B while at the same time there won't be other stronger properties in use like A and B differing in cardinality or topology or metric.

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    $\begingroup$ Is this a common usage? Do you have any evidence to back up that claim? I can't say that I've ever seen it in print, and the number of times that I've heard anything like it in verbal communication in my career can probably be counted on one hand. $\endgroup$
    – Xander Henderson
    Commented Sep 19, 2018 at 23:09
  • $\begingroup$ @XanderHenderson That's a valid question; my impression is that it isn't an uncommon thing to say but I don't have any concrete evidence of this. I don't think I've seen it in writing ever. $\endgroup$
    – JoshuaZ
    Commented Sep 20, 2018 at 16:50
  • $\begingroup$ To strengthen this answer: the "another" in the OP's quoted sentence makes sense only at the very meta level where we are distinguishing between different names for the same object; since this is not the intention, I'd say that inserting "another" in said sentence is even wrong (as opposed to merely introducing ambiguity), for this explicitly excludes the possibility that $A=B.$ $\endgroup$
    – ryang
    Commented Sep 15, 2023 at 5:33

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