I understand the method used to prove injectivity or surjectivity, however, I am confused as to how to handle a function that presents itself as a set $(m,n)$.
In the example, $f:Z \rightarrow Z \times Z$, $f(x) = (x^2, x + 1)$
We know that $x^2$ is not injective as we can find for example $f(-1) = f (1) = 1$.
However, we also know that the second part of the equation is injective as we can deduct that:
$x + 1 = y + 1$
$x = y$
The form $(m, n)$ of the function is what is really confusing me.
- Would it be correct to conclude that the function is injective since it will produce a set of unique $(m,n)$ values for each $x$?
If so, how would we define such a function to be surjective since there will exists coordinates for which there is no corresponding $x$? Would I be correct to conclude that the function is not surjective?
Also, if a function was in the form $f(x) = z + 5$, would it be correct to conclude that the function is not well defined, and thus we can't determine if it's surjective\injective due to the fact that it's based on a ambiguous variable $z$?