# Injective functions vs. correspondences

Injective functions are not the same thing as injective correspondences, correct? Injective functions are a subset of the injective correspondences.

For example in $y^2 = x$, y is not a function of x, but this is still a correspondence between $\mathbb{R} \rightarrow \mathbb{R}$. Can we say that the correspondence, $C:X \rightarrow Y$ is injective? (Since, if $C(x_1) = C(x_2)$ then $x_1 = x_2$.) Can we say it is also a surjective correspondence since every y-value has at least one (mostly two) pre-images?

Or am I way off base applying these definitions to things that are not functions?

Last question. "Mappings" are, as the Wikipedia suggests, the same thing as functions. (For some reason I thought mappings were correspondences. In fact, it is in my old notes. My old notes are wrong, right? )

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No, it's still perfectly valid to apply the definition of injectivity and surjectivity to relations; see en.wikipedia.org/wiki/… . – Qiaochu Yuan Nov 10 '10 at 20:42
@a little don: Some books use "injective correspondence" to refer to "bijective functions". I would personally not call a "multivalue function/relation" like $y^2=x$ (thinking of this as giving you values of $y$ for every $x$) a "corresopndence". And... what is your $C$ in the second paragraph? As for "mapping", again you are dealing with a question of nomenclature preference. Some people use 'mapping into' for injective functions. (There are even some old algebra books that use "isomorphism" for one-to-one homomorphism, and "isomorphism onto" for bijective homomorphism). – Arturo Magidin Nov 10 '10 at 20:43
I was thinking that "C" could be the name for the relation/correspondence. Does my notation imply that C must be a function? – futurebird Nov 10 '10 at 20:49

## 1 Answer

What you're describing is universally known as a "relation." It is true that you can have an injective relation which is not a function. Unfortunately, there are different uses for the term "correspondence" -- used alone, it can denote a general relation, but "one-to-one correspondence" can also denote a bijection. The safest thing to do is probably to avoid using the word altogether.

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"The safest thing to do is probably to avoid using the word altogether." Good. That I can do. This was making me very confused. – futurebird Nov 10 '10 at 20:46
Come to think of it, the use of the term "one to one correspondence" to mean "bijection" makes the word "correspondence" even more confusing. I think you are very right that it is better to avoid it. – futurebird Nov 10 '10 at 20:52