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I read that the arrow notation $x \rightarrow y$ was invented in the 20th century. Who introduced it?

Each map needs both an explicit domain and an explicit codomain (not just a domain, as in previous formulations of set theory, and not just a codomain, as in type theory). -- Lawvere and Rosebrugh Sets for Mathematics, 2003

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    @MichaelJoyce, thanks for markup - still learning Tex2012-05-14
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    Why did you remove the "for functions" on the title? That *is* what you are asking about, given the quote, right? Arrows are used for a variety of purposes in different settings, so it's important to be clear which setting you are talking about. For example, "$x\to y$" can be used to denote rewriting rules; or implications in the setting of propositional calculus. If you wanted to include things like "functors", then you should add it, not remove all context.2012-05-14
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    @ArturoMagidin, I understand your comment but, because - at least today, if not when the notation was introduced - arrows are not necessarily functions (ie, total functional relations). Arrows also refer to functors which carry more information than functions.2012-05-14
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    Then somebody telling you who introduced the notation for material implication would be an acceptable answer? I doubt it sincerely. And by the time the very notion of "functor" was introduced, the arrow was already established as notation for *functions*, and it was adopted for "functor" as analogy, since a functor is a function between categories. By removing context, you make your question more ambiguous, not better.2012-05-14
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    Actually I would be interested in who first used notation for material implication. Sometimes that's written as a double arrow. It's all arrows. Certainly Aristotle didn't use the notation, so I'd like to know any and all applications2012-05-14
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    The double arrow, $\implies$, is for the metamathematical statement, not for material implication which is a propositiona logic/mathematical statement. They are not interchangeable, so "sometimes that's written as a double arrow" is either incorrect, or describes incorrect usage. If you are interested in that as well, then your quote, without **any** further explanation, is misleading. Either way, you are not being clear and are just hoping readers will divine what's in your mind. Is it really that hard to add your own words and say *exactly* what you want?2012-05-14
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    In any event, [Jeff Miller's pages](http://jeff560.tripod.com/) is always a good place to start. In this case there's some information [here](http://jeff560.tripod.com/operation.html) at the very bottom, last section on **Arrow notation**.2012-05-14
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    @t.b.thanks for the link2012-05-14
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    @ArturoMagidin, can you please explain? on p.198 of R&L *Sets for Mathematics*, they write: "the operation \implies applied to a pair of statements B, D, gives another statement B \implies D, which is usually read "B implies D" or "if B then D". It is to be distinguished from B |- D, which is a statement *about* statements" ...I always assumed the latter means meta-statement?2012-05-14
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    $\to$ is an operation of propositional calculus. Statements are not part of propositional calculus, so nothing you are quoting is about the propositional calculus connective $\to$. $\longrightarrow$ is not the same as $\Longrightarrow$, just like $\vdash$ is not the same as $\models$.2012-05-14
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    Do short and long arrows mean different concepts?2012-05-14
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    @alancalvitti: Not usually.2012-05-14

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I found some information HERE

Saunders Mc Lane, in Categories for the working mathematician (Springer-Verlag, 1971, p. 29), says: "The fundamental idea of representing a function by an arrow first appeared in topology about 1940, probably in papers or lectures by W. Hurewicz on relative homotopy groups. (Hurewicz, W.: "On duality theorems," Bull. Am. Math. Soc. 47, 562-563) His initiative immediately attracted the attention of R. H. Fox and N. E. Steenrod, whose ... paper used arrows and (implicitly) functors... The arrow f: : X —> Y rapidly displaced the occasional notation f(X) (subset of ) Y for a function.

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    precisely [ http://www.ams.org/journals/bull/1941-47-07/S0002-9904-1941-07497-5/S0002-9904-1941-07497-5.pdf ]2014-01-24
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    The question was asked later at MO, and people found occurences predating 1940. https://mathoverflow.net/questions/194377/when-was-the-arrow-notation-for-functions-first-introduced2017-06-25