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A one-to-one correspondence is an alternative name for a bijection between two sets, but to what does the term 'correspondence' alone refer? As far as I can see, it seems to be another term for 'relation', but I think there must be a difference.

user50229
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  • By use (though perhaps not by actual definition), a 1-1 correspondence is a bijection, and from here I deduce that correspondence from $,A,$ to $,B,$ must be an onto function from $,A,$ to $,B,$ (the order here is important)...I think. – DonAntonio Apr 28 '13 at 11:44
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    Correspondences come up routinely in mathematical economics. One generally calls a map $f:X \to 2^Y$ a 'correspondence' or a set-valued map. These are important, say, in optimization where the set of arguments that maximize a given objective function will generally not be a function, but rather a correspondence. – Pete Caradonna Feb 26 '17 at 20:37

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Not that I ever use this regularly or believe everything that is written in it, The encyclopedic Dictionary of mathematics agrees with your guess. There may not have to be a difference after all: there are lots of synonymous words in mathematics.

What I read leads me to believe that "correspondence" is an older and dustier term for "relation".

If you asked me the answer without the benefit of resources, I would have told you my impression was that it is a synonym for "1-1 correspondence" and that the 1-1 part was usually added for emphasis.

rschwieb
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  • It's strange though that if correspondence is just a relation, that adding the qualification '1-1' alone turns this object into a bijection, as there was no suggestion of even being a function in the term 'correspondence'. – user50229 Apr 28 '13 at 11:53
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    @user50229 Yeah, but as we know well, mathematics does not always apply terminological conventions consistently or well. It may be that in history different people have used "correspondence" to mean "relation", "function", "onto relation", "onto function" and "bijection" or something entirely different. – rschwieb Apr 28 '13 at 11:56
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    @user50229 I'm not sure about the history of "1-1" either. Certainly today we just think that it sends distinct things to distinct things... but it seems plausible this could also be used in the sense that "everything is paired up between the two sets", and then onto would be implied, and functionality would be implied. – rschwieb Apr 28 '13 at 11:59
  • Good points, thanks – user50229 Apr 28 '13 at 12:02
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Correspondence is a synonim for the notion of relation. On the other hand, a one-to-one function (a subfamily of relations) is a synonim of injective function. Then, "one-to-one correspondence" literaly would mean an injective relation. However when we use it, we always mean a bijection.

Therefore, even though by "correspondence" we mean relation, when we use that term together with "one-to-one" we get a bijection.

As an analogy, take the derivative denoted as $\frac{dy}{dx}$. Even though it is represented as a quotion of two differentials, it is just a notation for a new notion (see this).

Emo
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