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Let $f \colon A \to B$ be a function from a set $A$ to a set $B$. In English, which of the following expressions is correct?

  1. The function $f$ associates an element of $A$ with (or to) an element of $B$;
  2. The function $f$ associates an element of $B$ with (or to) an element of $A$.

(I am not using quantifiers like "every" or "some" on purpose, because in natural language they can create some ambiguity)

In the literature and online, I have found both kinds of expression, but I am not sure if they are both correct, since I am not a native English-speaker. For instance, Wikipedia's page about function in mathematics uses both kinds of expression.

Disclaimer. My question is more about English than mathematics, but it requires a basic knowledge in mathematics to be answered, this is why I post it here and not in other Q&A forums.

jjagmath
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Taroccoesbrocco
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    I prefer 2. In fact, I am not sure the English of 1. corresponds to what function $f$ does. – 311411 Jan 26 '22 at 17:41
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    Either can be acceptable under the right interpretations, however it must be emphasized that each element of $A$ will each individually have exactly one element of $B$ associated to it, some elements of $B$ might be mapped to by multiple elements of $A$, and not every element of $B$ needs to be mapped to. – JMoravitz Jan 26 '22 at 17:41
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    From Nathan Jacobson's Lectures in Abstract Algebra, I: "A (single-valued) mapping $\alpha$ of a set $S$ into a set $T$ is a correspondence that associates with each $s \in S$ a single element $t \in T$" – 311411 Jan 26 '22 at 17:45
  • When drawing visual representations of functions, it is common to have labeled dots on the left corresponding to elements of $A$, labeled dots on the right corresponding to elements of $B$, and arrows pointed from the elements on the left to the elements on the right with the pointy part of the arrow on the right side. One can think of functions as machines that transform an input into an output. For that reason, I prefer the first, as the syntactical structure implies a sort of motion and that the elements of $A$ are those which are being affected moreso than the elements of $B$. – JMoravitz Jan 26 '22 at 17:46
  • How "every" create ambiguity? – jjagmath Jan 26 '22 at 17:47
  • @JMoravitz It doesn't sound strange to you in the case of a many-to-one function? Say $f(x)=x^2$. The function $f$ associates element $-2$ of $A$ with element $4$ of $B$. – 311411 Jan 26 '22 at 17:50
  • @jjagmath - Especially in 2, if I write "$f$ associates some element of $B$ with (or to) every element of $A$, it is not clear if the logical structure of the sentence is $\forall \exists$ or $\exists \forall$. – Taroccoesbrocco Jan 26 '22 at 17:50
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    @311411 "The function $f$ associates the input of $-2$ from $A$ with the output of $4$ from $B$", sounds fine to my ear – JMoravitz Jan 26 '22 at 17:52
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    @JMoravitz Okay, interesting. I always thought the English was trying to indicate the single-valued-ness of relations which are functions, as in Jacobson's English above. – 311411 Jan 26 '22 at 17:54
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    Both to me seem a bit linguistically awkward, although 2 is technically correct and 1 is not correct. Better to be more direct: "The function $f$ associates to each element of $A$ an element of $B.$" Note this version mentions $A$ first, which seems much more natural than to write in such a way that $B$ is mentioned first. Probably needs a bit more, however, if you want to ensure single-valuedness is incorporated, but as written it's still true because in the version I gave it is not claimed that ONLY functions satisfy the stated property. – Dave L. Renfro Jan 26 '22 at 18:26
  • @all - Thank you for all your valuable comments. If everyone writes them as answers, I can upvote them and accept one of them (according my humble opinion). – Taroccoesbrocco Jan 26 '22 at 21:42
  • @jjagmath Further to Taroccoesbrocco's first comment: to my ear, "$f$ associates some element of $B$ with every element of $A$" strongly suggests ∃b∈B ∀a∈A, whereas "$f$ associates some element of $B$ with each element of $A$ might mean either the same or ∀a∈A ∃b∈B (my automatic reading). In any case, the real issue here is hanging mixed quantifiers rather than 'each/every/any' per se. – ryang Feb 26 '23 at 10:58

1 Answers1

2

Disclaimer: Much of what follows has been remarked upon in the comments already.

The first statement seems wrong or at least ambiguous to me. To associate an element of $A$ 'with' an element of $B$ makes it sounds like elements of $A$ and $B$ are being matched up in some way, so it could be a bijection rather than just simply a function.

Also to 'associate' elements of $A$ with elements of $B$ is ambiguous. For example, can $a\in A$ be 'associated' with more than one element in $B$? If so, it is not a well-defined function. To 'associate an element of $A$ to an element of $B$' kind of looks like 'given an element of $B$, we obtain an element of $A$', which of course is the wrong way round. For this reason, the second statement looks better.

I would personally phrase it as follows:

The function $f:A\to B$ assigns a unique element $f(a)\in B$ to each element $a\in A$.

user829347
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  • So, all the expressions using "associate" are ambiguous or misleading (when talking about a function)? – Taroccoesbrocco Feb 01 '22 at 00:57
  • @Taroccoesbrocco I would say so, yes, although others may reasonably disagree. With a function $f:A\to B$ you want to quite explicitly convey a sense of direction from $A$ to $B$. Thinking of the everyday usage of the word 'associate', it is kind of like a symmetric relation that applies in both directions equally e.g. two friends are 'associated with each other'. – user829347 Feb 01 '22 at 01:02
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    Re: your suggestion: perhaps simply "exactly one element" instead of "a unique element"? The latter phrasing might, to some ears, suggest injectivity. – ryang Apr 16 '22 at 06:06