# Why is a "reverse surjection" incorrect?

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 $2y$

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

## 1 Answer

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 $x$.