[Math] Does an injective function imply two sets have same cardinality

Question

In the book "A first course in abstract algebra" by John B. Fraleigh, he states in the introduction chapter (which deals with sets and relations) that for two sets $X$ and $Y$, they have the same cardinality if there exists an one-to-one (injective) function between the two sets.

I'm not convinced about that since, if I understand the definitions correct, a one-to-one correspondence does not necessarily mean that the range is the same as the codomain, i.e. even if all $x \in X$ maps uniquely to $Y$, there might be elements in $Y$ that never gets mapped to (thus not same cardinality).

Have I missed something here or is Fraleigh wrong in that statement?

Answer 1

Using Google Books, the correct definition from the book is this:

0.13 Definition Two sets $X$ and $Y$ have the same cardinality if there exists a one-to-one function mapping $X$ onto $Y$, that is if there exists a one-to-one correspondence between $X$ and $Y$.

Notice the requirement that not only the function is one-to-one, but it is also required to be onto, so the range is the same as the codomain.