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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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    I was thinking that "C" could be the name for the relation/correspondence. Does my notation imply that$C$must be a function?2010-11-10

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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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    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.2010-11-10