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In mathematics, it's perfectly fine if we have a function which maps multiple elements of a set $X$ to the same element of a set $Y$.

Why would it be incorrect to map one element of $X$ to multiple elements of $Y$?

For example, something like $f(x)=y$ and $2y$

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  • $\begingroup$ It is perfectly correct, but it is called a relation, not a function. Or if you prefer, it is a function from $X$ to $\mathcal{P}(Y)$. $\endgroup$
    – J.-E. Pin
    May 2, 2015 at 15:42
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    $\begingroup$ Because a function is not defined this way. But mathematicians also study these kind of objects en.wikipedia.org/wiki/Multivalued_function $\endgroup$
    – Tryss
    May 2, 2015 at 15:42
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    $\begingroup$ Well, because it isn't a function any more. What you are describing is called a relation. $\endgroup$
    – James
    May 2, 2015 at 15:43
  • $\begingroup$ Or you can call it a function from $X$ to the set of subsets of $Y$. $\endgroup$ May 2, 2015 at 15:47
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    $\begingroup$ Why is it incorrect to call a dog a geranium? $\endgroup$
    – WillO
    May 2, 2015 at 15:47

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Well, it depends on what you mean with "correct".

It would no longer be a function, but a relation. A function is by definition a relation where each element of the domain is related to exactly one element of the codomain.

Note that this property is also what allows you to use the convenient notation $f(x)$ for that unique element in the codomain that is related to the element $x$ of the domain. For a general relation, it would not be clear which of the elements related to $x$ you mean. Therefore you'll generally need to use set notations or quantified logical expressions (i.e. "for all"/"there exists") when speaking about the values related to $x$.

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